Expressiveness of Communication in Answer Set Programming

Under c onsider ation for public ation in The ory and Pr actic e of L o gic Pr o gr amming 1 Expr essiveness of Communic ation in A nswer Set Pr o gr amming KIM BA UTERS, STEVEN SCHOCKAER T Dept. of Applie d Mathematics and Computer Scienc e, Krijgslaan 281 (WE02), Universiteit Gent, 9000 Gent, Belgium ( e-mail: (kim.bauters, steven.schockaert)@ugent.be ) JER OEN JANSSEN, DIRK VERMEIR Dept. of Computer Scienc e, V rije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium ( e-mail: (jeroen.janssen, dirk.vermeir)@vub.ac.be ) MAR TINE DE COCK Dept. of Applie d Mathematics and Computer Scienc e, Krijgslaan 281 (WE02), Universiteit Gent, 9000 Gent, Belgium ( e-mail: martine.decock@ugent.be ) submitte d 23 F ebruary 2011; revise d 26 August 2011; ac c epte d 5 Septemb er 2011 Abstract Answ er set programming (ASP) is a form of declarative programming that allows to succinctly formulate and eﬃciently solv e complex problems. An intuitiv e extension of this formalism is comm unicating ASP , in whic h multiple ASP programs collaborate to solv e the problem at hand. How ever, the expressiv eness of communicating ASP has not b een thoroughly studied. In this paper, w e presen t a systematic study of the additional expressiv eness oﬀered by allowing ASP programs to communicate. First, w e consider a simple form of communication where programs are only allow ed to ask questions to each other. F or the most part, w e delib erately only consider simple programs, i.e. programs for whic h computing the answer sets is in P . W e ﬁnd that the problem of deciding whether a literal is in some answer set of a comm unicating ASP program using simple comm unication is NP -hard. In other words: w e mov e up a step in the p olynomial hierarch y due to the abilit y of these simple ASP programs to communicate and collab orate. Second, we modify the communication mechanism to also allow us to fo cus on a sequence of communicating programs, where each program in the sequence may successively remov e some of the remaining mo dels. This mimics a netw ork of leaders, where the ﬁrst leader has the ﬁrst sa y and may remov e models that he or she ﬁnds unsatisfactory . Using this particular comm unication mechanism allows us to capture the entire p olynomial hierarc hy . This means, in particular, that communicating ASP could be used to solve problems that are ab ov e the second level of the p olynomial hierarch y , suc h as some forms of abductive reasoning as well as PSP ACE-complete problems suc h as STRIPS planning. KEYWORDS : logic programming, answer set programming, multi-agen t reasoning 2 K. Bauters et al. 1 In tro duction Answ er set programming (ASP) is a form of non-monotonic reasoning based on the stable mo del semantics (Gelfond and Lifzc hitz 1988). ASP has prov en successful as an elegan t and con venien t v ehicle for commonsense reasoning in discrete domains and to enco de combinatorial optimization problems in a purely declarative wa y . It has b een applied in, for example, plan generation (Lifschitz 2002), diagnosis (Eiter et al. 1999) and biological netw orks (Gebser et al. 2010). Being an active ﬁeld of research, a large b o dy of extensions hav e b een prop osed that impro ve up on the basics of ASP and oﬀer, for example, cardinality constrain ts (Niemel¨ a and Simons 2000) or nested expressions (Lifschitz et al. 1999). Not all of these extensions pro vide an increase in terms of computational expressiveness and some are merely con venien t syntactic sugar. One particularly in teresting extension of ASP is called communicating ASP . It allo ws for a num ber of ASP programs to communicate, i.e. share information about their kno wledge base, giving them the ability to co op erate with each other to solve the problem at hand. Each ASP program inv olved, called a comp onent program, has individual b eliefs and reasoning capabilities. One of the b eneﬁts of this ex- tension is that it eases the declarative formulation of a problem originating in a m ulti-agent context. Quite a num ber of diﬀerent ﬂav ours of communicating ASP ha ve b een proposed in the literature, b oth in the narro w domain of ASP and in the broader domain of logic programming, where eac h of these pap ers presents in- triguing examples that highligh t the usefulness of communicating ASP , for example (Dell’Acqua et al. 1999), (Brain and De V os 2003), (Ro elofsen and Seraﬁni 2005), (De V os et al. 2005) and (V an Nieu wen b orgh et al. 2007). In particular, all the examples inv olv e multi-dimensional problems ( e.g. a p olice inv estigation with mul- tiple witnesses) where each agent only has the knowledge contained in one or a few of the dimensions ( e.g. the witness only sa w the burglar enter the building). A standard example to illustrate the usefulness in this area is sho wn in Figure 1. 1 The ﬁgure depicts tw o p eople who are lo oking at a b ox. The b ox is called magic b ecause neither agent can mak e out its depth. The information the agen ts know is further limited b ecause parts of the b ox are blinded. By coop eration, b oth agen ts can pinp oint the exact lo cation of the ball. Indeed, Mr . 2 sees a ball on his left side. F rom this information Mr . 1 knows that there is a ball and that it m ust therefore be on his left (since he cannot see a ball on his right). This knowledge can b e rela y ed bac k to Mr . 2 . Both agen ts now know the exact p osition and depth of the ball. Complexit y results from (Brewk a et al. 2007) sho w that computing the answer sets of a communicating logic program is in NP . In general, how ev er, only few results exist regarding the expressiveness of suc h comm unicating ASP programs. In addition, for many of the known results, it is not clear whether an increase in expressiv eness is due to the type of comp onent programs considered ( i.e. the expres- siv eness of the individual programs inv olved) or due to the use of communication. In comm unicating ASP , the notion of an answer set can be deﬁned in diﬀerent 1 Illustration from (Ro elofsen and Seraﬁni 2005) used with p ermission from the authors. Expr essiveness of Communic ation in ASP 3 1 Mr. 2 Mr. Fig. 1. A magic b ox. w ays; we refer to the sp eciﬁc deﬁnition that is used as the comm unication mecha- nism. Answ er set seman tics is based on the idea of stable minimal mo dels. When dealing with agen ts that can comm unicate, it becomes unclear ho w we should in- terpret the notion of minimality . One option is to assume global minimality , i.e. we minimise ov er the conclusions of all the agents in the netw ork. Another option is to assume minimality on the level of a single agent. Since in general it is not p ossible to ﬁnd a mo del that is minimal for all individual agents, the order in which we minimise o ver the agents matters. The main contributions of this paper are as follows. W e presen t a systematic study of the additional expressiv eness oﬀered b y allo wing ASP programs to comm unicate, where, for the most part, we delib erately only consider comp onent programs that are simple, i.e. programs for which computing the answer sets is in P . Clearly any step w e mov e up in the p olynomial hierarc hy is then due to the abilit y of these simple ASP programs to communicate and collab orate with each other. W e show that the problem of deciding whether a literal is in any answ er set of a communicating ASP program using simple communication is in NP . W e also provide a sim ulation using a netw ork of simple programs that is capable of mo deling any program with negation-as-failure. These results are extended with complexity results for when the net work consists of disjunctiv e programs. In all cases, we examine the complexit y of determining whether an answ er set exists as well as whether a literal is true in an y (brav e reasoning) or all answ er sets (cautious reasoning). F urthermore, w e in tro duce the notion of m ulti-focused answer sets of communicating ASP programs, whic h allo ws us to successively focus ( i.e. minimise) on diﬀerent agents. As it turns out, using m ulti-focused answ er set programs it is p ossible to express any problem in PSP ACE. This means in particular that communicating ASP could b e used to solve problems that are ab ov e the second level of the p olynomial hierarch y , suc h as some forms of ab ductive reasoning (Eiter and Gottlob 1995) as well as PSP A CE-complete problems such as STRIPS planning (Bylander 1994). W e show that these results hold for any choice of program in the netw ork, b eing it either simple, normal or disjunctiv e, and we pro vide the complexit y of determining whether a multi-focused answ er set exists. This pap er aggregates and extends our w ork from (Bauters et al. 2010; Bauters et al. 2011). The ﬁrst work is extended with completeness results and an elab orate o verview of the sim ulation of negation-as-failure. F urthermore, w e pro vide more detailed complexit y results including the complexity of determining whether an answ er set exists and the complexity of determining whether some literal is true in some or all of the answer sets. The focused answ er sets in tro duced in Section 5 4 K. Bauters et al. are from (Bauters et al. 2011) and are more general than the corresp onding notion that w e in tro duced in (Bauters et al. 2010). Additional results are pro vided, suc h as the complexity of determining whether a multi-focused answer set exists, as w ell as new results concerning the use of disjunctive programs as comp onen t programs. W e also provide the pro ofs of all results in the app endix. 2 Bac kground on Answer set programming W e ﬁrst recall the basic concepts and results from ASP that are used in this pap er. T o deﬁne ASP programs, we start from a coun table set of atoms and we deﬁne a liter al l as an atom a or its classical negation ¬ a . If L is a set of literals, w e use ¬ L to denote the set {¬ l | l ∈ L } where, by deﬁnition, ¬¬ a = a . A set of literals L is c onsistent if L ∩ ¬ L = ∅ . An extende d liter al is either a literal or a literal preceded b y not which w e call the negation-as-failure op erator. Intuitiv ely we sa y that not l is true when we hav e no proof to supp ort l . F or a set of literals L , w e use not ( L ) to denote the set { not l | l ∈ L } . A disjunctive rule is an expression of the form γ ← ( α ∪ not ( β )) where γ is a set of literals (in terpreted as a disjunction, denoted as l 1 ; . . . ; l n ) called the head of the rule and ( α ∪ not ( β )) (in terpreted as a conjunction) is the bo dy of the rule with α and β sets of literals. When the b o dy is empty , the rule is called a fact . When the head is empty , the rule is called a c onstr aint . In this pap er, we do not consider constraints as they can readily b e simulated using extended literals. 2 A p ositive disjunctive rule is a disjunctiv e rule without n egation-as-failure in the bo dy , i.e. with β = ∅ . A disjunctive pr o gr am P is a ﬁnite set of disjunctive rules. The Herbr and b ase B P of P is the set of atoms app earing in program P . A (partial) interpr etation I of P is any consistent set of literals I ⊆ ( B P ∪ ¬B P ). I is total iﬀ I ∪ ¬ I = B P ∪ ¬B P . A normal rule is a disjunctive rule with exactly one literal l in the head. A normal pr o gr am P is a ﬁnite set of normal rules. A simple rule is a normal rule without negation-as-failure in the b o dy . A simple pr o gr am P is a ﬁnite set of simple rules. The satisfaction relation | = is deﬁned for an in terpretation I as I | = l iﬀ l ∈ I , otherwise I 6| = l . F or an interpretation I and L a set of literals, we deﬁne I | = L iﬀ ∀ l ∈ L · I | = l . An interpretation I is a mo del of a p ositive disjunctiv e rule r = γ ← α , denoted I | = r , if I 6| = ¬ γ or I 6| = α , i.e. the b o dy is false or at least one of the literals in the head can b e true. An in terpretation I of a p ositive disjunctiv e program P is a mo del of P iﬀ ∀ r ∈ P · I | = r . Answ er sets are deﬁned using the imme diate c onse quenc e op er ator T P for a simple program P w.r.t. an interpretation I as T P ( I ) = I ∪ { l | (( l ← α ) ∈ P ) ∧ ( α ⊆ I ) } . (1) W e use P ? to denote the ﬁxp oint which is obtained b y repeatedly applying T P start- ing from the empt y interpretation, i.e. the least ﬁxp oint of T P w.r.t. set inclusion. An in terpretation I is an answer set of a simple program P iﬀ I = P ? . 2 W e can simulate a constrain t ( ← body ) by ( fail ← not fail , body ) with fail a fresh atom. Expr essiveness of Communic ation in ASP 5 The r e duct P I of a disjunctive program P w.r.t. the in terpretation I is deﬁned as P I = { γ ← α | ( γ ← α ∪ not ( β )) ∈ P , β ∩ I = ∅} . W e say that I is an answer set of the disjunctive program P when I is a minimal mo del of P I w.r.t. set inclusion. In the speciﬁc case of normal programs, answ er sets can also b e characterised in terms of ﬁxp oin ts. Sp eciﬁcally , it is easy to see that in this case the reduct P I is a simple program. W e sa y that I is an answer set of a normal program P iﬀ  P I  ? = I , i.e. if I is the answer set of the reduct P I . 3 Comm unicating programs The underlying in tuition of communication b etw een ASP programs is that of a function call or, in terms of agents, asking questions to other agents. This commu- nication is based on a new kind of literal ‘ Q : l ’, as in (Giunchiglia and Seraﬁni 1994; Ro elofsen and Seraﬁni 2005; Brewk a and Eiter 2007). If the literal l is not in the answ er set of program Q then Q : l is false; otherwise Q : l is true. The semantics presen ted in this section are closely related to the minimal semantics of (Brewk a and Eiter 2007) and esp ecially the semantics of (Buccafurri et al. 2008). Let P be a ﬁnite set of program names. A P -situate d liter al is an expression of the form Q : l with Q ∈ P and l a literal. F or R ∈ P , a P -situated literal Q : l is called R -lo c al if Q = R . F or a set of literals L , we use Q : L as a shorthand for { Q : l | l ∈ L } . F or a set of P -situated literals X and Q ∈ P , w e use X Q to denote { l | Q : l ∈ X } , i.e. the pro jection of X on Q . A set of P -situated literals X is c onsistent iﬀ X Q is consistent for all Q ∈ P . By ¬ X we denote the set { Q : ¬ l | Q : l ∈ X } where w e deﬁne Q : ¬¬ l = Q : l . An extende d P -situate d liter al is either a P -situated literal or a P -situated literal preceded by not . F or a set of P -situated literals X , we use not ( X ) to denote the set { not Q : l | Q : l ∈ X } . F or a set of extended P -situated literals X we denote by X pos the set of P -situated literals in X , i.e. those extended P -situated literals in X that are not preceded by negation-as-failure, while X neg = { Q : l | not Q : l ∈ X } . A P -situate d disjunctive rule is an expression of the form Q : γ ← ( α ∪ not ( β )) where γ is a set of literals, called the head of the rule, and ( α ∪ not ( β )) is called the b o dy of the rule with α and β sets of P -situated literals. A P -situated disjunctive rule Q : γ ← ( α ∪ not ( β )) is called R -lo cal whenev er Q = R . A P -c omp onent dis- junctive pr o gr am Q is a ﬁnite set of Q -local P -situated disjunctive rules. Henceforth w e shall use P both to denote the set of program names and to denote the set of actual P -comp onent disjunctiv e programs. A c ommunic ating disjunctive pr o gr am P is then a ﬁnite set of P -comp onent disjunctive programs. A P -situate d normal rule is an expression of the form Q : l ← ( α ∪ not ( β )) where Q : l is a single P -situated literal. A P -situate d simple rule is an expression of the form Q : l ← α , i.e. a P -situated normal rule without negation-as-failure. A P - c omp onent normal (r esp. simple) pr o gr am Q is a ﬁnite set of Q -lo cal P -situated normal (resp. simple) rules. A c ommunic ating normal (r esp. simple) pr o gr am P is then a ﬁnite set of P -comp onent normal (resp. simple) programs. In the remainder of this pap er w e drop the P -preﬁx whenev er the set P is clear from the con text. Whenever the name of the component disjunctive program Q is 6 K. Bauters et al. clear, we write l instead of Q : l for Q -lo cal situated literals. Note that a communicat- ing disjunctive (resp. normal, simple) program with only one comp onent program th us trivially corresp onds to a classical disjunctive (resp. normal, simple) program. Finally , for notational con venience, we write communicating program when it is clear from the con text whether the program is a communicating simple program or a comm unicating normal program. Example 1 Consider the communicating normal program P = { Q, R } with the follo wing situ- ated rules: Q : a ← R : a Q : b ← Q : c ← Q : c R : a ← Q : a R : b ← not Q : c. Q : a , Q : b , Q : c , R : a and R : b are situated literals. The situated simple rules on the top line are Q -lo cal since we resp ectively ha ve Q : a , Q : b and Q : c in the head of these rules. The situated normal rules on the b ottom line are R -lo cal. Hence Q = { a ← R : a, b ← , c ← c } and R = { a ← Q : a, b ← not Q : c } . Similar as for a classical program, we can deﬁne the Herbr and b ase for a com- p onen t program Q as the set of atoms B Q = { a | Q : a or Q : ¬ a app earing in Q } , i.e. the set of atoms o ccurring in the Q -lo cal situated literals in Q . W e then deﬁne B P =  Q : a | Q ∈ P and a ∈ S R ∈P B R  as the Herbrand base of the comm unicat- ing program P . Example 2 Giv en the communicating normal program P = { Q, R } from Example 1 we ha ve that B Q = { a, b, c } , B R = { a, b } and B P = { Q : a, Q : b, Q : c, R : a, R : b, R : c } . W e sa y that a (partial) interpretation I of a comm unicating disjunctive program P is an y consistent subset I ⊆ ( B P ∪ ¬B P ). Given an interpretation I of a com- m unicating disjunctiv e program P , the reduct Q I for Q ∈ P is the comp onent disjunctiv e program obtained by deleting • each rule with an extended situated literal ‘ not R : l ’ such that R : l ∈ I ; • each remaining extended situated literal of the form ‘ not R : l ’; • each rule with a situated literal ‘ R : l ’ that is not Q -lo cal such that R : l / ∈ I ; • each remaining situated literal ‘ R : l ’ that is not Q -lo cal. Note that this deﬁnition actually com bines t wo t yp es of reducts together. On the one hand, we remov e the negation-as-failure according to the given knowledge. On the other hand, we also remov e situated literals that are not Q -lo cal, again according to the given knowledge. The underlying in tuition of the reduct remains unc hanged compared to the classical case: we take the information into account whic h is enco ded in the guess I and we simplify the program so that we can easily v erify whether or not I is stable, i.e. whether or not I is a minimal mo del of the reduct. Analogous to the deﬁnition of the reduct for disjunctiv e programs (Gelfond and Lifschitz 1991), the reduct of a comm unicating disjunctive program th us deﬁnes a w ay to reduce a program relative to some guess I . The reduct of a comm unicating Expr essiveness of Communic ation in ASP 7 disjunctiv e program is a communicating disjunctiv e program (without negation-as- failure) that only contains comp onent disjunctiv e programs Q with Q -lo cal situated literals. That is, each remaining comp onent disjunctive program Q corresp onds to a classical disjunctiv e program. Example 3 Let us once again consider the communicating normal program P = { Q, R } from Example 1. Giv en I = { Q : a, Q : b, R : a, R : b } w e ﬁnd that Q I = { a ← , b ← , c ← c } and R I = { a ← , b ←} . W e can easily treat Q I and R I separately since they now corresp ond to classical programs. Deﬁnition 1 W e say that an in terpretation I of a communicating disjunctive program P is an answer set of P if and only if ∀ Q ∈ P · ( Q : I Q ) is the minimal model w.r.t. set inclu- sion of Q I . In other w ords: an interpretation I is an answer set of a comm unicating disjunctiv e program P if and only if for every comp onent program Q we hav e that the pro jection of I on Q is an answer set of the comp onen t program Q I under the classical deﬁnition. In the speciﬁc case of a comm unicating normal program P w e can equiv alen tly sa y that I is an answer set of P if and only if w e ha ve that ∀ Q ∈ P · ( Q : I Q ) =  Q I  ? . Example 4 The communicating normal program P = { Q, R } from Example 1 has t wo answer sets, namely { Q : b, R : b } and { Q : a, Q : b, R : a, R : b } . Note that while most approaches do not allow self-references of the form Q : a ← Q : a , in our approach this p oses no problems as it is semantically equiv alent to Q : a ← a . Also note that our semantics allow for “mutual inﬂuence” as in (Brewk a and Eiter 2007; Buccafurri et al. 2008) where the b elief of an agent can b e sup- p orted by the agen t itself, via belief in other agen ts, e.g. { Q : a ← R : a, R : a ← Q : a } . F urthermore we wan t to p oint out that the b elief b et ween agents is the b elief as iden tiﬁed in (Lifsc hitz et al. 1999), i.e. the situated literal Q : l is true in our ap- proac h whenever “ ¬ not Q : l ” is true in the approach in tro duced in (Lifsc hitz et al. 1999) for nested logic programs and treating Q : l as a fresh atom. Before we introduce our ﬁrst prop osition, we generalise the immediate conse- quence op erator for (classical) normal programs to the case of com m unicating sim- ple programs. Sp eciﬁcally , the operator T P is deﬁned w.r.t. an interpretation I of P as T P ( I ) = I ∪ { Q : l | ( l ← α ) ∈ Q, Q ∈ P , α ⊆ I } where α is a set of P -situated literals. It is easy to see that this operator is monotone. T ogether with a result from (T arski 1955) w e kno w that this operator has a least ﬁxp oin t. W e use P ? to denote this ﬁxp oint obtained b y rep eatedly applying T P starting from the empty in terpretation. Clearly , this ﬁxp oint can b e computed in p olynomial time. F urthermore, just like the immediate consequence op erator for (classical) normal programs, this generalised op erator only derives the information that is absolutely necessary , i.e. the ﬁxp oint P ? is globally minimal. 8 K. Bauters et al. Pr op osition 1 Let P b e a communicating simple program. W e then hav e that: • there alwa ys exists at least one answer set of P ; • there is alw a ys a unique answer set of P that is globally minimal; • we can compute this unique globally minimal answer set in p olynomial time. Example 5 Consider the comm unicating simple program P with the rules Q : a ← R : a R : a ← Q : a Q : b ← . This communicating simple program has t wo answer sets, namely { Q : a, Q : b, R : a } and { Q : b } . W e hav e that P ? = { Q : b } , i.e. { Q : b } is the answer set that can b e computed in p olynomial time. Intuitiv ely , this is the answ er set of the communicat- ing simple program P where we treat ev ery situated literal as an ordinary literal. F or example, if we replace the situated literal Q : a (resp. Q : b , R : a ) by the literals q a (resp. q b, r a ) we obtain the simple program q a ← r a q b ← ra ← q a whic h has the unique answer set { q b } , with q b the literal that replaced Q : b . Note that the pro cedure inv olving the generalised ﬁxp oin t do es not allow us to derive the second answer set. In general, no p olynomial pro cedure will b e able to verify whether there is some answ er set in which a given literal is true (unless P = NP ). Although ﬁnding an answer set of a communicating simple program can b e done in p olynomial time, w e will see in the next section that brav e reasoning (the problem of determining whether a given situated literal Q : l o ccurs in any answ er set of a comm unicating simple program) is NP -hard. Consequently , cautious reasoning (the problem of determining whether a given literal Q : l o ccurs in all answer sets of a comm unicating simple program) is coNP -hard. 4 Sim ulating Negation-as-F ailure with Communication The addition of communication to ASP programs can provide added expressiv e- ness o ver simple programs and a resulting increase in computational complexit y for brav e reasoning and cautious reasoning. T o illustrate this observ ation, in this section we sho w that a comm unicating simple program can simulate normal pro- grams. 3 F urthermore, we illustrate that, surprisingly , there is no diﬀerence in terms of computational complexity b etw een communicating simple programs and com- m unicating normal programs; a communicating simple program can b e constructed whic h simulates any given communicating normal program. 3 Recall that simple programs are P -complete and normal programs are NP -complete (Baral 2003). Expr essiveness of Communic ation in ASP 9 F or starters, we recall some of the notions of complexity theory . The complexity classes ∆ P n , Σ P n and Π P n are deﬁned as follo ws, for i ∈  (Papadimitriou 1994): ∆ P 0 = Σ P 0 = Π P 0 = P ∆ P i +1 = P Σ P i Σ P i +1 = NP Σ P i Π P i +1 = co  Σ P i +1  where NP Σ P i (resp. P Σ P i ) is the class of problems that can b e solved in p olyno- mial time on a non-deterministic machine (resp. deterministic mac hine) with an Σ P i oracle, i.e. assuming a pro cedure that can solve Σ P i problems in constant time. F or a general complexity class C , a problem is C -hard if any other problem in C can b e eﬃciently reduced to this problem A problem is said to be C -complete if the problem is in C and the problem is C -hard. Deciding the v alidity of a QBF φ = ∃ X 1 ∀ X 2 ... Θ X n · p ( X 1 , X 2 , · · · X n ) with Θ = ∃ if n is odd and Θ = ∀ oth- erwise, is the canonical Σ P n -complete problem. Deciding the v alidity of a QBF φ = ∀ X 1 ∃ X 2 ... Θ X n · p ( X 1 , X 2 , · · · X n ) with Θ = ∀ if n is o dd and Θ = ∃ otherwise, is the canonical Π P n -complete problem. Moreo ver, these results also hold when we restrict ourselves to problems with p ( X 1 , X 2 , · · · X n ) in disjunctive normal form, except when the last quantiﬁer is an ∃ . 4 Bra ve reasoning as well as answer set exis- tence for simple, normal and disjunctive programs is P -complete, NP -complete and Σ P 2 -complete, resp ectiv ely (Baral 2003). Cautious reasoning for simple, normal and disjunctiv e programs is coP -complete, coNP -complete and co Σ P 2 -complete (Baral 2003). In this section we start by giving an example of the transformation that allo ws us to sim ulate (communicating) normal programs using comm unicating simple pro- grams. A formal deﬁnition of the simulation is giv en b elow in Deﬁnition 2. The correctness is pro ven by Prop ositions 2 and 3. Example 6 Consider the comm unicating normal program P with the rules Q 1 : a ← not Q 2 : b Q 2 : b ← not Q 1 : a. Note that if w e w ere to take Q 1 = Q 2 then this example corresponds to a normal program. In our simulation, the communicating normal program P is transformed 4 Given a QBF with the last quantiﬁer an ∃ and a formula in disjunctive normal form, we can reduce the problem in p olynomial time to a new QBF without the last quantiﬁer. T o do this, for every v ariable quantiﬁed by this last quantiﬁer w e remov e those clauses in which b oth the quantiﬁed v ariable and its negation occur (contradiction) and then remov e all o ccurrences of the quantiﬁed v ariables in the remaining clauses as well as the quantiﬁer itself. The new QBF is then v alid if and only if the original QBF is v alid. 10 K. Bauters et al. in to the following communicating simple program P 0 = { Q 0 1 , Q 0 2 , N 1 , N 2 } : Q 0 1 : a ← N 2 : ¬ b † N 1 : a † ← Q 0 1 : a Q 0 2 : b ← N 1 : ¬ a † N 2 : b † ← Q 0 2 : b Q 0 1 : ¬ a † ← N 1 : ¬ a † N 1 : ¬ a † ← Q 0 1 : ¬ a † Q 0 2 : ¬ b † ← N 2 : ¬ b † N 2 : ¬ b † ← Q 0 2 : ¬ b † . The transformation creates t w o t yp es of comp onent programs or ‘worlds’ , namely Q 0 i and N i . The comp onent program Q 0 i is similar to Q i but o ccurrences of ex- tended situated literals of the form not Q i : l are replaced by N i : ¬ l † , with l † a fresh literal. The non-monotonicity asso ciated with negation-as-failure is simulated b y in tro ducing the rules ¬ l † ← N i : ¬ l † and ¬ l † ← Q 0 i : ¬ l † in Q 0 i and N i , resp ectively . Finally , we add rules of the form l † ← Q 0 i : l to N i , creating an inconsistency when N i b eliev es ¬ l † and Q 0 i b eliev es l . The resulting communicating simple program P 0 is an equiv alent program in that its answer sets corresp ond to those of the original communicating normal program, y et without using negation-as-failure. Indeed, the answer sets of P are { Q 1 : a } and { Q 2 : b } and the answer sets of P 0 are { Q 0 1 : a } ∪  Q 0 2 : ¬ b † , N 2 : ¬ b † , N 1 : a †  and { Q 0 2 : b } ∪  Q 0 1 : ¬ a † , N 1 : ¬ a † , N 2 : b †  . Note that the simulation given in Example 6 can in fact b e simpliﬁed. Indeed, in this particular example there is no need to ha v e t w o additional component programs N 1 and N 2 since Q 1 and Q 2 do not share literals. Also, in this particular example, we need not use ‘ a † ’ and ‘ b † ’ since the simulation would w ork just as w ell if we simply considered ‘ a ’ and ‘ b ’ instead. Nonetheless, for the generalit y of the simulation suc h tec hnicalities are necessary . Without adding an additional component program N i for every original comp onent program Q i the s im ulation would in general not work when tw o component programs shared literals, e.g. Q 1 : a and Q 2 : a . F urthermore, w e need to introduce fresh literals as otherwise the simulation would in general not w ork when w e had true negation in the original program, e.g. Q : ¬ a . W e no w give the deﬁnition of the sim ulation which works in the general case. Deﬁnition 2 Let P = { Q 1 , . . . , Q n } be a communicating normal program. The comm unicating simple program P 0 = { Q 0 1 , . . . , Q 0 n , N 1 , . . . , N n } with 1 ≤ i, j ≤ n that simulates P is deﬁned b y Q 0 i =  l ← α 0 pos ∪  N j : ¬ k † | Q j : k ∈ α neg  | ( l ← α ) ∈ Q i  (2) ∪  ¬ b † ← N i : ¬ b † | Q i : b ∈ E neg  (3) N i =  ¬ b † ← Q 0 i : ¬ b † | Q i : b ∈ E neg  (4) ∪  b † ← Q 0 i : b | Q i : b ∈ E neg  (5) with α 0 =  Q 0 j : l | Q j : l ∈ α  , E neg = S n i =1  S ( a ← α ) ∈ Q i α neg  and with α pos and α neg as deﬁned b efore. Note how this is a p olynomial transformation with at most 3 · | E neg | additional rules. This is imp ortan t when later we use the NP -completeness results from normal programs to show that communicating simple programs are Expr essiveness of Communic ation in ASP 11 NP -complete as well. Recall that b oth ¬ b † and b † are fresh literals that intuitiv ely corresp ond to ¬ b and b . W e use Q 0 i + to denote the set of rules in Q 0 i deﬁned b y (2) and Q 0 i − to denote the set of rules in Q 0 i deﬁned b y (3). The intuition of the simulation in Deﬁnition 2 is as follows. The sim ulation uses the property of m utual inﬂuence to mimic the c hoice induced by negation-as-failure. This is obtained from the interpla y b etw een rules (3) and (4). As such, w e can use the new literal ‘ ¬ b † ’ instead of the original extended (situated) literal ‘ not b ’, allow- ing us to rewrite the rules as w e do in (2). In order to ensure that the simulation w orks even when the program we wan t to simulate already contains classical nega- tion, w e need to sp ecify some additional b o okkeeping (5). As will become clear from Proposition 2 and Proposition 3, the abov e transforma- tion preserves the semantics of the original program. Since we can thus rewrite an y normal program as a comm unicating normal program, the imp ortance is tw ofold. On one hand, we reveal that communicating normal programs do not hav e any ad- ditional expressiv e p o wer o v er comm unicating simple programs. On the other hand, it follo ws that comm unicating simple programs allo w us to solve NP -complete prob- lems. Before w e show the correctness of the sim ulation in Deﬁnition 2, w e introduce a lemma. L emma 1 Let P = { Q 1 , . . . , Q n } and let P 0 = { Q 0 1 , . . . , Q 0 n , N 1 , . . . , N n } with P a comm uni- cating normal program and P 0 the communicating simple program that simulates P deﬁned in Deﬁnition 2. Let M b e an answer set of P and let the interpretation M 0 b e deﬁned as: M 0 = { Q 0 i : a | Q i : a ∈ M } ∪  Q 0 i : ¬ b † | Q i : b / ∈ M  ∪  N i : ¬ b † | Q i : b ∈ M  ∪  N i : a † | Q i : a ∈ M  . (6) F or each i ∈ { 1 , . . . , n } it holds that ( Q 0 i +) M 0 =  l ← α 0 | l ← α ∈ Q M i  with Q 0 i + the set of rules deﬁned in (2) with α 0 = { Q 0 i : b | Q i : b ∈ α } . Using this lemma, w e can prov e that M 0 as deﬁned in Lemma 1 is indeed an answ er set of the comm unicating simple program that sim ulates the comm unicating normal program P when M is an answer set of P . Pr op osition 2 Let P = { Q 1 , . . . , Q n } and let P 0 = { Q 0 1 , . . . , Q 0 n , N 1 , . . . , N n } with P a comm uni- cating normal program and P 0 the communicating simple program that simulates P as deﬁned in Deﬁnition 2. If M is an answer set of P , then M 0 is an answer set of P 0 with M 0 deﬁned as in Lemma 1. Next we in tro duce Lemma 2, which is similar to Lemma 1 in approach but whic h states the con verse. 12 K. Bauters et al. L emma 2 Let P = { Q 1 , . . . , Q n } and let P 0 = { Q 0 1 , . . . , Q 0 n , N 1 , . . . , N n } with P a comm uni- cating normal program and P 0 the communicating simple program that simulates P . Assume that M 0 is an answer set of P 0 and that ( M 0 ) N i is total w.r.t. B N i for all i ∈ { 1 , . . . , n } . Let M b e deﬁned as M = n Q i : b | Q 0 i : b ∈  ( Q 0 i +) M 0  ? o (7) F or each i ∈ { 1 , . . . , n } , it holds that ( Q 0 i +) M 0 =  l ← α 0 | l ← α ∈ Q M i  with α 0 = { Q 0 i : b | Q i : b ∈ α } . Pr op osition 3 Let P = { Q 1 , . . . , Q n } and let P 0 = { Q 0 1 , . . . , Q 0 n , N 1 , . . . , N n } with P a comm uni- cating normal program and P 0 the communicating simple program that simulates P . Assume that M 0 is an answer set of P 0 and that ( M 0 ) N i is total w.r.t. B N i for all i ∈ { 1 , . . . , n } . Then the interpretation M deﬁned in Lemma 2 is an answer set of P . It is imp ortant to note that Lemma 2 and, by consequence, Prop osition 3 re- quire (part of ) the answ er set M 0 to be total. This is a necessary requiremen t, as demonstrated b y the following example. Example 7 Consider the normal program R = { a ← not a } whic h has no answer sets. The corresp onding communicating simple program P 0 = { Q 0 , N } has the following rules: Q 0 : a ← N : ¬ a † N : ¬ a † ← Q 0 : ¬ a † Q 0 : ¬ a † ← N : ¬ a † N : a † ← Q 0 : a. It is easy to see that I = ∅ is an answer set of P 0 since we hav e Q 0 I = N I = ∅ . Notice that I do es not corresp ond with an answer set of R , which is due to I N = ∅ not b eing total and hence we cannot apply Prop osition 3. Regardless, it is easy to see that the requiremen t for the answer set to b e total can b e built into the sim ulation program. Indeed, it suﬃces to introduce additional rules to ev ery N i with 1 ≤ i ≤ n in the simulation deﬁned in Deﬁnition 2. These rules are  N i : a ← N i : a † , N i : a ← N i : ¬ a † | a † ∈ B N i  ∪ { N i : total ← β } with β =  N i : a | a † ∈ B N i  . Th us the requirement that (part of ) the answer set m ust be total can b e replaced by the requiremen t that the situated literals N i : total must b e true in the answer set. Hence, if we w ant to c heck whether a literal Q : l is true in at least one answ er set of a (comm unicating) normal program, it suﬃces to c hec k whether Q : l and N i : total can b e derived in the communicating simple program that simulates it. Clearly we ﬁnd that bra ve reasoning for communicating simple programs is NP -hard. What w e ha ve done so far is imp ortant for tw o reasons. First, we ha v e sho wn that Expr essiveness of Communic ation in ASP 13 the complexit y of brav e reasoning for comm unicating normal programs is no harder than bra v e reasoning for communicating simple programs. Indeed, the problem of bra ve reasoning for communicating normal programs can b e reduced in p olynomial time to the problem of brav e reasoning for communicating simple programs. Sec- ond, since normal programs are a special case of communicating normal programs and since we know that bra ve reasoning for normal programs is an NP -complete problem, w e hav e successfully shown that brav e reasoning for communicating sim- ple programs is NP -hard. In order to sho w that brav e reasoning for communicating simple programs is NP -complete, we need to additionally show that this is a prob- lem in NP . T o this end, consider the following algorithm to ﬁnd the answer sets of a comm unicating simple program P : guess an interp retation I ⊆ ( B P ∪ ¬B P ) verify that this interp retation is an answer set as follows: calculate the reduct Q I of each comp onent p rogram Q calculate the ﬁxp oint of each simple comp onent program Q I verify that Q : I Q = ( Q I ) ? fo r each comp onent program Q The ﬁrst step of the algorithm requires a choice, hence the algorithm is non- deterministic. Next w e determine whether this guess is indeed a communicating answ er set, which inv olves taking the reduct, computing the ﬁxp oin t and verify- ing whether this ﬁxp oint coincides with our guess. Clearly , verifying whether the in terpretation is an answ er set can b e done in p olynomial time and thus the algo- rithm to compute the answ er sets of a communicating simple program is in NP , and th us NP -complete, regardless of the num b er of comp onent programs. These same results hold for communicating normal programs since the reduct also remov es all o ccurrences of negation-as-failure. F or comm unicating disjunctive programs it is easy to see that the Σ P 2 -completeness of classical disjunctiv e ASP carries ov er to comm unicating disjunctiv e programs. Cautious reasoning is then coNP and co Σ P 2 for comm unicating normal programs and communicating disjunctive programs, resp ectively , since this decision problem is the complement of brav e reasoning. Finally , the problem of answer set existence is carried ov er from normal programs and disjunctive programs (Baral 2003) and is NP -hard and co Σ P 2 -hard, resp ectiv ely . Most of these complexity results corresp ond with classical ASP , with the results from communicating simple programs b eing no- table exceptions; indeed, for communicating simple programs the communication asp ect clearly has an inﬂuence on the complexit y . T able 1 summarises the main complexit y results. T o conclude, given a communicating normal program P , there is an easy (linear) translation that transforms P into a normal program P 0 suc h that the answer sets of P 0 corresp onds to the answ er sets of P . Pr op osition 4 Let P b e a communicating normal program. Let P 0 b e the normal program deﬁned 14 K. Bauters et al. T able 1. Completeness results for the main reasoning tasks for a comm unicating program P = { Q 1 , . . . , Q n } reasoning task → answ er set existence brav e reasoning cautious reasoning comp onen t programs ↓ simple program P NP coNP normal program NP NP coNP disjunctiv e program Σ P 2 Σ P 2 co Σ P 2 as follo ws. F or every Q : a ∈ B P w e add the following rules to P 0 : g uess ( Q a ) ← not not g uess ( Q a ) not guess ( Q a ) ← not g uess ( Q a ) ← g uess ( Q a ) , not Q a ← not guess ( Q a ) , Q a. (8) F urthermore, for ev ery normal comm unicating rule P whic h is of the form r = Q : a ← body , w e add the rule Q a ← body 0 to P 0 with Q a a literal. W e deﬁne body 0 , whic h is a set of extended literals, as follows: body 0 = { Q b | Q : b ∈ body } ∪ { g uess ( R c ) | R : c ∈ body , Q 6 = R } ∪ { not S d | ( not S : d ) ∈ body } . (9) W e hav e that M = { Q : a | Q a ∈ M 0 } is an answer set of P if and only if M 0 is an answ er set of P . Pr o of (sk etch) The essential diﬀerence betw een a normal program and a communicating program is in the reduct. More sp eciﬁcally , the diﬀerence is in the treatment of situated literals of the form R : l whic h are not Q -lo cal. Indeed, such literals can, lik e naf-literals, b e guessed and veriﬁed whether or not they are stable, i.e. veriﬁed whether or not the minimal mo del of the reduct corresp onds to the initial guess. It can readily b e seen that this b ehaviour is mimic ked by the rules in (8). The ﬁrst t wo rules guess whether or not some situated literal Q : a is true, while the last tw o rules ensure that our guess is stable; i.e. we are only allow ed to guess Q : a when we are later on actually capable of deriving Q : a . The purp ose of (9) is then to ensure that guessing of situated literals is only used when the situated literal in question is not Q -lo cal and is not preceded by negation-as-failure. 5 Multi-F ocused Answ er Sets Answ er set seman tics are based on the idea of stable minimal mo dels. When dealing with agents that can communicate, it b ecomes unclear how we should interpret the notion of minimality . One option is to assume global minimalit y , i.e. we minimise o ver the conclusions of all the agen ts in the net work. This is the approac h that Expr essiveness of Communic ation in ASP 15 w as taken in Section 4. Another option is to assume minimality on the level of a single agent. Since it is not alw ays p ossible to ﬁnd a model that is minimal for all individual agen ts, the order in which we minimise o ver the agen ts matters, as the next example illustrates. Example 8 An employ ee (‘ E ’) needs a new printer (‘ P ’). She has a few choices (loud or silent, st ylish or dull), preferring silent and stylish. Her manager (‘ M ’) insists that it is a silen t printer. Her b oss (‘ B ’) do es not wan t an exp ensive printer, i.e. one that is b oth silent and st ylish. W e can consider the communicating normal program P = { E , M , B } with: P : sty l ish ← not P : dul l P : dul l ← not P : sty l ish (10) P : silent ← not P : l oud P : l oud ← not P : silent (11) E : undesir ed ← P : dul l E : undesir ed ← P : l oud (12) M : undesired ← P : l oud (13) B : expensiv e ← P : sty l ish, P : sil ent. (14) The rules in (10) and (11) encode the four p ossible printers and the rules in (12), (13) and (14) enco de the inclinations of the emplo yee, manager and boss, respectively . The answ er sets of this program, i.e. those with global minimality , are M 1 = { P : sty l ish, P : silent, B : expensiv e } M 2 = { P : sty l ish, P : loud, E : undesir ed, M : undesired } M 3 = { P : dul l , P : l oud, E : undesir ed, M : undesir ed } M 4 = { P : dul l , P : sil ent, E : undesir ed } The answer sets that are minimal for agent B are M 2 , M 3 and M 4 , i.e. the answer sets that do not contain B : expensiv e . The only answ er set that is minimal for agen t E is M 1 , i.e. the one that do es not contain E : undesir ed . Hence when w e determine lo cal minimality for communicating ASP , the order in which w e consider the agents is important as it induces a priority o ver them, i.e. it makes some agents more imp ortant than others. In this example, if the b oss comes ﬁrst, the employ ee no longer has the choice to pick M 1 . This leav es her with the choice of either a dull or a loud printer, among which she has no preferences. Since the manager prefers a silen t printer, when w e ﬁrst minimise ov er ‘ B ’ and then minimise ov er ‘ M ’ and ‘ E ’ (w e may as w ell minimise ov er ‘ E ’ and then ‘ M ’, as ‘ E ’ and ‘ M ’ ha ve no conﬂicting preferences) w e end up with the unique answer set M 4 . In this section, we formalise suc h a communication mec hanism. W e extend the seman tics of communicating programs in such a w ay that it b ecomes p ossible to fo cus on a sequence of component programs. As suc h, we can indicate that w e are only in terested in those answer sets that are successiv ely minimal with respect to eac h resp ective component program. The underlying intuition is that of leaders and follow ers, where the dec isions that an agent can make are limited b y what its leaders ha ve previously decided. 16 K. Bauters et al. Deﬁnition 3 Let P b e a comm unicating normal program and { Q 1 , . . . , Q n } ⊆ P a set of com- p onen t programs. A ( Q 1 , . . . , Q n ) -fo cuse d answer set of P is deﬁned recursively as follo ws: • M is a ( Q 1 , . . . , Q n )-fo cused answ er set of P and there are no ( Q 1 , . . . , Q n − 1 )-fo cused answ er sets M 0 of P such that M 0 Q n ⊂ M Q n ; • a ()-fo cused answ er set of P is any answer set of P . In other w ords, we sa y that M is a ( Q 1 , . . . , Q n )-fo cused answer set of P if and only if M is minimal among all ( Q 1 , . . . , Q n − 1 )-fo cused answer sets w.r.t. the pro jection on Q n . Example 9 Consider the comm unicating normal program P = { Q, R, S } with the rules Q : a ← R : b ← S : c S : a ← Q : b ← not S : d R : a ← S : c S : c ← not S : d, not R : c Q : c ← R : c R : a ← S : d S : c ← not S : c, not R : c R : c ← not R : a The comm unicating normal program P has three answer sets, namely M 1 = Q : { a, b, c } ∪ R : { c } ∪ S : { a } M 2 = Q : { a, b } ∪ R : { a, b } ∪ S : { a, c } M 3 = Q : { a } ∪ R : { a } ∪ S : { a, d } . The only ( R, S )-fo cused answer set of P 9 is M 1 . Indeed, since { a } = ( M 3 ) R ⊂ ( M 2 ) R = { a, b } we ﬁnd that M 2 is not a ( R )-fo cused answer set. F urthermore { a } = ( M 1 ) S ⊂ ( M 3 ) S = { a, d } , hence M 3 is not an ( R, S )-fo cused answer set. Pr op osition 5 Let P b e a communicating simple program. W e then hav e: • there alwa ys exists at least one ( Q 1 , ..., Q n )-fo cused answ er set of P ; • we can compute this ( Q 1 , ..., Q n )-fo cused answ er set in p olynomial time. T o in vestigate the computational complexit y of m ulti-focused answer sets w e now sho w ho w the v alidity of quan tiﬁed b o olean form ulas (QBF) can b e chec k ed using m ulti-fo cused answer sets of communicating ASP programs. Deﬁnition 4 Let φ = ∃ X 1 ∀ X 2 ... Θ X n · p ( X 1 , X 2 , · · · X n ) b e a QBF where Θ = ∀ if n is even and Θ = ∃ otherwise, and p ( X 1 , X 2 , · · · X n ) is a formula of the form θ 1 ∨ . . . ∨ θ m in disjunctiv e normal form ov er X 1 ∪ . . . ∪ X n with X i , 1 ≤ i ≤ n , sets of v ariables and where eac h θ t is a conjunction of prop ositional literals. W e deﬁne Q 0 as follo ws: Q 0 = { x ← not ¬ x, ¬ x ← not x | x ∈ X 1 ∪ . . . ∪ X n } (15) ∪ { sat ← Q 0 : θ t | 1 ≤ t ≤ m } (16) ∪ {¬ sat ← not sat } . (17) Expr essiveness of Communic ation in ASP 17 F or 1 ≤ j ≤ n − 1 we deﬁne Q j as follo ws: Q j = { x ← Q 0 : x, ¬ x ← Q 0 : ¬ x | x ∈ ( X 1 ∪ . . . ∪ X n − j ) } (18) ∪ ( {¬ sat ← Q 0 : ¬ sat } if ( n − j ) is ev en { sat ← Q 0 : sat } if ( n − j ) is o dd . (19) The communicating normal program corresponding with φ is P φ = { Q 0 , . . . , Q n − 1 } . F or a QBF of the form φ = ∀ X 1 ∃ X 2 ... Θ X n · p ( X 1 , X 2 , · · · X n ) where Θ = ∃ if n is even and Θ = ∀ otherwise and p ( X 1 , X 2 , · · · X n ) once again a formula in disjunctiv e normal form, the simulation only changes sligh tly . Indeed, only the conditions in (19) are sw app ed. Example 10 Giv en the QBF φ = ∃ x ∀ y ∃ z · ( x ∧ y ) ∨ ( ¬ x ∧ y ∧ z ) ∨ ( ¬ x ∧ ¬ y ∧ ¬ z ), the communicating normal program P corresp onding with the QBF φ is deﬁned as follows: Q 0 : x ← not ¬ x Q 0 : y ← not ¬ y Q 0 : z ← not ¬ z Q 0 : ¬ x ← not x Q 0 : ¬ y ← not y Q 0 : ¬ z ← not z Q 0 : sat ← x, y Q 0 : sat ← ¬ x, y , z Q 0 : sat ← ¬ x, ¬ y , ¬ z Q 0 : ¬ sat ← not sat Q 1 : x ← Q 0 : x Q 1 : y ← Q 0 : y Q 1 : ¬ x ← Q 0 : ¬ x Q 1 : ¬ y ← Q 0 : ¬ y Q 1 : ¬ sat ← Q 0 : ¬ sat Q 2 : x ← Q 0 : x Q 2 : ¬ x ← Q 0 : ¬ x Q 2 : sat ← Q 0 : sat The communicating normal program in Example 10 can b e used to determine whether the QBF φ is v alid. First, note that the rules in (15) generate all p ossible truth assignmen ts of the v ariables, i.e. all p ossible prop ositional interpretations. The rules in (16) ensure that ‘ sat ’ is true exactly for those interpretations that satisfy the form ula p ( X 1 , X 2 , . . . , X n ). In tuitively , the comp onent programs { Q 1 , . . . , Q n − 1 } successively bind few er and few er v ariables. In particular, fo cusing on Q 1 , . . . , Q n − 1 allo ws us to consider the binding of the v ariables in X n − 1 , . . . , X 1 , resp ectively . Dep ending on the rules from (19), fo cusing on Q i allo ws us to verify that either some or all of the as- signmen ts of the v ariables in X n − j mak e the formula p ( X 1 , . . . , X n ) satisﬁed, given the bindings that hav e already b een determined by the preceding comp onents. W e no w prov e that the QBF φ is satisﬁable iﬀ Q 0 : sat is true in some ( Q 1 , . . . , Q n − 1 )- fo cused answ er set of the corresp onding program. Pr op osition 6 Let φ and P b e as in Deﬁnition 4. W e hav e that a QBF φ of the form φ = ∃ X 1 ∀ X 2 ... Θ X n · p ( X 1 , X 2 , · · · X n ) is satisﬁable if and only if Q 0 : sat is true in some ( Q 1 , . . . , Q n − 1 )-fo cused answer set of P . F urthermore, w e hav e that a QBF φ of the form φ = ∀ X 1 ∃ X 2 ... Θ X n · p ( X 1 , X 2 , · · · X n ) is satisﬁable if and only if Q 0 : sat is true in all ( Q 1 , . . . , Q n − 1 )-fo cused answ er sets of P . 18 K. Bauters et al. Cor ol lary 1 Let P b e a communicating normal program with Q i ∈ P . The problem of deciding whether there exists a ( Q 1 , . . . , Q n )-fo cused answ er set M of P suc h that Q i : l ∈ M (bra ve reasoning) is Σ P n +1 -hard. Cor ol lary 2 Let P b e a communicating normal program with Q i ∈ P . The problem of deciding whether all ( Q 1 , . . . , Q n )-fo cused answ er sets contain Q i : l (cautious reasoning) is Π P n +1 -hard. In addition to these hardness results, w e can also establish the corresp onding mem b ership results. Pr op osition 7 Let P b e a communicating normal program with Q i ∈ P . The problem of deciding whether there exists a ( Q 1 , . . . , Q n )-fo cused answ er set M of P suc h that Q i : l ∈ M (bra ve reasoning) is in Σ P n +1 . Since cautious reasoning is the complementary problem of brav e reasoning it read- ily follo ws that cautious reasoning is in co Σ P n =1 . No w that we hav e b oth hardness and mem b ership results, we readily obtain the following corollary . Cor ol lary 3 Let P b e a communicating normal program with Q i ∈ P . The problem of decid- ing whether Q i : l ∈ M with M a ( Q 1 , . . . , Q n )-fo cused answer set of P is Σ P n +1 - complete. The next corollary sho ws that the complexity remains the same when going from normal comp onen t programs to simple comp onent programs. Pr op osition 8 Let P b e a communicating simple program with Q i ∈ P . The problem of deciding whether there exists a ( Q 1 , . . . , Q n )-fo cused answ er set M of P suc h that Q i : l ∈ M (bra ve reasoning) is in Σ P n +1 . Finally , we also hav e a result for comm unicating disjunctiv e programs instead of comm unicating normal programs. Pr op osition 9 Let P be a communicating disjunctive program with Q i ∈ P . The problem of deciding whether Q i : l ∈ M with M a ( Q 1 , . . . , Q n )-fo cused answ er set of P is in Σ P n +2 . T able 2 summarises the mem b ership results for brav e reasoning that w ere discussed in this section. Due to the extra expressiveness and complexit y of m ulti-focused answer sets, it is clear that no translation to classical ASP is po ssible. P ossible future implementa- tions ma y , how ever, be based on a translation to other PSP ACE complete problems suc h as QBF formulas or mo dal logics. A translation to QBF formulas seems to b e the most natural, esp ecially since the proof of the complexit y of m ulti-fo cused Expr essiveness of Communic ation in ASP 19 T able 2. Membership results for bra ve reasoning with (multi-focused) answ er sets of the comm unicating program P = { Q 1 , . . . , Q n } form of communication → none situated literals multi-focused t yp e of comp onent program ↓ simple program P NP Σ P n +1 normal program NP NP Σ P n +1 disjunctiv e program Σ P 2 Σ P 2 Σ P n +2 answ er sets inv olves reducing QBF form ulas to m ulti-fo cused answer sets. Ho w ever, an y such translation falls beyond the scope is this paper and is the sub ject of future researc h. 6 Case Study: subset-minimal ab ductive diagnosis In this section w e w ork out an example that highligh ts the usefulness of multi- fo cused answ er sets. The use of m ulti-fo cused answer sets has already pro ven itself useful in mo deling problems where one can use a negotiation paradigm, e.g. in (Bauters 2011). Ho wev er, the main goal in (Bauters 2011) w as to sho w that such a paradigm is p ossible, rather than actually trying to enco de a problem that is known to b e more complex that Σ P 2 . Though a lot of interesting problems are indeed in P , NP or Σ P 2 , there are still some important problems that are ev en higher up in the p olynomial hierarc hy . One such a problem is a sp ecial form of ab ductive diagonistics. An abductive diagnostic problem is encoded as a triple h H, T , O i (Eiter et al. 1999), where H is a set of atoms referred to as hypotheses, T is an ASP pro- gram referred to as the theory and O is a set of literals referred to as observ ations. In tuitively , the theory T describes the dynamics of the system, the observ ations O describ e the observ ed state of the system and the h yp otheses H try to explain these observ ations within the theory . The goal in subset-minimal ab ductiv e diagnosis is to ﬁnd the minimal set of hypotheses that explain the observ ation. That is, we w ant to ﬁnd the minimal set of hypotheses suc h that O ⊆ M with M an answer set of T ∪ H . Subset-minimal ab ductiv e diagnostics ov er a theory consisting of a disjunctiv e program is a problem in Σ P 3 and hence w e cannot rely on classical ASP to ﬁnd the solutions to this problem. How ev er, as we will see in the next example, w e can easily solve this problem using multi-focused answer sets. Example 11 ( A dapte d fr om (Eiter et al. 1999) ) Consider an electronic circuit, as in Figure 2, where w e hav e a p ow er source, a con trol lamp, three hot-plates wired in parallel and a fuse to protect each hot- plate. It is kno wn that some of the fuses are sensitive to high current and ma y consequen tly blow, but it is not known which fuses. F urthermore, plate A sits near a source of water ( e.g. a tap). If water comes into contact with plate A, this causes a short circuit whic h blows the nearest fuse, i.e. fuse A, to preven t any damage. 20 K. Bauters et al. fuse A fuse B fuse C plate A plate B plate C bulb AC p ower Fig. 2. Schematics of the electronic circuit we wan t to diagnose. Up on insp ection, we ﬁnd that the control lamp is on and that plate A feels cold to the touch. W e wan t to ﬁnd the subset minimal diagnoses that would explain the problem, i.e. w e wan t to ﬁnd the minimal causes that can explain this situation. First we need to describ e the theory , i.e. the schematics. The theory describ es the dynamics of the system and thus also ho w the system ma y fail. W e can describ e the theory as follows. F or starters, a melted fuse can be caused by a high curren t, or, for fuse A, due to a hazardous water leak: Q : melte d A ; Q : melte d B ; Q : melte d C ← Q : high Q : melte d A ← Q : le ak . F urthermore, under a num ber of conditions the control light will b e oﬀ: Q : light oﬀ ← Q : p ower oﬀ Q : light oﬀ ← Q : br oken bulb Q : light oﬀ ← Q : melte d A , Q : melte d B , Q : melte d C . Then w e describ e under what conditions each plate will b e hot: Q : hot plateA ← not Q : melte d A , not Q : p ower oﬀ Q : hot plateB ← not Q : melte d B , not Q : p ower oﬀ Q : hot plateC ← not Q : melte d C , not Q : p ower oﬀ . W e no w encode the hypotheses. W e ha v e a num b er of causes, each of whic h ma y b y itself or in conjunction with other causes explain our observ ation. In total, w e ha ve four causes. The p ow er can b e oﬀ ( p ower oﬀ ), the ligh t bulb might b e broken ( br oken bulb ), there ma y hav e b een a high current ( high ) and/or a water leak may ha ve o ccurred ( le ak ). W e describ e all these hypotheses as follows: Q : p ower oﬀ ← not Q : no p ower oﬀ Q : no p ower oﬀ ← not Q : p ower oﬀ Q : br oken bulb ← not Q : no br oken bulb Q : no br oken bulb ← not Q : br oken bulb Q : high ← not Q : no high Expr essiveness of Communic ation in ASP 21 Q : no high ← not Q : high Q : le ak ← not Q : no le ak Q : no le ak ← not Q : le ak . It is easy to see that these rules in Q enco de all p ossible subsets of h yp otheses that ma y hav e o ccurred. W e then add rules to a separate comp onent program H which merely rela ys the information on the set of h yp otheses that w e c hose. The reason for this separate component program H is that we can no w minimise ov er the set of h yp otheses that is assumed, simply by fo cusing on H . H : p ower oﬀ ← Q : p ower oﬀ H : br oken bulb ← Q : br oken bulb H : high curr ent ← Q : high H : water le ak ← Q : le ak Finally , w e mo del the observ ation. W e observe that the control light is on and that plate A is cold. In other w ords, we obtain the rules (which enco de constraints): Q : c ontr adiction ← not Q : c ontr adiction , Q : light oﬀ Q : c ontr adiction ← not Q : c ontr adiction , Q : hot plateA whic h in tuitiv ely tell us that we cannot hav e that the light is oﬀ, nor can w e hav e that plate A is hot. The ( H )-fo cused answer sets give us the subset minimal ab ductive diagnoses. It is easy to see that the focus on H is needed to minimise ov er the hypotheses. The program P = { Q, H } has t w o ( H )-fo cused answ er sets M 1 and M 2 , both containing M shared = { Q : no p ower oﬀ , Q : no br oken bulb , Q : hot plateB , Q : hot plateC } : M 1 = M shared ∪ { Q : melted A, Q : no leak , Q : hig h, H : hig h } M 2 = M shared ∪ { Q : melted A, Q : l eak , Q : no hig h, H : leak } . Hence the minimal sets of h yp otheses that supp ort our observ ation, i.e. M H with M an ( H )-fo cused answ er set, are that either there was a high current (which melted fuse A) or there w as a water leak (which also melted fuse A). 7 Related W ork A large b o dy of research has b een devoted to com bining logic programming with m ulti-agent or multi-con text ideas, with v arious underlying reasons. One reason for suc h a combination is that the logic can b e used to describe the (rational) b ehaviour of the agen ts in a multi-agen t netw ork, as in (Dell’Acqua et al. 1999). Alternately , it can be used to com bine diﬀerent ﬂav ours of logic programming languages (Luo et al. 2005; Eiter et al. 2008). It can also b e used to externally solv e tasks for whic h ASP is not suited, while remaining in a declarativ e framew ork (Eiter et al. 2006). As a ﬁnal example, it can b e used as a form of co op eration, where multiple agen ts or contexts collab orate to solve a diﬃcult problem (De V os et al. 2005; V an Nieu wen b orgh et al. 2007). The approach in this pap er falls in the last category 22 K. Bauters et al. and is concerned with how the collab oration of diﬀerent ASP programs aﬀects the expressiv eness of the ov erall system. Imp ortan t work has b een done in the domain of multi-con text systems (MCS) and m ulti-agent ASP to enable collaboration b etw een the diﬀerent con texts/ASP programs. W e discuss some of the more prominent work in these areas. The w ork of (Roelofsen and Seraﬁni 2005) prop oses an extension of m ulti-con text systems or MCSs (Giunc higlia and Seraﬁni 1994) that allows MCSs to reason about absen t information, i.e. they in tro duce non-monotonicity in the context of MCSs. The idea of a MCS is that w e ha ve a n umber of contexts that eac h ha ve access to only a subset of the a v ailable information. Each context has a local mo del and reasoning capabilities, but there is also an information ﬂo w deﬁned b y the system b et ween the diﬀerent contexts. It is this idea that was later adopted in the ASP comm unity and in our pap er in particular. Our pap er has a comparable syn tax as (Ro elofsen and Seraﬁni 2005) but rather diﬀeren t seman tics. The semantics in (Roelofsen and Seraﬁni 2005) are closely re- lated to the well-founded semantics (Gelder et al. 1991), while our seman tics are closer in spirit to the stable mo del seman tics (Gelfond and Lifzchitz 1988). Another p oin t where our semantics diﬀer is that w e allo w a restricted circular explanation of why a literal is true, if that circular explanation is due to our reliance on other comp onen t programs. This particular form of circular reasoning has b een identiﬁed in (Buccafurri et al. 2008) as a requirement in the representation of social reasoning. The work in (Brewk a et al. 2007) further extends the work in (Roelofsen and Seraﬁni 2005) and addresses a n umber of problems and deﬁciencies. The paper is, to the b est of our kno wledge, the ﬁrst to oﬀer a syntactical rather than seman- tical description of communication in m ulti-context systems, making it easier to implemen t an actual algorithm. A n umber of interesting applications of contextual framew orks, including information fusion, game theory and so cial choice theory are highligh ted in the paper. Lastly , the pap er identiﬁes that the complexit y of the main reasoning task is on the second level of the p olynomial hierarch y . Along similar lines (Brewk a and Eiter 2007) combines the non-monotonicity from (Ro elofsen and Seraﬁni 2005) with the heterogeneous approach presented in (Giunc higlia and Seraﬁni 1994) in to a single framework for heterogenous non- monotonic multi-con text reasoning. The work in (Brewk a and Eiter 2007) intro- duces several notions of equilibria, including minimal and grounded equilibria. In our approac h, lo cal reasoning is captured by grounded equilibria (whic h do es not allo w circular explanations) while communicating with other comp onent programs is captured by the w eaker minimal equilibria. The w ork in (Brewk a and Eiter 2007) oﬀers v arious mem b ership results on deciding the existence of an equilibrium and is one of the ﬁrst to note that multi-con text systems, due to the nature of the bridge rules/situated literals, can be non-monotonic even if all the logics in the component programs themselv es are monotonic. W ork on a distributed solv er for heterogenous multi-con text system s commenced in (Dao-T ran et al. 2010). While solvers exist to compute m ulti-context systems lo cally , this is the ﬁrst w ork to consider an algorithm whic h is b oth distributed ( i.e. no shared memory) and mo dular ( i.e. computation starting from partial mo d- Expr essiveness of Communic ation in ASP 23 els). When the con text under consideration uses e.g. ASP , loop form ulas can b e devised which allo w bridge rules to b e compiled into lo cal classical theories. It is then p ossible to use SA T solv ers to compute the grounded equilibria of the het- erogenous multi-con text system. Later work in (Dresc her et al. 2011) impro v ed on the idea by oﬀering a mechanism to identify and break symmetries ( i.e. permuta- tions of b elief which result in identical knowledge). As suc h, the solv er need nev er visit t wo p oints in the search space that are symmetric, thus p otentially oﬀering a considerable speedup. Exp erimen tal results sho w that the solution space can indeed b e (signiﬁcantly) compressed. A similar idea might b e used to compute answer sets of a comm unicating ASP program in a distributed fashion. Indeed, such answer sets are closely related to the idea of minimal equilibria from (Brewk a and Eiter 2007). A few mo diﬁcations should nonetheless b e made. F or example, the Herbrand base needs to be redeﬁned in a wa y that is safe in suc h a distributed setting, e.g. by only taking situated literals in to account that occur in a given component program. Optimizations to the distributed algorithm also seem likely to b e applicable to the setting of communicating ASP . On the other hand, it do es not seem to b e straight- forw ard to extend these ideas to compute multi-focused answer sets in a distributed fashion. One of the most recent extensions to m ulti-context systems are managed m ulti- con text systems or mMCS (Brewk a et al. 2011). Normally , bridge rules can only b e used to pass along information which allows for e.g. selection and abstraction of information b etw een con texts. In an mMCS, how ev er, additional operations on kno wledge bases can b e freely deﬁned. F or example, op erations ma y b e deﬁned that remov e or revise information. Such ope rations are p erformed by the con text itself, i.e. by the legacy system that is used such as ASP , but mMCS allow to cop e with this additional functionality in a principled w ay . As one would exp ect, adding suc h complex operations increases the expressiveness of the resulting system considerably . Our w ork, on the other hand, only allo ws for information to be passed along. By v arying the w a y that the communication w orks, we achiev ed a comparable expressiv eness. W e now direct our atten tion to work done within the ASP communit y . The ideas presen ted in this pap er are related to HEX programs (Eiter et al. 2005) in which ASP is extended with higher-order predicates and external atoms. These external atoms allow to exchange kno wledge in a declarative wa y with external sources that may implement functionality which is incon venien t or imp ossible to encode using curren t answ er set programming paradigms. Application-wise, HEX is mainly prop osed as a to ol for non-monotonic seman tic web reasoning under the answ er set seman tics. Hence HEX is not primarily targeted at increasing the expressiveness, but foremost at extending the applicabilit y and ease of use of ASP . Tw o other imp ortant works in the area of multi-agen t ASP are (De V os et al. 2005) and (V an Nieuw en b orgh et al. 2007). In b oth (De V os et al. 2005) and (V an Nieu wen b orgh et al. 2007) a multi-agen t system is developed in whic h multiple agen ts/comp onent programs can communicate with each other. Most imp ortantly from the p oint of view of our work, b oth approaches use ASP and hav e agen ts that are quite expressive in their own right. Indeed, in (De V os et al. 2005) each agent 24 K. Bauters et al. is an Ordered Choice Logic Program (OCLP) (Brain and De V os 2003) and in (V an Nieu wen b orgh et al. 2007) each agent uses the extended answer set semantics. The framework introduced in (De V os et al. 2005) is called LAIMA. Each of the agen ts is an OCLP . The agents can communicate with who ever they wan t and cir- cular comm unication is allo wed (where agen t A tells something to agent B whic h tells something to A . . . ). How ever, only p ositive information can b e shared and the authors do not lo ok at the actual expressiv eness of the framew ork. In (V an Nieu wen b orgh et al. 2007) each agen t uses the extended answer set semantics. T he net work is a linear “hierarchical” netw ork ( i.e. information only ﬂows in one direc- tion), yet they emplo y the idea of a failure feedbac k mechanism. Intuitiv ely , a failure feedbac k mec hanism allo ws the previous agent in a netw ork to revise his conclusion when the conclusion leads to an unresolv able inconsistency for the next agent in the net work. It is this mechanism that gives rise to a higher expressiveness, namely Σ P n for a hierarc hical net work of n agents. Our w ork is diﬀerent in that we start from simple and normal ASP programs for the agen ts. Our comm unication mechanism is also quite simple and do es not rely on any kind of feedback. Regardless, we obtain a comparable expressiv eness. W e also mention (Dao-T ran et al. 2009) where recursive mo dular non-monotonic logic programs (MLP) under the ASP seman tics are considered. The main diﬀerence b et ween MLP and our work is that our communication mec hanism is parameter- less, i.e. the truth of a situated literal is not dep endent on parameters passed b y the situated literal to the target comp onent program. Our approach is clearly diﬀerent and w e cannot readily mimic the b eha viour of the netw orks presented in (Dao-T ran et al. 2009). Our expressiv eness results therefore do not directly apply to MLPs. Finally , there is an interesting resem blance b etw een m ulti-fo cused answ er sets and the work on multi-lev el integer programming (Jeroslow 1985). In multi-lev el in teger programming, diﬀerent agents control diﬀerent v ariables that are outside of the control of the other agents, yet are linked b y means of linear inequalities (constrain ts). The agents hav e to ﬁx the v alues of the v ariables they can control in a predeﬁned order, such that their own linear ob jective function is optimized. Similarly , in communicating ASP , literals b elong to diﬀerent comp onen t programs (agen ts), and their v alues are link ed through constraints, which in this case take the form of rules. Again the agen ts act in a predeﬁned order, but no w they try to minimise the set of literals they hav e to accept as b eing true, rather than a linear ob jective function. Though there is an in tuitive link, further research is required to make this link b etw een multi-focused answer sets and the work on multi-lev el in teger programming explicit. 8 Conclusions W e ha v e systematically studied the eﬀect of adding communication to ASP in terms of expressiveness and computational complexit y . W e start from simple programs, i.e. deﬁnite programs extended with true negation. Determining whether a literal b elongs to an answer set of a simple program is a problem in P . A netw ork of these simple programs, whic h w e call comm unicating simple programs, is how ev er expres- Expr essiveness of Communic ation in ASP 25 siv e enough to sim ulate normal programs. In other w ords, determining whether a literal b elongs to an answer set of a communicating simple program is NP -hard. F urthermore, comm unicating simple programs can also simulate communicating normal programs provided that the resulting answer sets are partially complete, th us showing that adding negation-as-failure to communicating simple programs do es not further increase the expressiveness. W e hav e introduced multi-focused answer sets for communicating programs. The underlying in tuition is that of leaders and follo wers, where the c hoices av ailable to the follo wers are limited by what the leaders ha ve previously decided. On a tec hnical lev el, the problem translates to establishing local minimality for some of the comp onent programs in the communicating program. Since in general it is not p ossible to ensure lo cal minimalit y for all component programs, an order m ust be deﬁned among comp onent programs on which to fo cus. The result is an increase in expressiveness, where the problem of deciding whether Q i : l ∈ M with M a ( Q 1 , . . . , Q n )-fo cused answ er set of P is Σ P n +1 -complete. In our w ork w e thus ﬁnd that the choice of the communication mec hanism is paramoun t w.r.t. the expres- siv eness of the ov erall system, in addition to the expressiveness of the individual agen ts. T able 2 highlights the mem b ership results for brav e reasoning obtained in Section 5. App endix A Result pro ofs Pr op osition 1 Let P b e a communicating simple program. W e then hav e that: • there alwa ys exists at least one answer set of P ; • there is alw a ys a unique answer set of P that is globally minimal; • we can compute this unique globally minimal answer set in p olynomial time. Pr o of W e can easily generalise the immediate consequence op erator for (classical) simple programs to the case of comm unicating simple programs. Sp eciﬁcally , the op erator T P is deﬁned w.r.t. an in terpretation I of P as T P ( I ) = I ∪ { Q : l | ( Q : l ← α ) ∈ Q, Q ∈ P , α ⊆ I } where α is a set of P -situated literals. It is easy to see that this operator is monotone. T ogether with a result from (T arski 1955) w e kno w that this operator has a least ﬁxp oin t. W e use P ? to denote this ﬁxp oint obtained b y rep eatedly applying T P starting from the empty in terpretation. Clearly , this ﬁxp oint can b e computed in p olynomial time. W e need to verify that P ? is indeed an answer set. Since P is a comm unicating simple program, we know that the reduct Q P ? will only remov e rules that contain situated literals R : l that are not Q -local with R : l / ∈ P ? . In other words, rules that are not applicable ( α 6⊆ P ? ) and that con tain non- Q -lo cal situated literals are remo ved. F urthermore, remaining situated literals of the form R : l that are not Q - lo cal ( i.e. those where R : l ∈ P ? ) are remo v ed from the bo dy of the remaining rules. 26 K. Bauters et al. Hence the remaining rules are all Q -lo cal. Notice that the op erator T P is clearly an extension of the op erator T Q . Indeed, for a comp onent simple program Q 0 that is Q 0 -lo cal it is easy to verify that if ( Q 0 ) ? = M 0 then (( P 0 ) ? ) Q 0 = M 0 with P 0 = { Q 0 } . It then readily follo ws, since all rules are Q -lo cal and therefore independent of all other comp onen t programs, that  Q P ?  ? = ( P ? ) Q for all Q ∈ P . So far we found that an answer set exists and that it can b e computed in polyno- mial time. All that rem ains to b e shown is that this answer set is globally minimal. This trivially follows from the wa y we deﬁned the op erator T P since it only makes true the information that is absolutely necessary , i.e. the information that follo ws directly from the facts in the communicating simple program. Hence this is the minimal amount of information that needs to b e derived for a set of situated lit- erals to b e a mo del of the comm unicating simple program at hand and thus the ﬁxp oin t P ? is the globally minimal answ er set. L emma 1 Let P = { Q 1 , . . . , Q n } and let P 0 = { Q 0 1 , . . . , Q 0 n , N 1 , . . . , N n } with P a comm uni- cating normal program and P 0 the communicating simple program that simulates P as deﬁned in Deﬁnition 2. Let M be an answer set of P and let the interpretation M 0 b e deﬁned as: M 0 = { Q 0 i : a | a ∈ M Q i , Q i ∈ P } ∪  Q 0 i : ¬ b † | b / ∈ M Q i , Q i ∈ P  ∪  N i : ¬ b † | b / ∈ M Q i , Q i ∈ P  ∪  N i : a † | a ∈ M Q i , Q i ∈ P  . (A1) F or each i ∈ { 1 , . . . , n } it holds that ( Q 0 i +) M 0 =  l ← α 0 | l ← α ∈ Q M i  with Q 0 i + the set of rules deﬁned in (2) with α 0 = { Q 0 i : b | Q i : b ∈ α } . Pr o of T o prov e this, we ﬁrst sho w that any rule of the form ( l ← α ) ∈ Q M i reapp ears in ( Q 0 i +) M 0 under the form ( l ← α 0 ) for any i ∈ { 1 , . . . , n } . The second step, showing that the con verse also holds, can then b e done in an analogous wa y . Supp ose ( l ← α ) ∈ Q M i for some i ∈ { 1 , . . . , n } . By the deﬁnition of the reduct w e know that there is some rule of the form ( l ← α ∪ not β ∪ γ ) ∈ Q i suc h that β ∩ M = ∅ and γ ⊆ M is a set of situated literals of the form Q j : d with i 6 = j , 1 ≤ j ≤ n . F rom Deﬁnition 2, we know that the comm unicating normal rule ( Q i : l ← α ∪ not β ∪ γ ) is transformed in to the rule ( Q 0 i : l ← α 0 ∪ β 0 ∪ γ 0 ) with α 0 = { Q 0 i : b | Q i : b ∈ α } , β 0 =  N k : ¬ c † | Q k : c ∈ β , k ∈ { 1 , . . . , n }  and γ 0 =  Q 0 j : d | Q j : d ∈ γ , j ∈ { 1 , . . . , n }  . W e show that, indeed, ( l ← α ) ∈ Q M i reapp ears in ( Q 0 i +) M 0 under the form ( l ← α 0 ). First, whenever Q k : c ∈ β , we know that Q k : c / ∈ M since β ∩ M = ∅ . F rom the construction of M 0 w e hav e that N k : ¬ c † ∈ M 0 . Similarly , since γ ⊆ M w e kno w from the construction of M 0 that Q 0 j : d ∈ M 0 whenev er Q j : d ∈ γ . Hence when determining the reduct ( Q 0 i +) M 0 , the extended situated literals in β 0 and γ 0 will be deleted. Expr essiveness of Communic ation in ASP 27 Finally , whenever α ∩ M 6 = ∅ we kno w from the construction of M 0 that Q 0 i : b ∈ M 0 whenev er Q i : b ∈ α . Clearly , when determining the reduct, none of these extended situated literals will b e deleted as they are Q 0 i -lo cal. Hence it is clear that the reduct of the communicating rule ( Q 0 i : l ← α 0 ∪ β 0 ∪ γ 0 ) is the rule Q 0 i : l ← α 0 . This completes the ﬁrst part of the proof. As indicated, the second part of the pro of is completely analogous. Pr op osition 2 Let P = { Q 1 , . . . , Q n } and let P 0 = { Q 0 1 , . . . , Q 0 n , N 1 , . . . , N n } with P a comm uni- cating normal program and P 0 the communicating simple program that simulates P as deﬁned in Deﬁnition 2. If M is an answer set of P , then M 0 is an answer set of P 0 with M 0 deﬁned as in Lemma 1. Pr o of This pro of is divided in to tw o parts. In part 1 w e only consider the comp onen t programs Q 0 i with i ∈ { 1 , . . . , n } and show that  ( Q 0 i ) M 0  ? = ( M 0 ) Q 0 i . In part 2 we do the same, but we only consider the comp onent programs N i with i ∈ { 1 , . . . , n } . As p er Deﬁnition 1 w e hav e then shown that M 0 is indeed an answ er set of P 0 . Consider a comp onent program Q 0 i with i ∈ { 1 , . . . , n } . By Deﬁnition 2 we ha ve that Q 0 i = ( Q 0 i +) ∪ ( Q 0 i − ) and th us ( Q 0 i ) M 0 = ( Q 0 i +) M 0 ∪ ( Q 0 i − ) M 0 . (A2) F or Q 0 i − we know by construction that it only contains rules that are of the form ( Q 0 i : ¬ b † ← N i : ¬ b † ) and that the only rules of this form are in Q 0 i − . Therefore, due to the deﬁnition of the reduct, w e hav e ( Q 0 i − ) M 0 =  ¬ b † ← | N i : ¬ b † ∈ M 0  and b ecause of the construction of M 0 , see (A1), w e obtain ( Q 0 i − ) M 0 =  ¬ b † ← | b / ∈ M Q i  . (A3) Hence ( Q 0 i − ) M 0 only contains facts ab out literals that, by construction of Q 0 i , do not o ccur in Q 0 i +. This means that from (A2) and (A3) we obtain  ( Q 0 i ) M 0  ? =  ( Q 0 i +) M 0  ? ∪  ¬ b † ← | b / ∈ M Q i  . (A4) F rom Lemma 1 we know that ( Q 0 i +) M 0 =  l ← α 0 | l ← α ∈ Q M i  where α 0 = { Q 0 i : b | Q i : b ∈ α } . Because of the deﬁnition of an answer set of a communicating program w e hav e M Q i =  Q M i  ? =  ( Q 0 i +) M 0  ? . (A5) Com bining this with (A4) we get  ( Q 0 i ) M 0  ? = M Q i ∪  ¬ b † | b / ∈ M Q i  = ( M 0 ) Q 0 i (deﬁnition of M 0 , see (A1)) 28 K. Bauters et al. This concludes the ﬁrst part of the pro of. In the second part of the pro of, we only consider the comp onen t programs N 0 i with i ∈ { 1 , . . . , n } . By construction of N i w e know that all the rules of the form ¬ b † ← Q 0 i : ¬ b † and b † ← Q 0 i : b are in N i and that all the rules in N i are of this form. W e hav e ( N i ) M 0 =  ¬ b † ← | Q 0 i : ¬ b † ∈ M 0  ∪  b † ← | Q 0 i : b ∈ M 0  whic h, due to the deﬁnition of M 0 can b e written as =  ¬ b † ← | b / ∈ M Q i  ∪  b † ← | b ∈ M Q i  from whic h it follows that  ( N i ) M 0  ? = ( M 0 ) N i . L emma 2 Let P = { Q 1 , . . . , Q n } and let P 0 = { Q 0 1 , . . . , Q 0 n , N 1 , . . . , N n } with P a comm uni- cating normal program and P 0 the communicating simple program that simulates P . Assume that M 0 is an answer set of P 0 and that ( M 0 ) N i is total w.r.t. B N i for all i ∈ { 1 , . . . , n } . Let M b e deﬁned as M = n Q i : b | Q 0 i : b ∈  ( Q 0 i +) M 0  ? o (A6) F or each i ∈ { 1 , . . . , n } , it holds that ( Q 0 i +) M 0 =  l ← α 0 | l ← α ∈ Q M i  with α 0 = { Q 0 i : b | Q i : b ∈ α } . Pr o of T o pro ve this, we ﬁrst show that any rule of the form ( l ← α 0 ) ∈ ( Q 0 i +) M 0 reapp ears in Q M i under the form l ← α for an y i ∈ { 1 , . . . , n } . W e then sho w that the conv erse also holds, which is rather analogous to the pro of of the ﬁrst part of Lemma 1. Due to some technical subtleties in the second part of the pro of, how ever, w e present the pro of in detail. Supp ose ( l ← α 0 ) ∈ ( Q 0 i +) M 0 . By the deﬁnition of the reduct of a comm unicating simple program we know that there is some communicating simple rule of the form ( l ← α 0 ∪ β 0 ∪ γ 0 ) ∈ Q 0 i + such that β 0 ⊆ M 0 is a set of situated literals of the form N k : ¬ c † and γ 0 ⊆ M 0 is a set of situated literals of the form Q 0 j : d with i 6 = j and 1 ≤ j, k ≤ n . F rom the deﬁnition of Q 0 i +, w e know that ( Q 0 i : l ← α 0 ∪ β 0 ∪ γ 0 ) corresp onds to a rule ( l ← α ∪ not β ∪ γ ) ∈ Q i where w e hav e that α = { Q i : b | Q 0 i : b ∈ α 0 } , β =  Q k : c | N k : ¬ c † ∈ β 0  and γ =  Q j : d | Q 0 j : d ∈ γ 0  . W e sho w that, indeed, ( l ← α 0 ) ∈ ( Q 0 i +) M 0 reapp ears in Q M i under the form ( l ← α ). First, since β 0 ⊆ M 0 , whenever N k : ¬ c † ∈ β 0 w e kno w that N k : ¬ c † ∈ M 0 . Since M 0 is a mo del (indeed, it is an answer set) it is an interpretation (and thus con- sisten t). Therefore, if N k : ¬ c † ∈ M 0 then surely N k : c † / ∈ M 0 . No w, if w e were to hav e Q 0 k : c ∈ M 0 , then applying the immediate consequence op erator on the rule N k : c † ← Q 0 k : c found in the comp onent program N k w ould force us to hav e N k : c † ∈ M 0 whic h results in a contradiction. Hence we ﬁnd that Q 0 k : c / ∈ M 0 . Expr essiveness of Communic ation in ASP 29 By Deﬁnition 2 w e kno w that Q 0 k = ( Q 0 k +) ∪ ( Q 0 k − ) and th us, by the deﬁnition of the reduct, we know that ( Q 0 k ) M 0 = ( Q 0 k +) M 0 ∪ ( Q 0 k − ) M 0 . Then w e ﬁnd that (( Q 0 k ) M 0 ) ? = (( Q 0 k +) M 0 ) ? ∪ (( Q 0 k − ) M 0 ) ? since all the rules in Q 0 k − hav e fresh lit- erals in the head and literals from N k in the b o dy and hence cannot in teract with the rules from Q 0 k + which only depend on information deriv ed from Q 0 k + and N k in their b o dies. Recall from the deﬁnition of an answer set of a communicating program that ∀ Q 0 k ∈ P 0 · ( Q 0 k : M 0 Q 0 k ) =  ( Q 0 k ) M 0  ? . Since w e already found that Q 0 k : c / ∈ M 0 w e must hav e Q 0 k : c / ∈  ( Q 0 i +) M 0  ? , or, because of the deﬁnition of M , that Q k : c / ∈ M . Hence when determining the reduct ( l ← α ∪ not β ∪ γ ) M , the extended situated literals in not β will b e deleted. In a similar wa y of reasoning, since γ 0 ⊆ M 0 and because γ =  Q j : d | Q 0 j : d ∈ γ 0  w e kno w from the construction of M that γ ⊆ M . Hence when determining the reduct, the situated literals in γ will be deleted. Finally , since α 0 ⊆ M 0 and b ecause { Q i : b | Q 0 i : b ∈ α 0 } ⊆ M we know from the construction of M that α ∈ M . Clearly , when determining the reduct, none of the situated literals in α will be deleted as they are Q i -lo cal. Hence the reduct of the comm unicating rule ( Q i : l ← α ∪ not β ∪ γ ) is the rule Q i : l ← α . This completes the ﬁrst part of the pro of. W e no w come to the second part. This time we sho w that an y rule of the form ( l ← α ) ∈ Q M i reapp ears in ( Q 0 i +) M 0 under the form ( l ← α 0 ) for any i ∈ { 1 , . . . , n } . Supp ose ( l ← α ) ∈ Q M i for some i ∈ { 1 , . . . , n } . By the deﬁnition of the reduct we kno w that there is some rule of the form ( l ← α ∪ not β ∪ γ ) ∈ Q i suc h that β ∩ M = ∅ and γ ⊆ M is a set of situated literals of the form Q j : d with i 6 = j , 1 ≤ j ≤ n . F rom Deﬁnition 2, w e know that the comm unicating normal rule ( Q i : l ← α ∪ not β ∪ γ ) is transformed into the rule ( Q 0 i : l ← α 0 ∪ β 0 ∪ γ 0 ) with α 0 = { Q 0 i : b | Q i : b ∈ α } , β 0 =  N k : ¬ c † | Q k : c ∈ β , k ∈ { 1 , . . . , n }  and γ 0 =  Q 0 j : d | Q j : d ∈ γ , j ∈ { 1 , . . . , n }  . W e show that, indeed, ( l ← α ) ∈ Q M i reapp ears in ( Q 0 i +) M 0 under the form ( l ← α 0 ) when M 0 N i is total w.r.t. B N i for all i ∈ { 1 , . . . , n } . First, since γ ⊆ M w e kno w from the construction of M that Q j : d ∈ M whenev er Q 0 j : d ∈ γ 0 . Also, when Q k : c ∈ β , w e know that Q k : c / ∈ M since β ∩ M = ∅ . F rom the construction of M we then know that Q 0 k : c / ∈ M 0 and since M 0 is an answer set w e readily obtain that N k : c † / ∈ M 0 due to the construction of N k . T ogether with the requiremen t that M 0 N k is total w.r.t. B N k w e then m ust hav e that N k : ¬ c † ∈ M 0 . Hence when determining the reduct ( Q 0 i +) M 0 , the extended situated literals in β 0 and γ 0 will b e deleted. Finally , whenever α ∩ M 6 = ∅ we kno w from the construction of M 0 that Q 0 i : b ∈ M 0 whenev er Q i : b ∈ α . Clearly , when determining the reduct, none of these extended situated literals will b e deleted as they are Q 0 i -lo cal. Hence it is clear that the reduct of the communicating rule ( Q 0 i : l ← α 0 ∪ β 0 ∪ γ 0 ) is the rule Q 0 i : l ← α 0 . This completes the second part of the pro of. Pr op osition 3 Let P = { Q 1 , . . . , Q n } and let P 0 = { Q 0 1 , . . . , Q 0 n , N 1 , . . . , N n } with P a comm uni- 30 K. Bauters et al. cating normal program and P 0 the communicating simple program that simulates P . Assume that M 0 is an answer set of P 0 and that ( M 0 ) N i is total w.r.t. B N i for all i ∈ { 1 , . . . , n } . Then the interpretation M deﬁned in Lemma 2 is an answer set of P . Pr o of Lemma 2 tells us that ( Q 0 i +) M 0 =  l ← α 0 | l ← α ∈ Q M i  where we ha ve α 0 = { Q 0 i : b | Q i : b ∈ α } . Hence w e ha ve  ( Q 0 i +) M 0  ? =  Q M i  ? since repeatedly applying the immediate consequence operator m ust conclude the same literals l due to the corresp ondence of the rules in the reducts and because of the wa y α 0 is deﬁned. Since w e deﬁned M as n Q i : b | Q 0 i : b ∈  ( Q 0 i +) M 0  ? o it follo ws immediately that M is an answer set of P since ∀ i ∈ { 1 , . . . , n } ·  Q M i  ? = M Q i (A7) whic h completes the pro of. Pr op osition 5 Let P b e a communicating simple program. W e then hav e: • there alwa ys exists at least one ( Q 1 , ..., Q n )-fo cused answ er set of P ; • we can compute this ( Q 1 , ..., Q n )-fo cused answ er set in p olynomial time. Pr o of W e kno w from Proposition 1 that w e can alw ays ﬁnd a globally minimal answ er of P in p olynomial time. Due to the w ay we deﬁned the immediate ﬁxpoint op erator T P this op erator only makes true the information that is absolutely necessary , i.e. the minimal amount of information that can be deriv ed (for each component program). It is then easy to see that no comp onent program can derive an y less information (w e ha ve no negation-as-failure) and thus that this globally minimal answ er set is also lo cally minimal and th us a ( Q 1 , ..., Q n )-fo cused answ er set of P . Pr op osition 6 Let φ and P b e as in Deﬁnition 4. W e hav e that a QBF φ of the form φ = ∃ X 1 ∀ X 2 ... Θ X n · p ( X 1 , X 2 , · · · X n ) is satisﬁable if and only if Q 0 : sat is true in some ( Q 1 , . . . , Q n − 1 )-fo cused answer set of P . F urthermore, w e hav e that a QBF φ of the form φ = ∀ X 1 ∃ X 2 ... Θ X n · p ( X 1 , X 2 , · · · X n ) is satisﬁable if and only if Q 0 : sat is true in all ( Q 1 , . . . , Q n − 1 )-fo cused answ er sets of P . Expr essiveness of Communic ation in ASP 31 Pr o of W e give a pro of b y induction. Assume w e hav e a QBF φ 1 of the form ∃ X 1 · p ( X 1 ) with P 1 = { Q 0 } the communicating normal program corresp onding with φ 1 according to Deﬁnition 4. If the formula p 1 ( X 1 ) of the QBF φ 1 is satisﬁable then we know that there is a ()-fo cused answer set M of P 1 suc h that Q 0 : sat ∈ M . Otherwise, w e know that Q 0 : sat / ∈ M for all ()-answ er sets M of P 1 . Hence the induction h yp othesis is v alid for n = 1. Assume the result holds for an y QBF φ n − 1 of the form ∃ X 1 ∀ X 2 . . . Θ X n − 1 · p n ( X 1 , X 2 , . . . , X n − 1 ). W e show in the induction step that it holds for any QBF φ n of the form ∃ X 1 ∀ X 2 . . . Θ X n · p n − 1 ( X 1 , X 2 , . . . , X n ). Let P = { Q 0 , . . . , Q n − 1 } and P 0 =  Q 0 0 , . . . , Q 0 n − 2  b e the communicating normal programs that correspond with φ n and φ n − 1 , resp ectiv ely . Note that the component programs Q 2 , . . . , Q n − 1 are deﬁned in exactly the same w ay as the comp onent programs Q 0 1 , . . . , Q 0 n − 2 , the only diﬀerence b eing the name of the comp onen t programs. What is of imp ortance in the case of φ n is therefore only the additional rules in Q 0 and the new comp o- nen t program Q 1 . The additional rules in Q 0 merely generate the corresp onding in terpretations, where we now need to consider the possible interpretations of the v ariables from X n as w ell. The rules in the new comp onent program Q 1 ensure that Q 1 : x ∈ M whenever Q 0 : x ∈ M and Q 1 : ¬ x ∈ M whenever Q 0 : ¬ x ∈ M for every M an answ er set of P and x ∈ ( X 1 ∪ . . . ∪ X n − 1 ). Depending on n being even or o dd, w e get tw o distinct cases: • if n is even, then we hav e ( sat ← Q 0 : sat ) ∈ Q 1 and we kno w that the QBF φ n has the form ∃ X 1 ∀ X 2 . . . ∀ X n · p n ( X 1 , X 2 , . . . , X n ). Let us consider what happ ens when we determine the ( Q 1 )-fo cused answer sets of P . Due to the construction of Q 1 , we know that M 0 Q 1 ⊂ M Q 1 can only hold for tw o answer sets M 0 and M of P if M 0 and M corresp ond to iden tical interpretations of the v ariables in X 1 ∪ . . . ∪ X n − 1 . F urthermore, M 0 Q 1 ⊂ M Q 1 is only possible if Q 1 : sat ∈ M while Q 1 : sat / ∈ M 0 . No w note that giv en an in terpretation of the v ariables in X 1 ∪ . . . ∪ X n − 1 , there is exactly one answer set for each choice of X n . When we hav e M 0 with Q 1 : sat / ∈ M 0 this implies that there is an interpretation suc h that, for some c hoice of X n , this particular assignment of v alues of the QBF do es not satisfy the QBF. Similarly , if we ha ve M with Q 1 : sat ∈ M then the QBF is satisﬁed for that particular c hoice of X n . Determining ( Q 1 )-fo cused answer sets of P will eliminate M since M 0 Q 1 ⊂ M Q 1 . In other words, for iden tical in terpretations of the v ariables in X 1 ∪ . . . ∪ X n − 1 , the answer set M 0 enco des a counterexam- ple that shows that for these interpretations it do es not hold that the QBF is satisﬁed for all choices of X n . F o cusing thus eliminates those answer sets that claim that the QBF is satisﬁable for the v ariables in X 1 ∪ . . . ∪ X n − 1 . When we cannot ﬁnd such M 0 Q 1 ⊂ M Q 1 this is either because none of the in terpretations satisfy the QBF or all of the interpretations satisfy the QBF. In b oth cases, there is no need to eliminate any answer sets. W e thus eﬀectively mimic the requiremen t that the QBF φ n should hold for ∀ X n . 32 K. Bauters et al. • if n is o dd, then ( ¬ sat ← Q 0 : ¬ sat ) ∈ Q 1 and we know that the QBF φ n has the form ∃ X 1 ∀ X 2 . . . ∃ X n · p n ( X 1 , X 2 , . . . , X n ). As b efore, we know that M 0 Q 1 ⊂ M Q 1 can only hold for tw o answer sets M 0 and M of P if M 0 and M corresp ond to identical interpretations of the v ariables in X 1 ∪ . . . ∪ X n − 1 . Ho wev er, this time M 0 Q 1 ⊂ M Q 1 is only p ossible if Q 1 : ¬ sat ∈ M while Q 1 : ¬ sat / ∈ M 0 . If we hav e M with Q 1 : ¬ sat ∈ M then the QBF is not satisﬁed for that partic- ular c hoice of X n , whereas when M 0 with Q 1 : ¬ sat / ∈ M 0 this implies that there is an interpretation such that, for some c hoice of X n , this particular assignmen t of the v ariables do es satisfy the QBF. Determining ( Q 1 )-fo cused answer sets of P will eliminate M since M 0 Q 1 ⊂ M Q 1 . F or identical interpretations of the v ariables in X 1 ∪ . . . ∪ X n − 1 , the answer set M 0 enco des a counterexample that sho ws that for these interpretations there is some choice of X n suc h that the QBF is satisﬁed. F o cusing thus eliminates those answer sets that claim that the QBF is not satisﬁable for the v ariables in X 1 ∪ . . . ∪ X n − 1 . When w e cannot ﬁnd suc h M 0 Q 1 ⊂ M Q 1 this is either because none of the in terpretations satisfy the QBF or all of the interpretations satisfy the QBF. In b oth cases, there is no need to eliminate any answer sets. W e eﬀectively mimic the requiremen t that the QBF φ n should hold for ∃ X n . F or a QBF of the form ∀ X 1 ∃ X 2 . . . Θ X n · p ( X 1 , X 2 , . . . , X n ), with Θ = ∃ if n is ev en and Θ = ∀ otherwise, the pro of is analogous. In the base case, w e know that a QBF φ 1 of the form ∀ X 1 · p ( X 1 ) is satisﬁable only when for ev ery ()- fo cused answer set M of P 1 = { Q 0 } w e ﬁnd that Q 0 : sat ∈ M . Otherwise, we know that there exists some ()-fo cused answers sets M of P 1 suc h that Q 0 : sat / ∈ M . Hence the induction h yp othesis is v alid for n = 1. The induction step is then entirely analogous to what w e hav e prov en b efore, with the only diﬀerence b eing that the cases for n b eing ev en or o dd are sw app ed. Finally , since the ﬁrst quantiﬁ er is ∀ , w e need to v erify that Q 0 : sat is true in ev ery ( Q 1 , . . . , Q n − 1 )-fo cused answ er set of P . Pr op osition 7 Let P b e a communicating normal program with Q i ∈ P . The problem of deciding whether there exists a ( Q 1 , . . . , Q n )-fo cused answ er set M of P suc h that Q i : l ∈ M (bra ve reasoning) is in Σ P n +1 . Pr o of W e show the pro of by induction on n . In the case where n = 1, w e need to guess a ( Q 1 )-fo cused answ er set M of P which can clearly b e done in p olynomial time. W e no w need to verify that this is indeed a ( Q 1 )-fo cused answer set which is a problem in coNP . Indeed, verifying that M is not a ( Q 1 )-fo cused answer set can b e done using the follo wing pro cedure in NP : • guess an in terpretation M 0 • verify that M 0 is an answ er set of P • verify that M 0 Q 1 ⊂ M Q 1 . Expr essiveness of Communic ation in ASP 33 Hence, to ﬁnd a ( Q 1 )-fo cused answer set, w e guess an interpretation, verify that it is an answer set in polynomial time, and w e subsequen tly use an NP oracle to decide whether this answ er set is ( Q 1 )-fo cused, i.e. the problem is in Σ P 2 . Assume that there exists an algorithm to compute the ( Q 1 , . . . , Q n − 1 )-fo cused answ er sets of P that is in Σ P n . In a similar fashion, we can guess a ( Q 1 , . . . , Q n )-fo cused answer set and v erify there is no ( Q 1 , . . . , Q n )-fo cused answer set M 0 of P such that M 0 Q n ⊂ M Q n using a Σ P n oracle, i.e. the algorithm is in Σ P n +1 . Pr op osition 8 Let P b e a communicating simple program with Q i ∈ P . The problem of deciding whether there exists a ( Q 1 , . . . , Q n )-fo cused answ er set M of P suc h that Q i : l ∈ M (bra ve reasoning) is in Σ P n +1 . Pr o of W e know from Prop osition 2 that one normal program can b e simulated by a com- m unicating simple program with tw o comp onent programs. Since only the program Q 0 in the sim ulation in Deﬁnition 4 includes negation-as-failure, it suﬃces to add a single simple component program in order to sim ulate the negation-as-failure. Since the num ber of comp onent programs is of no imp ortance in Prop osition 7, the result readily follo ws. Pr op osition 9 Let P be a communicating disjunctive program with Q i ∈ P . The problem of deciding whether Q i : l ∈ M with M a ( Q 1 , . . . , Q n )-fo cused answ er set of P is in Σ P n +2 . 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Expressiveness of Communication in Answer Set Programming

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