Model-checking ATL under Imperfect Information and Perfect Recall Semantics is Undecidable

Mo del-c hec king AT L under Imp erfect Information and P erfect Recall Seman tics is Undecidable C˘ at˘ alin Dima a , F erucio Laurent ¸ iu T ¸ iplea b a LAC L, Universit´ e Pari s Est- Cr ´ eteil, 61 av. du G-r al de Gaul le, 940 10 Cr´ eteil, F r anc e b Dep artment of Computer Scienc e, “Al.I.Cuza” University of Ia ¸ si, Ia¸ si 7 00506 , R omania Abstract W e prop ose a formal pro of of the undecidabilit y of t he mo del c hec king problem for alternating- time temp oral logic under imp erfect information and p erfect recall seman tics. This problem w as a nnounced to b e undecidable according to a p ersonal communic ation on multi-pla y er games with imperfect infor mation, but no formal pro of w as ev er published. Our pro of is based on a direct reduction fro m the non-halting problem for T uring mac hines. Keywor ds: Alternating-time temp oral logic, imp erfect information, p erfect recall, mo del c hec king, decidabilit y 1. In tr o duction The Alternating-time T emp o ral Logic ( AT L ) hav e b een intro duced in [1 ] a s a logic to reason ab out strategic abilities of agents in mu lti-agent systems. AT L extends C T L b y replacing the path quan tifiers ∀ and ∃ b y c o op er ation mo dal i ties ⟪ A ⟫ , whe re A is a team of agen ts. A form ula ⟪ A ⟫ ϕ expresses that the team A ha s a collectiv e strategy to enforce ϕ . The seman tics of AT L is defined o v er c oncurr ent game structur es ( C GS ) [1] whic h are transition systems whose states are lab eled b y atomic prop ositions and for whic h a set of agents is specified. E ac h agent ma y ha v e inc omplete/imp erfe ct information ab out the state of the sy stem in the sense that the agen t may not b e able to difference b et w een some states. When the agen t is able to observ e the entire state labeling, we sa y that he has c omplete/p erfe ct inform ation . A t r a nsition from a state to another one is p erformed by an action tuple consisting of an action for eac h agen t in the sys tem. The actio n an agent is allo w ed to p erform at a state is c hosen fro m a give n set of actioned allow ed t o b e p erformed b y the agen t at that state and ma y dep end on the curren t state (this is called imp erfe ct r e c al l ) or on the whole history of ev en ts that ha v e happened ( t his is called p erfe ct r e c al l ). Com bining imp erfect o r p erfect informat io n with imp erfect or p erfect recall w e obtain fo ur t yp es of concurren t game structures and, consequen tly , four t yp es o f seman tics for AT L . Email addr esses: dima @univ -paris 12.fr (C˘ at˘ alin Dima), fltipl ea@in fo.ua ic.ro (F erucio Laurent ¸ iu T ¸ iplea) Pr eprint submitte d to Elsevier Septemb er 13, 2018 A series of pap ers hav e b een addressed the mo del-c heck ing problem for AT L [1, 3 , 2]. Based on unpublished w ork of Y annak akis [4], the mo del c heck ing problem for AT L with imp erfect information a nd p erfect recall seman tics w as announced to b e undecidable in [1]. Since then, man y authors hav e men tioned this result but, unfortunately , no formal pro of w as eve r published (see also [2]). In this pap er w e prop ose a formal pro of of this pro blem. Our pro of is based on a direct sim ulation o f T uring mac hines b y concurren t game structures under imp erfect in- formation and p erfect recall, whic h allo ws for a reduction of the non-halting problem for T uring ma chines to the mo del chec king problem fo r AT L under imp erfect information and p erfect recall semantic s. Moreo v er, the strategies used b y agen ts to sim ula t e the T uring mac hine are primitiv e recursiv e. This sho ws that the undecidabilit y o f mo del c hec king AT L under imp erfect information and p erfect recall seman tics is mainly due to the imp erfect information agen ts ha v e abo ut the system states. While our pro o f is giv en for the de dicto strategies from [1], the same construction w orks also fo r the d e r e strategies fr o m [3, 5]. 2. Alternating-time T emp oral Logic W e recall in this section the syn tax and seman tics of the alternating- time tempo r a l logic. W e will mainly fo llow the approac h in [2] and fix first a few notations. N stands for the se t of positive in tegers (natural nu m b ers) and P de notes the pow erset operator . Giv en a set V , V + denotes the f ree semi-group and V ∗ denotes the f ree monoid g enerated b y V under concatenation. λ stands f or the empt y w ord (the unit y of V ∗ ). The notatio n f ∶ X ⇀ Y means that f is a pa rtially defined function from X to Y . AT L syntax. The syn tax o f AT L is given b y the grammar ϕ ∶ ∶= p ∣ ¬ ϕ ∣ ϕ ∧ ϕ ∣ ⟪ A ⟫ # ϕ ∣ ⟪ A ⟫ ◻ ϕ ∣ ⟪ A ⟫ ϕU ϕ where p ranges o v er a finite non-empt y set of atomic pr op ositions Π, A is a non-empt y subset of a finite set Ag of agents , and # , ◻ , and U are the standard temp oral op erators next , glob al ly , and until , r esp ective ly . Note that, in order to define combinations o f temp oral op erators inside the coalition op erators, the we ak-until op erator should b e giv en as a primitiv e op erato r [6], since it cannot b e deriv ed from the ab o v e o p erators. Ho w ev er our result holds also fo r this restricted syn tax. AT L semantics. AT L is in terpreted o ver c oncurr ent gam e structur es ( C GS ) [1]. Suc h a structure consists o f a set of states lab eled by a tomic prop ositions and a set of agen ts. Eac h agen t may p erform some actions and at least one action is av aila ble to the agen t at eac h state. His decision in c ho osing whic h action should b e p erformed at some state ma y b e based on his capabilit y of observing all or some atomic prop ositions at the current state, usually called p erfe ct or imp erfe ct in formation , and on his full o r part ia l history , us ually called p erfe ct or imp erfe ct r e c a l l . 2 In what f ollo ws w e fo cus on C G S unde r imp erfect information and perfect r ecall and adopt the formal approach in [2]. A C GS under imp erfe ct information is a tuple G = ( Ag , S, Π , π , ( ∼ i ∣ i ∈ Ag ) , Act, d, → ) , where: • Ag = { 1 , . . . , k } is a finite non-empt y set of agents ; • S is a finite non-empt y set of states ; • Π is a finite non-empt y set of atomic pr op ositions ; • π ∶ S → P ( Π ) is the state-lab eling function ; • ∼ i is an equiv alence relation on S , for an y agent i ; • Act is a finite non-empt y set of actions ; • d ∶ Ag × S → P ( Act ) − { ∅} giv es the set of actions av ailable to agen ts at eac h state, satisfying d ( i, s ) = d ( i, s ′ ) fo r any agen t i and states s a nd s ′ with s ∼ i s ′ ; • → ∶ S × Act k ⇀ S is the (part ially defined) tr ansition function satisfying, for an y s ∈ S and ( a 1 , . . . , a k ) ∈ Act k , the f o llo wing prop ert y: → ( s, ( a 1 , . . . , a k )) is defined iff a i ∈ d ( i, s ) for a n y agent i . W e will write s ( a 1 ,...,a k ) Ð Ð Ð Ð Ð → s ′ , whenev er → ( s, ( a 1 , . . . , a k )) = s ′ . If s and s ′ are ∼ i -equiv alen t (i.e., s ∼ i s ′ ) then w e sa y that s and s ′ are i n distinguish- able from the agen t i ’s p oin t of view (due to his partial view on the states). Eac h ∼ i is comp onen t- wise extended to sequenc es of states. Th us, f o r α, α ′ ∈ S + w e write α ∼ i α ′ and sa y that α and α ′ are ∼ i -e quivalent if α = s 0 ⋯ s n and α ′ = s ′ 0 ⋯ s ′ n for some n ∈ N , and s j ∼ i s ′ j for all 0 ≤ j ≤ n . A p erfe ct r e c al l str ate gy for an ag ent i in a C GS G is a function σ ∶ S + → Act whic h is c omp atible with d and ∼ i , i.e., • σ ( αs ) ∈ d ( i, s ) , for an y α ∈ S ∗ and s ∈ S ; • σ ( α ) = σ ( α ′ ) , for an y α , α ′ ∈ S + with α ∼ i α ′ . A p erfe ct r e c al l str ate gy f or a te am A of agents is a family σ A = ( σ i ∣ i ∈ A ) of p erfect recall strategies for the agen ts in A . If σ A is a p erfect recall strategy for the agen ts in A , αs ∈ S ∗ S , and a = ( a 1 , . . . , a k ) ∈ Act k , then w e write a ∈ σ A ( αs ) if the follow ing prop erties hold: • a i ∈ d ( i, s ) , for a n y i ∈ Ag − A ; • a i ∈ σ i ( αs ) , for an y i ∈ A . 3 Giv en a state s of G and σ A as ab ov e, define out G ( s, σ A ) as b eing the set o f all infinite sequence s of states λ = s 0 s 1 s 2 ⋯ suc h that s 0 = s and, for any j ≥ 0, there exists a ∈ σ A ( s 0 ⋯ s j ) with s j a Ð → s j + 1 . F or λ = s 0 s 1 s 2 ⋯ an infinite sequen ce o f states a nd j ≥ 0, λ [ j ] denotes the j -th state in the sequenc e, λ [ j ] = s j The imp erfe ct information p erfe ct r e c al l semantics for AT L , denoted ⊧ iR , is defined as follo ws ( G is a C GS under imp erfect information and s is a state of G ): • ( G , s ) ⊧ iR p if p ∈ π ( s ) ; • ( G , s ) ⊧ iR ¬ ϕ if ( G , s ) / ⊧ iR ϕ ; • ( G , s ) ⊧ iR ϕ ∧ ψ if ( G , s ) ⊧ iR ϕ and ( G , s ) ⊧ iR ψ ; • ( G , s ) ⊧ iR ⟪ A ⟫ # ϕ if there exists a p erfect recall strat egy σ A suc h t hat ( G , λ [ 1 ]) ⊧ iR ϕ , for an y λ ∈ ou t G ( s, σ A ) ; • ( G , s ) ⊧ iR ⟪ A ⟫ ◻ ϕ if there exists a p erfect recall strategy σ A suc h that ( G , λ [ j ]) ⊧ iR ϕ , for an y λ ∈ ou t G ( s, σ A ) and an y j ≥ 0 ; • ( G , s ) ⊧ iR ⟪ A ⟫ ϕU ψ if there exists a p erfect recall strategy σ A suc h that for any λ ∈ out G ( s, σ A ) there exists j ≥ 0 with ( G , λ [ j ]) ⊧ iR ψ and ( G , λ [ k ]) ⊧ iR ϕ for all 0 ≤ k < j . The mo del che cki n g pr oblem for AT L fo rm ulas under imp erfect informat io n and p erfect recall seman tics is to decide, give n an AT L formula ϕ , a concurren t game structure G under imp erfect information, and a state s of G , whether ( G , s ) ⊧ iR ϕ . Computation tr e es. The pro of of our main result in the next section will b e based on c omputation tr e es asso ciated to C GS s. These are sp ecial cases of lab eled trees, whic h are structures T = ( V , E , v 0 , l 1 , l 2 ) , where • ( V , E , v 0 ) is a tree whose set o f no des is V , whose set of edges is E , and whose ro ot is v 0 ; • l 1 is the no de-lab eling function; • l 2 is the edge-lab eling function. Paths in a lab eled tree T = ( V , E , v 0 , l 1 , l 2 ) are defined inductiv ely as usual a s sequences of no des: • v 0 is a pa th in T ; • if v 0 ⋯ v n is a pa th in T and ( v n , v ) ∈ E , then v 0 ⋯ v n v is a path in T . If v is a node of T , then path T ( v 0 , v ) stands for the unique pat h from the ro ot v 0 to v in T . The n umber of no des on a path τ is the length of τ , denoted ∣ τ ∣ . The lab eling function l 1 is homomorphically extended to paths, that is, l 1 ( τ 1 τ 2 ) = l 1 ( τ 1 ) l 1 ( τ 2 ) . L evels in a lab eled tr ee T = ( V , E , v 0 , l 1 , l 2 ) are sets of no des of T defined inductiv ely as follo ws: 4 • l ev el T ( 0 ) = { v 0 } ; • l ev el T ( n + 1 ) = { v ∈ V ∣( ∃ v ′ ∈ l ev el T ( n ))( ( v ′ , v ) ∈ E )} , for any n ≥ 0. lev el T ( n ) is referred t o as the level n in T . Giv en a CGS G , a state s of G , a coalit ion A of agen ts, and a p erfect recall strategy σ A for agen ts in A , define inductive ly the s -r o ote d c omputation tr e es of G under σ A as follo ws: • any tree with exactly one no de (its ro ot) lab eled b y s is an s -ro oted computation tree of G under σ A ; • if T = ( V , E , v 0 , l 1 , l 2 ) is an s -ro oted computation tree of G under σ A , v is a no de of T , and l 1 ( v ) a Ð → s ′ for some a ctio n- tuple a ∈ σ A ( l 1 ( path T ( v 0 , v ))) and state s ′ suc h that no edge fro m v is lab eled by a , then the tr ee T ′ obtained as follo ws is an s -ro o ted computation tree of G : – T ′ is o btained from T b y adding a new no de v ′ lab eled by s ′ and an edge ( v , v ′ ) lab eled by a . If T ′ is obtained from T as ab ov e, w e will also write T ⇒ G , σ A T ′ or T a ⇒ G , σ A T ′ if w e w an t to specify the action tuple a as w ell. Remark 1. It i s e asy to se e that, for any atomic pr op osition p , the fol lowing pr op erty holds true: • ( G , s ) ⊧ iR ⟪ A ⟫ ◻ p if and only if ther e exists a p erfe ct r e c al l str ate gy σ A such that p ∈ π ( l 1 ( v )) , for any s -r o ote d c omputation tr e e T of G under σ A , and a ny no de v of T . 3. Undecidabilit y of Mo del Chec king A T L iR W e will prov e in this section that the mo del c hec king pro blem for AT L iR is undecidable. The pro of techniq ue is b y reduction from the non-halting problem for deterministic T uring mac hines. Giv en a deterministic T u ring mac hine M , w e construct a concurren t game structure under imp erfect information G with three ag ents Ag = { 1 , 2 , 3 } , a state s init of G , and a n AT L formula ⟪{ 1 , 2 }⟫ ◻ ok , where ok is an atomic prop o sition, suc h that M do es not halt on t he empty word if and only if ( G , s init ) ⊧ iR ⟪{ 1 , 2 }⟫ ◻ ok . The deterministic T uring machine s w e consider are tuples M = ( Q, Σ , q 0 , B , δ ) , where Q is a finite set of states, Σ is a finite tap e alphab et, q 0 is the initial state, B ∈ Σ is the blank sym b ol, and δ ∶ Q × Σ ⇀ Q × Σ × { L, R } is a part ia lly defined transition function, where “ L ” sp ecifies a “left mov e” and “ R ” sp ecifies a “ r igh t mov e”. A configuration of M is a w ord a 1 ⋯ a i − 1 q a i ⋯ a n , where all a ’s are from Σ and q is a state. Suc h a configura - tion sp ecifies that M is in state q , it s read/write head p oints t o the i th cell of the tap e, and the j th cell holds a j if j ≤ n , a nd B , otherwise. The initial configuration is q 0 B . The transition relation on configura tions, denoted ⇒ M , is defined as usual. F or instance, a 1 ⋯ a i − 1 q a i ⋯ a n ⇒ M a 1 ⋯ q ′ a i − 1 a ′ i ⋯ a n if i > 1 and δ ( q , a i ) = ( q ′ , a ′ i , L ) . 5 The T uring mac hine M halts on t he empty w ord if, starting with the initial configura- tion, the mac hine reac hes a configuration a 1 ⋯ a i − 1 q a i ⋯ a n for which δ ( q , a i ) is undefined or i = 1 and δ ( q , a i ) = ( q ′ , a ′ i , L ) for some q ′ and a ′ i . Intuition first. The main idea of the construction is to enco de the configurations of the T ur- ing mac hine horizontally in the leve ls of the computation tree. A configuration a 1 ⋯ a i − 1 q a i ⋯ a k of M will b e sim ula t ed in A b y some lev el in some computation tree lik e in F igure 1 (where i = 2 a nd k = 3 ) . The no des of this tree are represen ted by circles. The lab el of a no de s init s ′ lb s a 1 s ′ tr s q ,a 2 s ′ tr s a 3 Figure 1: Level cor resp onding to a 1 q a 2 a 3 is carried inside the circle represen ting the no de. The no de lab eled s ′ lb sp ecifies the left b order o f M ’s tap e, the no de lab eled s ′ tr is a cell separator also used to tra nsfer information b et w een paths o f computation trees, t he no des lab eled s a 1 and s a 3 sp ecify the con tent of the first a nd third cell, resp ectiv ely , and the no de lab eled s q ,a 2 sp ecifies b oth the con ten t of the second cell and the fact that M is in state q and its read/write head p oin ts to the second cell. The generation of the initial configuration q 0 B of M is sim ulated b y the computation tree in Figur e 2. All states in this tree ale lab eled b y ok ; the no de lab eled s g en has one more lab el, namely p 1 (this lab el is g r a phically represen ted because it will b e particularly imp ortant in defining the a g en ts strategies). As w e will see later, the tw o maximal paths in this tree are ∼ 2 -equiv alen t. This allo ws, together with the strategy w e will use, for the sync hronization in the last computation step of these paths. s init s ′ init s lb s ′ lb ( i, ( q 0 ) , i ) ( i, i, i ) ( i, i, br 1 ) s g en ∣ p 1 s B s q 0 ,B ( i, ( q 0 ) , i ) ( i, i, i ) ( i, i, br 2 ) Figure 2: Genera ting the initial configuratio n q 0 B of M 6 The lev els enco ding configura tions of the T uring mac hine will b e enco ded on the even p ositions in a computation tree, the o dd lev els b eing used for correctly represen ting transi- tions of the T uring mac hine. Some no des in the leve ls of ev en index will then enco de tap e cells, while some other no des will b e used for tra nsferring information b et wee n adja cen t cells. Some examples presen ting this idea are giv en in the follo wing, b efore the formal construction and pro of. A computation step a 1 q a 2 a 3 ⇒ M a 1 a ′ 2 q ′ a 3 in the T uring machine is sim ulated b y ex- tending the computation tree in Figur e 1 as in Fig ure 3. The sync hronizatio n betw een the fourth and fifth paths is possible because, as w e will see , these paths are ∼ 1 -equiv alen t. Similarly , the sync hronization b et w een the fifth a nd sixth paths is p o ssible b ecause these paths are ∼ 2 -equiv alen t. s init s ′ lb s ′ lb s ′ lb ( i, i, i ) ( i, i, i ) s a 1 s a 1 s a 1 ( i, i, i ) ( i, i, i ) s ′ tr s ′ tr s ′ tr ( i, i, i ) ( i, i, i ) s q ,a 2 s a ′ 2 s a ′ 2 ( i, i, i ) (( q , q ′ , R ) , i, i ) s ′ tr s q ,q ′ ,R s ′ tr ( i, ( q , q ′ , R ) , i ) (( q , q ′ , R ) , i, i ) s a 3 s a 3 s q ′ ,a 3 ( i, ( q , q ′ , R ) , i ) ( i, i, i ) Figure 3: Simulation o f a 1 q a 2 a 3 ⇒ M a 1 a ′ 2 q ′ a 3 The sim ulation represen ted in these t wo figures pro ceeds a s follo ws: in the observ able history cor r esp o nding to the path ending in s q ,a 2 , the only p ossibility for agen t 1 to put the system in a state whic h satisfie s ok at the next leve l is to take a ctio n ( q , q ′ , R ) , whic h corresp onds to the transition δ ( q , a 2 ) = ( q ′ , a ′ 2 , R ) in the T uring mac hine. Due to identic observ abilit y for agent 1 , the same a ction has to b e play ed b y agen t 1 in t he history whic h ends in state s ′ tr whic h is next to the rig h t of stat e s q ,a 2 . The effect of this action in state s ′ tr (com bined with an idle action for agen t 2) is to bring the system in state s q ,q ′ ,R . In this state, it’s upto agen t 2 t o try to satisfy ok a t the next step, and he can only do t his b y applying t he a ction ( q , q ′ , R ) . The effect of this action in state s q ,q ′ ,R is t o bring the system bac k in stat e s ′ tr . But the same action has to b e pla yed by ag ent 2 in the history whic h ends in state s a 3 on lev el 3 of the tree, due to identical observ abilit y . This play will lead the system t o state s q ′ ,a 3 . On the other hand, in state s a 1 , in order to ensure ok , b oth agen ts m ust play idle, whic h lea ve s the sy stem in state s a 1 . Iden tical observ abilit y will then ensure that agent 1 has to pla y idle also in state s ′ tr whic h is next to the righ t of state s a 1 , and a gen t 2 has to play idle in state s ′ lb on 3rd and 4t h leve ls. 7 The effect of all these is that lev el 4 on this tree enco des the configuration a 1 q ′ a ′ 2 a 3 , whic h results from applying the transition δ ( q , a 2 ) = ( q ′ , a ′ 2 , R ) to the configuration a 1 q a 2 a 3 . States s g en and s tr are use d for “creating” all the no des that sim ulate tap e cells. In a computation tree whic h satisfies t he goa l ◻ ok , these are the only states to ha v e tw o sons. Figure 4 presen ts the sim ulation of the computation step a 1 q a 2 a 3 ⇒ M q ′ a 1 a ′ 2 a 3 Note here that the rˆ ole of ag ents 1 and 2 are inte rc hanged because it is a left transition. s init s ′ lb s ′ lb s ′ lb ( i, i, i ) ( i, i, i ) s a 1 s a 1 s q ′ ,a 1 (( q , q ′ , L ) , i, i ) ( i, i, i ) s ′ tr s q ,q ′ ,L s ′ tr (( q , q ′ , L ) , i, i ) ( i, ( q , q ′ , L ) , i ) s q ,a 2 s a ′ 2 s a ′ 2 ( i, i, i ) ( i, ( q , q ′ , L ) , i ) s ′ tr s ′ tr s ′ tr ( i, i, i ) ( i, i, i ) s a 3 s a 3 s a 3 ( i, i, i ) ( i, i, i ) Figure 4: Simulation o f a 1 q a 2 a 3 ⇒ M q ′ a 1 a ′ 2 a 3 And in Figure 5, a sim ulation of the computation q 0 B ⇒ M aq 1 B ⇒ M q 2 ab is sho wn. Construction of a game structur e asso ciate d to M . The concurren t game structure under imp erfect information G = ( Ag , S, Π , π , Act, ( ∼ i ∣ i ∈ Ag ) , d, → ) t ha t sim ulates the determin- istic T ur ing machine M is based on three agen ts, i.e. Ag = { 1 , 2 , 3 } . Its s et S of states, together with their meaning, consists of: • s init (the initial state); • s ′ init (cop y of s init ); • s lb (sp ecifies the left b order of M ’s tap e); • s ′ lb (cop y of s lb ); • s g en (initiates the generation of a new blank cell of M ’s t a p e); • s tr (initiates the generation of a new cell separator); • s ′ tr (used for t r a nsferring information b etw een to equiv alen t runs); • s a , for an y a ∈ Σ (specifies tha t some tap e cell holds a ); • s q ,a , for any state q ∈ Q and a ∈ Σ (sp ecifies that M is in state q and the read/write head p oin ts a cell holding sym b ol a ); 8 s init s ′ init s lb s ′ lb s ′ lb s ′ lb s ′ lb s ′ lb ( i, i, i ) ( i, i, i ) ( i, i, i ) ( i, i, i ) ( i, ( q 0 ) , i ) ( i, i, i ) ( i, i, br 1 ) s g en ∣ p 1 s B s q 0 ,B s a s a s a s q 2 ,a (( q 1 , q 2 , L ) , i, i ) ( i, i, i ) ( i, i, i ) (( q 0 , q 1 , R ) , i, i ) ( i, ( q 0 ) , i ) ( i, i, br 1 ) s tr ∣ p 2 s ′ tr s q 0 ,q 1 ,R s ′ tr s q 1 ,q 2 ,L s ′ tr (( q 1 , q 2 , L ) , i, i ) ( i, ( q 1 , q 2 , L ) , i ) ( i, ( q 0 , q 1 , R ) , i ) (( q 0 , q 1 , R ) , i, i ) ( i, i, br 1 ) s g en ∣ p 1 s B s q 1 ,B s b s b ( i, i, i ) ( i, ( q 1 , q 2 , L ) , i ) ( i, ( q 0 , q 1 , R ) , i ) ( i, i, br 1 ) ( i, i, br 2 ) ( i, i, br 2 ) ( i, i, br 2 ) Figure 5: Simulation of the co mputation q 0 B ⇒ M aq 1 B ⇒ M q 2 ab . • s q ,q ′ ,X , for any q , q ′ ∈ Q and X ∈ { L, R } suc h tha t δ ( q , a ) = ( q ′ , a ′ , X ) for some a and a ′ (sp ecifies that the mac hine M enters state q ′ from state q by an X -mov e); • s er r (“error” state used to collect all “un w an ted” transitions the agen ts mus t a v oid bringing the system in this state). The set of atomic prop ositions is Π = { p 1 , p 2 , ok } and the lab eling function π is: π ( s ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ { ok } , if s ∈ S − { s g en , s tr , s er r } { p 1 , ok } , if s = s g en { p 2 , ok } , if s = s tr ∅ , if s = s er r F or the sak e of simplicit y , all states but s er r will b e called ok - s tates (b eing lab eled b y ok ). The relation ∼ 3 is the iden tit y . The equiv alenc e relations ∼ 1 and ∼ 2 are defined b y s ∼ i s ′ iff ( p i ∈ π ( s ) ⇔ p i ∈ π ( s ′ )) , 9 for an y i = 1 , 2. That is, s and s ′ are ∼ i -equiv alen t if the agen t i observ es p i either in b oth states s and s ′ or in none of them. The set Act of a ctio ns consists o f: • id le , whic h is meant to say that the agen t doing it is no t “in charge of ” accomplishing some lo cal ob jectiv e (this action will b e abbreviated by i in our pictures and whenev er no confusion ma y arise); • ( q 0 ) , whic h is an action mean t to set up the initial state of M ; • ( q , q ′ , X ) , for any q , q ′ ∈ Q and X ∈ { L, R } with δ ( q , a ) = ( q ′ , a ′ , X ) for some a, a ′ ∈ Σ. Suc h a n action sim ula tes the passing of M from q t o q ′ b y a n X -mov e; • br 1 and br 2 , whic h are t w o “branching” actions. The ag en ts 1 and 2 are allo w ed to p erfo rm an y action but br 1 and br 2 , while the third agen t can only p erform br 1 , br 2 , and idl e . More precisely , d ( i, s ) = Act − { br 1 , br 2 } for an y i ∈ { 1 , 2 } and state s , d ( 3 , s ) = { br 1 , br 2 } if s ∈ { s init , s g en , s tr } , and d ( 3 , s ) = idl e , o therwise. Note that the agen ts’ actions are designed suc h that d ( i, s ) = d ( i, s ′ ) f o r an y ag ent i and states s and s ′ with s ∼ i s ′ . The transition relation of the game structure is a s follows: • s init ( i,i,br 1 ) Ð Ð Ð Ð → s ′ init and s init ( i,i,br 2 ) Ð Ð Ð Ð → s g en and s init c Ð → s er r , for an y c different from the ab ov e action tuples; • s ′ init ( i,i,i ) Ð Ð Ð → s lb and s ′ init c Ð → s er r , for a ny c / = ( i, i, i ) ; • s lb ( i, ( q 0 ) ,i ) Ð Ð Ð Ð → s ′ lb and s lb c Ð → s er r , for an y c / = ( i, ( q 0 ) , i ) ; • s ′ lb ( i,i,i ) Ð Ð Ð → s ′ lb and s ′ lb c Ð → s er r , for an y c / = ( i, i, i ) ; • s g en ( i,i,br 1 ) Ð Ð Ð Ð → s B and s g en ( i,i,br 2 ) Ð Ð Ð Ð → s tr and s g en c Ð → s er r , for any c differen t from the ab ov e action tuples; • s tr ( i,i,br 1 ) Ð Ð Ð Ð → s ′ tr and s tr ( i,i,br 2 ) Ð Ð Ð Ð → s g en and s tr c Ð → s er r , for an y c differen t from the ab o v e action tuples; • for any a ∈ Σ, t he transitions at s a are: – s a ( i,i,i ) Ð Ð Ð → s a ; – s B ( i, ( q 0 ) ,i ) Ð Ð Ð Ð → s q 0 ,B ; – s a ( i, ( q, q ′ ,R ) ,i ) Ð Ð Ð Ð Ð Ð → s q ,a , for an y action ( q , q ′ , R ) ; – s a (( q,q ′ ,L ) ,i,i ) Ð Ð Ð Ð Ð Ð → s q ,a , for a ny action ( q , q ′ , L ) ; 10 – s a c Ð → s er r , for a ny c different from a ny of the a b o v e actions; • for any q ∈ Q and a ∈ Σ, the transitions at s q ,a are: – s q ,a (( q,q ′ ,R ) ,i,i ) Ð Ð Ð Ð Ð Ð → s a ′ , if δ ( q , a ) = ( q ′ , a ′ , R ) ; – s q ,a (( q,q ′ ,L ) ,i,i ) Ð Ð Ð Ð Ð Ð → s a ′ , if δ ( q , a ) = ( q ′ , a ′ , L ) ; – s q ,a c Ð → s er r , for an y c differen t f r o m an y of the ab ov e actions; • the transitions at s ′ tr are: – s ′ tr ( i,i,i ) Ð Ð Ð → s ′ tr . – s ′ tr (( q,q ′ ,R ) ,i,i ) Ð Ð Ð Ð Ð Ð → s q ,q ′ ,R , for an y action ( q , q ′ , R ) ; – s ′ tr ( i, ( q, q ′ ,L ) ,i ) Ð Ð Ð Ð Ð Ð → s q ,q ′ ,L , for an y action ( q , q ′ , L ) ; – s ′ tr c Ð → s er r , for a ny c different from a ny of the ab ov e actions; • s q ,q ′ ,R ( i, ( q, q ′ ,R ) ,i ) Ð Ð Ð Ð Ð Ð → s ′ tr and s q ,q ′ ,L (( q,q ′ ,L ) ,i,i ) Ð Ð Ð Ð Ð Ð → s ′ tr and s q ,q ′ ,X c Ð → s er r , for any X and any c differen t f rom an y of the ab ov e actions. Pr o of of the c orr e ctness of the c onstruction. Let M b e a deterministic T uring mac hine. Without loss of generalit y we may assume that M , starting in state q 0 , will nev er reac h again q 0 . First, we pro ve tha t if M do es not ha lt on the empty word then ( G , s init ) ⊧ iR ⟪{ 1 , 2 }⟫ ◻ ok . According to Remark 1, it suffices to sho w that, if M do es not halt on the empt y w ord, then there exists a strategy σ = ( σ 1 , σ 2 ) for the agen ts 1 and 2 in G suc h t ha t an y s init -ro ot ed computatio n tree of G under σ has only no des lab eled b y ok - states. In order to define σ w ith the prop ert y ab ov e, w e classify the non-empty sequences of states of G as follows: • a sequence α ∈ S + is of typ e 1 if α = s init s ′ init α ′ , where α ′ ∈ S ∗ ; • a sequenc e α ∈ S + is of typ e 2 if α = s init s g en α ′ , where α ′ ∈ S ∗ . T yp e 2 sequences of states can b e f urt her classified according to the num b er of states s g en and s tr they con tain: – a seque nce α is o f typ e 2 ( i )( i − 1 ) , where i ≥ 1, if α = s init ( s g en s tr ) i − 1 s g en α ′ , where α ′ ∈ S ∗ do es not con tain s g en and s tr ; – a sequence α is of typ e 2 ( i )( i ) , where i ≥ 1, if α = s init ( s g en s tr ) i α ′ , where α ′ ∈ S ∗ do es not con ta in s g en and s tr . Of course, there are sequences α ∈ S + whic h are neither of t yp e 1 nor of t yp e 2. A path τ of a computation t r ee o f G will b e called of typ e x if l 1 ( τ ) is of type x , where x is as ab ov e. The follo wing claim follo ws easily from definitions. 11 Claim 1. L et α and α ′ b e two non-empty se quenc es of states. Then, the fol lowing pr op er- ties h old: 1. If α is of ty p e 1 and α ′ is of typ e 2, then α / ∼ 1 α ′ ; 2. If α is of ty p e 1 and α ′ is of typ e 2 and α ∼ 2 α ′ , then α ′ is of typ e 2 ( 1 )( 0 ) ; 3. If α and α ′ ar e of typ e 2, h a v e a di ff er ent numb er of s g en or s tr states, and α ∼ 1 α ′ , then α is of typ e 2 ( i )( i − 1 ) and α ′ is of typ e 2 ( i ) ( i ) , or vic e-versa; 4. If α and α ′ ar e of typ e 2, h a v e a di ff er ent numb er of s g en or s tr states, and α ∼ 2 α ′ , then α is of typ e 2 ( i )( i ) a nd α ′ is of typ e 2 ( i + 1 )( i ) , or vic e-versa. No w, define a strategy σ = ( σ 1 , σ 2 ) as follo ws: • σ 1 ( s init ) = σ 1 ( α ) = idl e , for any ty p e 1 sequence α ∈ S + ; • σ 2 ( s init ) = σ 2 ( α ) = idl e , for an y type 1 sequence α ∈ S + differen t from s init s ′ init s lb , and σ 2 ( s init s ′ init s lb ) = ( q 0 ) ; • σ 1 ( αs q ,a ) = ( q , q ′ , R ) = σ 1 ( α ′ s ′ tr ) , for an y αs q ,a of ty p e 2 ( i )( i − 1 ) and an y α ′ s ′ tr of t yp e 2 ( i )( i ) for whic h i ≥ 1 a nd the fo llo wing prop ert y holds: – ∣ αs q ,a ∣ = 3 + ( 2 j − 1 ) = ∣ α ′ s ′ tr ∣ for some j ≥ 1, and the agen t 1 sim ulating the first j steps of M deduces that the curren t configuratio n of M is of t he form uq av , where ∣ u ∣ = i − 1, and δ ( q , a ) = ( q ′ , a ′ , R ) , for some q ′ and a ′ ; • σ 1 ( αs a ) = ( q , q ′ , L ) = σ 1 ( α ′ s q ,q ′ ,L ) , for any αs a of t yp e 2 ( i )( i − 1 ) a nd any α ′ s q ,q ′ ,L of t yp e 2 ( i )( i ) for whic h i ≥ 1 a nd the fo llo wing prop ert y holds: – ∣ αs a ∣ = 3 + 2 j = ∣ α ′ s q ,q ′ ,L ∣ for some j ≥ 1, and the agen t 1 sim ulating the first j steps of M deduces that the curren t configuratio n of M is of the form uaq bv , where ∣ u ∣ = i − 1, and δ ( q , b ) = ( q ′ , b ′ , L ) , for some q ′ and b ′ ; • σ 2 ( αs q ,q ′ ,R ) = ( q , q ′ , R ) = σ 2 ( α ′ s a ) , for an y αs q ,q ′ ,R of type 2 ( i )( i ) a nd an y α ′ s a of t yp e 2 ( i + 1 )( i ) for whic h i ≥ 1 a nd the following prop ert y holds: – ∣ αs q ,q ′ ,R ∣ = 3 + 2 j = ∣ α ′ s a ∣ for some j ≥ 1, and the agent 2 simulating the first j steps of M deduces that the current configuration of M is of the form uq av , where ∣ u ∣ = i − 1, and δ ( q , a ) = ( q ′ , a ′ , R ) , for some q ′ and a ′ ; • σ 2 ( αs ′ tr ) = ( q , q ′ , L ) = σ 2 ( α ′ s q ,a ) , f o r an y αs ′ tr of t yp e 2 ( i )( i ) and an y α ′ s q ,a of t yp e 2 ( i + 1 )( i ) for whic h i ≥ 1 and the follo wing prop erty holds: – ∣ αs ′ tr ∣ = 3 + ( 2 j − 1 ) = ∣ α ′ s q ,a ∣ for some j ≥ 1, and the agen t 2 sim ulating the first j steps of M deduces tha t the curren t configuration of M is of the form u aq bv , where ∣ u ∣ = i − 1, and δ ( q , b ) = ( q ′ , b ′ , L ) , for some q ′ and b ′ ; • σ 2 ( s init s g en s B ) = ( q 0 ) ; 12 • σ 1 ( α ) = idl e and σ 2 ( α ′ ) = idl e for all the other cases. The strategies σ 1 and σ 2 are bo th compatible with d , σ 1 is compatible with ∼ 1 , and σ 2 is compatible with ∼ 2 . An y tree with exactly one no de (its ro ot) lab eled b y s init is an s init -ro ot ed computation tree of G under σ and its no des are all lab eled b y ok -states. Assume that T is an s init -ro ot ed computation tree o f G under σ and all its no des are lab eled b y ok - stat es. It is easy to see that T ma y only hav e t yp e 1, t yp e 2 ( i )( i − 1 ) , or t yp e 2 ( i )( i ) paths, fo r some i ≥ 1. An y extension T ′ of T (i.e., T ⇒ G , σ T ′ ) adds new no des to T whic h cannot b e lab eled by s er r b ecause M do es not halt (see the definition of σ ). Therefore, an y s init -ro ot ed computation t r ee of G under σ has all its no des lab eled by ok -states. Con v ersely , w e sho w that M do es not halt on the empt y w ord if all s init -ro ot ed com- putation trees o f G under some strat egy σ f or { 1 , 2 } hav e only no des lab eled b y ok - states. Let σ b e a strategy with the prop erty ab o v e a nd consider an s init -ro ot ed computatio n tree T = ( V , E , v 0 , l 1 , l 2 ) under σ . A no de v of T will b e called of typ e x if l 1 ( path T ( v 0 , v )) is of t yp e x ( x is 1, 2, 2 ( i ) ( i − 1 ) , or 2 ( i )( i ) , for some i ≥ 1). W e then define a pa rtial o rdering ≺ T on the no des of T as the least partial ordering with the follo wing prop erties: • if v and v ′ are no des on the same lev el of T and l 1 ( v ′ ) ∈ { s g en , s tr } , then v ≺ T v ′ ; • if v and v ′ are no des on the same leve l of T and there exist u on the path from ro ot to v and u ′ on the pa th from ro ot to v ′ with u ≺ T u ′ , then v ≺ T v ′ Some prop erties of T a nd its lev el sets are listed in the seque l. Claim 2. L et T = ( V , E , v 0 , l 1 , l 2 ) b e an s init -r o ote d c omputation tr e e of G unde r σ , and n ≥ 1 . Then: 1. l ev el T ( n ) has at most n + 1 no des, and e ach of them is either of typ e 1 , or o f ty p e 2, or of typ e 2 ( i )( i − 1 ) , or of typ e 2 ( i )( i ) , for some i ≥ 1 ; 2. l ev el T ( n ) c ontains a t most one no de of typ e 1; 3. l ev el T ( n ) c on tain s at most one no de of typ e 2 ( i )( i − 1 ) and at most one no de of typ e 2 ( i )( i ) , for e ach i ≤ ⌈ n / 2 ⌉ ; 4. for any v , v ′ ∈ l ev el T ( n ) , v ≺ T v ′ if and only if one of the fol lo w ing pr op erties hold: (a) v = v ′ ; (b) v is of typ e 1 ; (c) v is of typ e 2 ( i ) ( i ′ ) , v ′ is of typ e 2 ( j )( j ′ ) , and i < j o r, if i = j then i ′ < j ′ . 5. ≺ T is a total or dering on l ev el T ( n ) . Proo f. All the prop erties in Claim 2 can b e pro v ed b y induction on n ≥ 1 and mak e use of the fact that all no des of T are lab eled b y ok -states. Th us, if v is a no de on the leve l n of T and it is not lab el b y s g en or s tr , then it ma y hav e at most one descendan t v ′ on 13 the lev el n + 1 (b y σ , eac h of the agents 1 and 2 has exactly one c hoice at l 1 ( v ) , and b y d 3 , the a g en t 3 has exactly one c hoice a s well at l 1 ( v ) ). Moreov er, v ′ and v hav e the same t yp e. If v is lab eled by s g en , then its t yp e is 2 ( i )( i − 1 ) for some i ≥ 1, and it ma y ha v e a t most t wo descendan ts v ′ and v ′′ on the lev el n + 1 (by σ , eac h of the agen ts 1 and 2 ha s exactly one c hoice at l 1 ( v ) , but the ag ent 3 has tw o c hoices). O ne of this descendan ts is of type 2 ( i )( i − 1 ) , while the other is o f type 2 ( i )( i ) and it is lab eled b y s tr . Similarly , if v is lab eled by s tr , then its t yp e is 2 ( i )( i ) for some i ≥ 1, a nd it may ha v e at most tw o descendan ts v ′ and v ′′ on the lev el n + 1. One of this descendan ts is of t yp e 2 ( i )( i ) , while the other is of type 2 ( i + 1 )( i ) and it is lab eled b y s g en . Com bining these remarks with the fact that l ev el T ( 1 ) may con t a in at most tw o no des, one of them la b eled b y s ′ init (whic h is of type 1) and the other b y s g en , w e obtain (1), (2), and (3) in the Claim. (4) f ollo ws from the definition of ≺ T and the ab ov e prop erties, and (5 ) follows from (4). ◻ If lev el T ( n ) = { v 1 , . . . , v n + 1 } of an s init -ro ot ed computation tree T of G under σ has exactly n + 1 no des, then w e sa y that it is c omple te . Moreo ve r, if w e assume that v 1 ≺ T ⋯ ≺ T v n + 1 , then w e may view l ev el T ( n ) a s a sequence of no des, v 1 ⋯ v n + 1 . Claim 3. L et T = ( V , E , v 0 , l 1 , l 2 ) b e an s init -r o ote d c omputation tr e e of G unde r σ , and n ≥ 1 such that l ev el T ( n ) is c omplete and its se quenc e of no des is v 1 ⋯ v n + 1 . Then, the fol lowing p r op erties hold : 1. l ev el T ( m ) is c omplete, for any m ≤ n ; 2. v 1 is of typ e 1 , v 2 i is of typ e 2 ( i )( i − 1 ) , and v 2 i + 1 is of typ e 2 ( i )( i ) , for al l i ≥ 1 with 2 i ≤ n ; 3. (a) l 1 ( path T ( v 0 , v 1 )) ∼ 2 l 1 ( path T ( v 0 , v 2 )) ; (b) l 1 ( path T ( v 0 , v 2 i )) ∼ 1 l 1 ( path T ( v 0 , v 2 i + 1 )) , for al l i ≥ 1 with 2 i ≤ n ; (c) l 1 ( path T ( v 0 , v 2 i + 1 )) ∼ 2 l 1 ( path T ( v 0 , v 2 ( i + 1 ) )) , f o r al l i ≥ 1 with 2 i + 1 ≤ n ; 4. l 1 ( v 1 ⋯ v n + 1 ) is of the one of the fo l lowing form s: (a) s ′ init s g en , if n = 1 ; (b) s lb s B s tr , if n = 2 ; (c) s ′ lb s a 1 s ′ tr ⋯ s a j − 1 s ′ tr s q ,a j s tr ′ s a j + 1 ⋯ s ′ tr s a m s ′ tr s g en , if n > 2 is o dd, wher e a 1 , . . . , a m ∈ Σ , q ∈ Q , m = ( n − 1 )/ 2 , and 1 ≤ j ≤ m (for j = 1 , s a 1 b e c omes s q ,a 1 , a nd for j = m , a m b e c omes s q ,a m ); (d) s ′ lb s a 1 s ′ tr ⋯ s a j − 1 s ′ tr s a j s q ,q ′ ,X s a j + 1 ⋯ s ′ tr s a m − 1 s ′ tr s B s tr , if n > 2 is even, wher e a 1 , . . . , a m − 1 ∈ Σ , q , q ′ ∈ Q , X ∈ { L, R } , m = n / 2 , and 1 ≤ j ≤ m − 1 ; 5. ther e exists an s init -r o ote d c omputation tr e e T ′ of G under σ such that T ∗ ⇒ G , σ T ′ and lev el T ′ ( n + 1 ) is c omplete. Mor e over, if the se quenc e of no des of l ev el T ( n ) has the form (4a) ((4b), (4c) , (4d)), then lev el T ′ ( n + 1 ) has the form (4b) ((4c), (4d), (4c), r esp e ctively). 14 Proo f. (1), (2), and (3) can b e prov ed in a similar w ay to the statemen ts in Claim 2. W e pro v e (4) and (5) together. It is easy to show tha t l 1 ( v 1 ⋯ v n + 1 ) has the fo rm (4a) if n = 1. As l 1 ( path T ( v 0 , v 1 )) ∼ 2 l 1 ( path T ( v 0 , v 2 )) and T has only ok -states, the strategy σ 2 should select only idl e as the only choice for agent 2 a t l 1 ( v 1 ) a nd l 1 ( v 2 ) . σ 1 should select idl e for agent 1 at l 1 ( v 1 ) and l 1 ( v 2 ) , while the agen t 3 has t he only c hoice idle at l 1 ( v 1 ) and t wo c hoices, br 1 and br 2 , at l 1 ( v 2 ) . Therefore, w e can extend T b y adding a new descendan t v ′ 1 of v 1 and t wo new descendan ts v ′ 2 and v ′′ 2 of v 2 , b y the rules l 1 ( v 1 ) ( i,i,i ) Ð Ð Ð → l 1 ( v ′ 1 ) = s lb , l 1 ( v 2 ) ( i,i,br 1 ) Ð Ð Ð Ð → l 1 ( v ′ 2 ) = s B , l 1 ( v 2 ) ( i,i,br 2 ) Ð Ð Ð Ð → l 1 ( v ′′ 2 ) = s tr . W e obtain a new s init -ro ot ed computation tree T ′ of G under σ whose lev el 2 satisfies (4) and (5). Assume n = 2 and l 1 ( v 1 , v 2 , v 3 ) = s lb s B s tr . As l 1 ( path T ( v 0 , v 1 )) ∼ 2 l 1 ( path T ( v 0 , v 2 )) and T has only ok -states, the strategy σ 2 should select only ( q 0 ) as the o nly c hoice for agen t 2 at l 1 ( v 1 ) and l 1 ( v 2 ) . The agents 1 has the only c hoice idle at l 1 ( v 1 ) and l 2 ( v 2 ) (by σ 1 ), and the a gen t 3 has the same c hoice at these states (b y d 3 ). Therefore, w e can add a new descendan t v ′ 1 of v 1 and a new desce ndan t v ′ 2 of v 2 b y the rules l 1 ( v 1 ) = s lb ( i, ( q 0 ) ,i ) Ð Ð Ð Ð → l 1 ( v ′ 1 ) = s ′ lb and l 1 ( v 2 ) = s B ( i, ( q 0 ) ,i ) Ð Ð Ð Ð → l 1 ( v ′ 2 ) = s q 0 ,B . There are t wo choices at l 1 ( v 3 ) , namely ( i, i, br 1 ) a nd ( i, i, br 2 ) , allowing to add t w o de- scendan ts v ′ 3 and v ′′ 3 of v 3 on the next lev el. Moreov er, l 1 ( v ′ 3 ) = s ′ tr and l 1 ( v ′′ 3 ) = s g en . As a conclusion, T can b e extende d to a new tree T ′ whose sequence of no des on lev el 3 are v ′ 1 v ′ 2 v ′ 3 v ′′ 3 and l 1 ( v ′ 1 v ′ 2 v ′ 3 v ′′ 3 ) = s ′ lb s q 0 ,B s ′ tr s g en whic h is the form (4c). Moreov er, (5) holds to o. Assume n > 2 odd, l 1 ( v 1 ⋯ v n + 1 ) of the form (4c), and j > 1 ( t he case j = 1 can b e discusse d in a similar w a y). W e hav e that l 1 ( v 2 j ) = s q ,a j and l 1 ( v 2 j − 1 ) = l 1 ( v 2 j + 1 ) = s ′ tr . Due to the fact that l 1 ( path T ( v 0 , v 2 j )) ∼ 1 l 1 ( path T ( v 0 , v 2 j + 1 )) and T has only ok -states, σ 1 should select an action o f the form ( q , q ′ , R ) or ( q , q ′ , L ) f or agent 1 as a c hoice at l 1 ( v 2 j ) a nd l 1 ( v 2 j + 1 ) ( q ′ ∈ Q and this choice is obtained from t he transition function of M ). Assume that this choice is ( q , q ′ , R ) a nd δ ( q , a j ) = ( q ′ , a ′ j , R ) ( t he other case is similar to this). Each of the agen ts 2 and 3 has exactly one choice at l 1 ( v 2 j ) and l 1 ( v 2 j + 1 ) , namely idl e . Therefore, T can b e extended by adding t w o new descendan ts v ′ 2 j and v ′ 2 j + 1 b y the rules l 1 ( v 2 j ) = s q ,a j (( q,q ′ ,R ) ,i,i ) Ð Ð Ð Ð Ð Ð → l 1 ( v ′ 2 j ) = s a ′ j and l 1 ( v 2 j + 1 ) = s ′ tr (( q,q ′ ,R ) ,i,i ) Ð Ð Ð Ð Ð Ð → l 1 ( v ′ 2 j + 1 ) = s q ,q ′ ,R . F or the no des v i with i / ∈ { 2 j, 2 j + 1 , n + 1 } , there is exactly one choice for each agen t, namely idl e , and therefore, a new desc endan t v ′ i of v i can b e added b y the rule l 1 ( v i ) ( i,i,i ) Ð Ð Ð → l 1 ( v ′ i ) = l 1 ( v i ) . F or the no de v n + 1 w e ma y reason as in the case n = 2 ab ov e. Tw o descendan ts v ′ n + 1 and v ′′ n + 1 can b e added, with l 1 ( v ′ n + 1 ) = s B and l 1 ( v ′′ n + 1 ) = s tr . In this w a y , w e obtain a new tree T ′ whose lev el n + 1 satisfies (4) and (5). The case “ n > 2 ev en and l 1 ( v 1 ⋯ v n + 1 ) of the for m (4 d)” can b e treated analogously to the ab ov e o ne. ◻ 15 Consider further the homomorphism h ∶ S → ( Q ∪ Σ ) ∗ giv en by: h ( s ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ a, if s = s a q a, if s = s q ,a λ, otherwise W e shall write h ( lev el T ( n )) for h ( v 1 ⋯ v n + 1 ) , where v 1 ⋯ v n + 1 is the sequence of no des asso ciated to complete lev el l ev el T ( n ) of some s init -ro ot ed computation tree T of G under σ . Claim 4. L et T = ( V , E , v 0 , l 1 , l 2 ) b e an s init -r o ote d c omputation tr e e of G unde r σ , and n ≥ 3 o dd such that l ev el T ( n ) is c omplete. The n : 1. h ( l ev el T ( n )) ∈ Σ ∗ Q ΣΣ ∗ ; 2. ther e exists an s init -r o ote d c omputation tr e e T ′ of G under σ such that T ∗ ⇒ G , σ T ′ , lev el T ′ ( n + 2 ) is c omplete, and h ( l ev el T ( n )) ⇒ M h ( lev el T ′ ( n + 2 )) . Proo f. F rom the definition of h , Claim 3 , and by insp ecting the pro of of Claim 3 . ◻ It is straigh tforw a rd to see that there exists an s init -ro ot ed computation tr ee T o f G under σ whose l ev el T ( 3 ) is complete. Moreov er, by Claim 3, we ha ve h ( l ev el T ( 3 )) = q 0 B (that is, the initial configuration of M ). Then, com bining with Claim 4, we obtain t hat M do es not halt on the empty w ord if all s init -ro ot ed computation tr ees of G under some strategy σ for { 1 , 2 } ha v e only no des lab eled b y ok - states. Our discussion ab ov e leads to: Theorem 1. The mo del che cking pr oblem for AT L iR is unde cidable. 4. Conclusions The proo f abov e sho ws that the strategies used b y the agen ts 1 and 2 to sim ulate the deterministic T uring mac hine M a r e primitiv e recursiv e. Therefore, the crucial elemen ts whic h allo w t o simulate M are the equiv alence relations ∼ 1 and ∼ 2 . These equiv alence relations are “ in ter- r elat ed” and are used to transfer information from one computation path can be transferred to another computation pat h. A deep er a nalysis of the na t ur e o f the observ ational equiv alence relations asso ciated to agen ts in a C GS w o uld b e in teresting. 16 References References [1] R. Alur, Th. A. Henzinger, O. K upferman. Alternating-time T emp or al L o gic , Journal of the ACM 49, 2002, 672–713. Preliminary v ersion a pp eared in the Pro c. of t he 38th IEEE Symp o sium on F oundatio ns of Computer Science (FOCS ’97), 1 997, 100-109. [2] N. Bulling, J. Dix, W. Jamroga. Mo del Che ck i n g L o gics of S tr ate gic Ability: Com- plexity , in “ Sp ecification and V erification of Multi-Agen t Systems” (M. D astani, K. Hindriks, J.-J. Mey er, eds.), Springer-V erlag, 2 010 (to app ear). [3] P .-Y. Sc hobb ens. A lternating-time L o gic w ith Imp erfe ct R e c al l , Electronic Notes in theoretical Computer Science 85(2), 2004. 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