SAT Has No Wizards

Sat Has No Wiza rds Silv ano Di Zenzo Department of Computer Science, U niversit y of Rome Abstract An (enco d ed) decision pr oblem o ver Σ is a pair ( E , F ) wh ere E =w ords that encode instances of the pr ob lem, F =words to b e ac- cepted. W e use str ing s in a te c hnical sense, b orrow ed from Com- putabilit y . With an y NP problem ( E , F ) w e asso ciate a set of strings | Log E ( F ) | called the redu ced log ogram of F relativ e to E , w h ic h con- v eys structural inform ation on E , F , and how F is em b edded in E . W e define notions of in ternal ind ep endence of d ecision p roblems in terms of | Log E ( F ) | . The k ernel K er ( P ) of a program P that solv es ( E , F ) is t he set of those strings in | Log E ( F ) | that are actually u sed b y P in making decisions. Th ere are strict r elations b et ween K er ( P ) and the complexit y of P . W e d ev elop an application to S AT that relies u p on a prop ert y of strong in ternal indep endence of S AT . W e show that S AT cannot h a ve in its reduced logogram certain strings that, w h en p resen t, serv e as collect ive certificates. As a consequence, all the pr ograms that s olv e S AT ha v e the same ke rn el K er ( P ) = | Log C N F ( S AT ) | . 1 In tr o duction W e dev elop an application to S AT , to b e p ositioned in curren t stream of in terest in the structure o f Bo olean satisfiabilit y [1 ]. W e use str ing s in a tec hnical sense, b orrow ed from Computabilit y [2]. An encoded decision pr oblem ov er Σ is a pair ( E , F ) where E =words that enco de instances of the problem, F =w ords to b e accepted. On input x , a decision progra m P fo r ( E , F ) either accepts x ( if x is in F ) or r ej ects x (if x is in E − F ) o r else discar d s x (f or x outside E ). Our fundamen tal construct is a set o f strings Log E ( F ) called log og r am of F relative to E that conv eys structural information on E , F , and ho w F is em b edded in E . W e mostly use the reduced v ersion | Log E ( F ) | , consisting of those strings in Log E ( F ) that do not include other strings in Log E ( F ). The 1 k er nel K er ( P ) of a program P that solve s ( E , F ) is the set of those strings in | Log E ( F ) | that are actually used by P in making decisions. There are strict relationships b et we en t he comp osition in terms of strings of the k ernel K er ( P ) of a program solving ( E , F ) and the complexit y of P . Our a pplicatio n to S AT use s a prop erty of in ternal indep endence of a decision problem that w e call “strong in ternal indep endence.” Think of a computation in whic h the result of an y computation step do es not c hange the results that are p ossible for subsequen t steps. In ternal indep endence is defined in terms of a relation of entang lement ⊒ E b et w een sets of strings relativ e to reference set E . Our main results are the following. W e show that ( C N F , S AT ) exhibits the strong in ternal indep endence prop erty: In tuitiv ely , no “en tanglemen t at distance” b etw een strings in | Log C N F ( S AT ) | is p ossible. Besides, w e sho w that problem ( C N F , S AT ) cannot hav e, in its reduced logogram, certain collectiv e certificates that we call w iz ar ds . As a consequenc e, the decision programs P that solv e ( C N F , S AT ) all hav e the same k ernel K er ( P ) = | Log C N F ( S AT ) | . 2 Certific ate s of Mem b ers h ip as String s W e first recall notions rega rding the certificates of membership in NP theory . As next step, w e illustrate p ossible use of strings to represen t certificates. W e conclude the section r eviewing basic a lg ebraic prop erties of strings. Let G ⊆ Σ ∗ × Σ ∗ so that G is a relation on w ords o v er Σ. Let D om ( G ) and C od ( G ) b e first a nd second pro jection o f G . A relation G which is b oth p olynomial-time decidable and p olynomially balanced is an NP relatio n. L is in NP if and only if there exists a n NP relation G suc h tha t L = D om ( G ). W e in terc hange problems with languages: ( E , F ) ∈ N P and F ∈ N P amoun t to the same. Let ( E , F ) b e a n NP problem. Then t here exists a sequence y 1 , y 2 ,.. of w o rds (o v er some appropria te alphab et) called sol utions or else cer tif icates of member ship for problem ( E , F ). F or an y problem instance x ∈ E we ha v e that x can p ossibly b e satisfied b y some of t he y i s . W e also hav e “unsatisfiable” instances. What “ satisfaction” means op erat io nally is prop er of pro blem ( E , F ). Cardinalit y function α ( n ) of an NP problem: W e may arra nge notatio ns so that all solutions that can p ossibly satisfy an x o f size n are b etw een y 1 and y α ( n ) . Asso ciated with solutions y 1 , y 2 ,.. there is a decomp osition of target set F in to subsets F i called r eg ions , where F i is the set of tho se x ′ s tha t a re 2 satisfied b y y i . Regions satisfy the ob vious relation F = ∪ i F i . 2.1 Generalized Certificates In this pap er w e replace certificates with g ener al iz ed cer tif icates . These are represen ted b y str ing s , defined to b e functions N → Σ with finite domain ( N =p ositiv e in tegers). In lo ose words, a string g being included (or sub- sumed) in a w ord x is that whic h remains b y canceling zero or more letters in x , while lea ving blanks in places of letters. Note that w ords are certain sp ecial strings, thus the solutions y 1 , y 2 ,.. contin ue to b e certificates. This generalization a llows us to in tro duce certain more general certificates that w e call w iz ar ds . W e assume t ha t satisfiabilit y , b eing a prop ert y exhibited by certain w ords, is accompanied b y characteristic sig ns , that w e think as distinctiv e marks, or signatures, b eing someho w inscrib ed within the w o rd x under study . A detailed discussion w ould yield strings as prop er f ormalization of suc h notions as “mark” or “signature.” Th us, we assume that signs are strings inte rsp ersed in x . Since strings represen t w ords in short ha nd, w e call their set a l og og r am . 2.2 Strings W e define Σ ∞ to b e the set of all strings ov er Σ. Lo ok at g ∈ Σ ∞ as a prescription that a w ord x ov er Σ ma y or ma y not satisfy . If D om ( g ) is an initial segmen t of N then g is an ordinary w ord: Th us, w ords are certain sp ecial strings. The length (or size) | g | is the greatest n um b er in D om ( g ). Σ ∞ is partially ordered. Give n f , g ∈ Σ ∞ , g is an extension of f , written f ≤ g , a s so on as D om ( f ) ⊆ D om ( g ) and g tak es same v alues as f in D om ( f ). If f ≤ g and g ≤ f then f = g . If f ≤ g but not g ≤ f , write f < g a nd sa y g is a prop er extension of f , or else f is a prop er r estr iction of g . The empt y partia l function N → Σ , noted ⊥ , is the v o id string, and D om ( ⊥ ) = ∅ . An y f in Σ ∞ is an extension o f ⊥ , thus (Σ ∞ , ≤ ) has a least elemen t ⊥ . Tw o strings f and g are compatible as so on as f ( x ) = g ( x ) for any x in D om ( f ) ∩ D om ( g ). If f , g are disjoint, whic h is to sa y D om ( f ) ∩ D om ( g ) = ∅ , then f and g are certainly compatible. The meet f ∧ g of an y pair f , g is the restriction of f (or g ) to that p ortion of the inters ection D om ( f ) ∩ D om ( g ) where f and g agree. The j oin of tw o compatible strings f , g , noted f + g , is the least string whic h is an extension of b oth f and g . Thus f , g ≤ f + g and D om ( f + g ) = D om ( f ) ∪ D om ( g ). Equipp ed with meet and jo in, Σ ∞ is an up w ard directed complete meet-semilattice [3]. 3 3 En t ang lemen t among Strings The cylinders defined b elow ar e as in Computabilit y (t he formalism is sligh tly differen t). The logogra m is a new comer in Computer Science. En ta nglemen t is a key concept to deal with in ternal structure of computat io nal problems. 3.1 Cylinders Giv en H ⊆ Σ ∞ w e define E xp ( H ) = { x ∈ Σ ∗ : ∃ a ∈ H ( x ≥ a ) } (1) Th us, E xp ( H )=set of all w ords that include strings fro m H . Call E xp ( H ) absolu te expansion , equiv alently , absol ute cy linder asso ciated with H . Note that E xp ( H ) is the union of the elemen tary cylinders E xp ( g ) for g ∈ H . Giv en any recursiv e set of words E , w e write Σ ∞ ( E ) for the set o f all strings that happ en to b e included in words of E , thus Σ ∞ ( E ) = { g ∈ Σ ∞ : E xp ( g ) ∩ E 6 = ∅} (2) Σ ∞ ( E ) is t he set of those strings g in Σ ∞ whose asso ciated cylinder E xp ( g ) in tersects E . W e think o f E as the set of words o v er Σ that enco de instances of some fixed reference computational problem Π. (Whenev er w e talk o f a reference set E there is implicit reference to some fixed abstract decision problem Π as we ll a s to a program P solving Π.) F or H ⊆ Σ ∞ ( E ) w e write E xp E ( H ) = E H = { x ∈ E : ∃ a ∈ H ( x ≥ a ) } = E ∩ E xp ( H ) (3) Th us, E H is the set of t hose words in E t ha t contain strings from H . E H is the expansion of H relative to base E . W e actually regard E H as a relativized cylinder, equiv alen t ly , as b eing a cylinder relativ e to a reference set E . Note that for E = Σ ∗ w e regain the absolute expansion of set H . Giv en H ⊆ Σ ∞ ( E ), corresp ondence E xp E : H → E H exhibits prop erties: E H ∪ E K = E H ∪ K , E H ∩ E K = E H + K (4) Th us, unions and inters ections of sets that are cylinders relativ e to refer- ence set E are cylinders in E . Also no t e that, for any H , K ∈ Σ ∞ ( E ), H ⊆ K ⇒ E H ⊆ E K (5) E xp E ( E xp E ( H )) = E xp E ( E H ) = E H (6) 4 3.2 Logograms In this section we in tro duce the l og og r am of a set of words F relativ e to a reference set E . Given F ⊆ E , we define Log E ( F ) = { g ∈ Σ ∞ ( E ) : ∀ x ∈ E ( x ≥ g ⇒ x ∈ E F ) } (7) Since ordinary w ords are strings, F can b e regarded as a set of strings, hence E F is defined. E F is the relativ e cylindrification of F in E , and this in turn is the set of all w ords in E that are prefixed b y w ords in F . Remar k If F is a cylinder in E , i.e., F = E H for some H ⊆ Σ ∞ ( E ), then E F = F by Equation 6. Remar k F or E = Σ ∗ Equation 7 giv es absolu te l og og r am . The main prop ert y of corresp ondence Log E : E → Σ ∞ ( E ) is the f o llo wing. Giv en A, B ⊆ E , Log E ( A ∪ B ) ⊇ Log E ( A ) ∪ Log E ( B ) . (8) Let us understand this inclusion. Log E ( A ∪ B ) is the set of all strings that, for x in E , are able to trigger ev en t x ∈ E A ∪ B = E A ∪ E B . A string that triggers x ∈ E A certainly b elongs to Log E ( A ∪ B ). Analogously , a string that triggers x ∈ E B certainly b elongs t o Log E ( A ∪ B ). Th us, Log E ( A ) ∪ Log E ( B ) certainly is a subset o f Log E ( A ∪ B ). How ev er, there can b e strings f whose inclusion in a w ord x ∈ E is a sufficien t condition for ev ent x ∈ E A ∪ E B but not for x ∈ E A or x ∈ E B . Th us, in the general case Log E ( A ∪ B ) is not the same set a s Log E ( A ) ∪ Log E ( B ). 3.3 En tanglemen t The presence of certain strings in a w ord ma y entail that of certain others. Giv en H , K ⊆ Σ ∞ ( E ), we write K ⊒ E H if the follow ing happ ens: Ev ery w o rd in E whic h includes strings from K also includes strings from H . (Think of strings in K as spies, o r else symptoms, fo r presenc e in a n input string x of strings from H .) If H ⊒ E K and K ⊒ E H then w e write H ≡ E K and sa y t ha t H , K are isoexpansiv e relativ e to E . Clearly , ≡ E is a n equiv alence relation. It is easily seen that H ≡ E K if and only if E H = E K . F or E = Σ ∗ w e rewrite ⊒ E as ⊒ and ≡ E as ≡ . Note tha t f ⊒ E g if and only if ev ery word x (within E ) whic h includes f also includes g . W e men tio n a few easy facts. (i) If f ≤ g then f ⊒ E g for an y p ossible E . (ii) In the general case f ⊒ E g do es not imply f ≤ g . (It is we ll p ossible that this holds for sp ecific sets E . F or example, if E = Σ ∗ then f ⊒ E g if and 5 only if f ≤ g .) (iii) If f , g are incompatible, then it cannot b e that f ⊒ E g . (iv) Giv en an y H , K ⊆ Σ ∞ ( E ), H ⊆ K ⇒ H ⊒ K ⇒ H ⊒ E K (9) W e ask: Is there an y easy piece of algebra linking expansion, logogram, en t a nglemen t? T o get an answ er, w e define a Galois connection that will pro vide us with a closure op era t ion in Σ ∞ ( E ), not ed H → H αβ . W e will see t hat H and H αβ are iso expansiv e relativ e to E . What more, there can b e distinct subsets K , I ,.. of H b eing iso expansiv e ( mo d E ) to H αβ while p ossibly exhibiting differen t computationa l b eha viors. W e define our connection to b e a pair ( α, β ) of corresp ondences b et wee n sets of strings and sets of w ords. The first corresp ondence α carries a set o f strings H ⊆ Σ ∞ ( E ) into a corr esp o nding set of w o rds H α ⊆ E . The second carries a set of w ords A ⊆ E in to a set o f strings A β ⊆ Σ ∞ ( E ) according to H ⊑ E K ⇒ H α ⊇ K α (10) A ⊆ B ⇒ A β ⊒ E B β (11) H ⊑ E H αβ , A ⊆ A β α (12) The connection is formally defined through the explicit expressions: H α = E H (13) A β = Log E ( A ) (14) W e emphasize that A is any subset o f E . Th us, giv en an y subset A of the reference set E the function A β = Log E ( A ) is defined. Ho w ev er, not all subsets A of E happ en to b e the conjuga te set H α of some set H ⊆ Σ ∞ ( E ). If that happ ens, w e say that A is closed. Note that A closed implies E A = A . Theorem 1. ( α , β ) is a Galois c onne ction. Pr o of. W e m ust deriv e Equations 10-12 from Equations 1 3-14. (I) Let H , K ⊆ Σ ∞ ( E ) b e giv en, and assume H ⊑ E K . Let g ∈ K , and let x b e an y w ord in E suc h tha t x ≥ g . Then x ∈ E K hence x ∈ K α . Since H ⊑ E K , there exists f ∈ H suc h tha t x ≥ f . Then x is in E H hence x ∈ H α . Equation 10 is pro v ed. (I I) Next, w e prov e Equation 11. Let A, B ⊆ E and assume A ⊆ B . W e m ust pro v e that if a word x ∈ E includes a string g from A β then x also includes a string f from B β . Let g ∈ Log E ( A ) so that g ∈ A β b y Equation 14 . 6 Th us, for a ll x ∈ E w e ha v e x ≥ g ⇒ x ∈ E A . But A ⊆ B , hence x ∈ E A ⇒ x ∈ E B b y Equation 5. Th us, for all x ∈ E w e ha v e x ≥ g ⇒ x ∈ E B . By Equation 14 this is to sa y g ∈ Log E ( B ) = B β . W e hav e sho wn that A β ⊆ B β . Equ ation 11 follows b y virtue of Equation 9. (I I I) Next, w e pro v e the first of Equations 12. Let g ∈ H αβ and let x ∈ E b e an y string in E suc h that x ≥ g . W e ha v e H α = E H hence H αβ = Log E ( E H ). Th us, g ∈ Log E ( E H ). By Equation 7 w e ha v e x ∈ E xp E ( E H ), and then, b y virtue of Equation 6, x ∈ E H . W e conclude t ha t there exists f ∈ H suc h that x ≥ f . (IV) Next, w e prov e the second of Equations 12. It follo ws from Equations 13, 14 that A β α = E Log E ( A ) . On other hand one has A ⊆ E Log E ( A ) for a ny A ⊆ E . Indeed, A ⊆ E A and E A = E Log E ( A ) from definitions, taking in to accoun t Equation 6. W e prov ed the second of Equations 12. Theorem 2. The fol lowing e quations hold: H ⊆ H αβ , A ⊆ A β α (15) H α = H αβ α , A β = A β αβ (16) Pr o of. F rom theory of G alois connection [4]. Theorem 3. The map H → H αβ is a closur e op er ation in Σ ∞ ( E ) , and A → A β α is a closur e op er ation in E . Only for a closed A do w e ha v e tha t , for all x in E , x ∈ A if and only if there exists a g ∈ Log E ( A ) suc h that x ≥ g . If and only if A is a closed subset of the reference set E w e define the reduced k ernel | Log E ( A ) | . Regarding the reduced kerne l | Log E ( A ) | of a closed set A ⊆ E , w e explicitly note that, for an y x in E , x ∈ A if and only if there is g ∈ | Log E ( A ) | such that x ≥ g . 4 The Kernel of a Decisio n Program W e b egin with a few remarks on the nature of the strings that happ en to o ccur in the reduced logog r am | Log E ( F ) | . F or an y NP decision problem ( E , F ) w e a ssume F to b e a relative cylinder in E . (It is kno wn that S AT is a cylinder [5].) The strings in | Log E ( F ) | ar e certificates of mem b ership for F relativ e to E : F or w ords in E , to include one or more strings from | Log E ( F ) | is necessary and sufficien t for mem b ership in F . In principle, w e cannot exclude that | Log E ( F ) | may con tain strings that b eha v e as collectiv e 7 witnesses , also called wizards. (There exist problems, e.g. P R I M E S , where | Log E ( F ) | has wizards.) In that case a progra m P solving ( E , F ) mig ht do calculations that are functionally equiv a lent to testing input x for wizards. Let P solve problem ( E , F ). The computations that P p erfor ms are functionally equiv alen t to sequences of tests done on input x . This is part of Scott’s view of computatio ns [6] [7 ]. (The term “test” is o urs: Dana Scott uses “toke n” or else “piece of information” according to con t ext.) Note that Scott’s t heory is consis ten t with our dev elopmen t s as soon as we iden tify Scott’s to k ens with strings. In this view what P actually do es is searc hing the input x for strings in | Log E ( F ) | . That yields a view of computatio ns as sequence s of tests in d isg uise . Let program P solve problem ( E , F ). The t ests in | Log E ( F ) | are those that P can use: They are so to sp eak at disp osal for a pro g ram P . Whic h of these tests are actually used by P is a differen t story . W e define the k er nel of program P , noted K er ( P ), to b e the set of the strings from | Log E ( F ) | that P actually uses for ma king decisions. The strings in K er ( P ) are uniquely iden tified b y the algorithm that P implemen ts. The comp osition o f K er ( P ) in terms of strings can also b e determined throug h experimen ts with the executable of P . A concept of great relev ance fo r sequel is that of a complete subset of the reduced logo gram | Log E ( F ) | of decision problem ( E , F ): W e define a set H ⊆ | Log E ( F ) | to b e complete for problem ( E , F ) as so o n as, for any x ∈ E , one has x ∈ F ⇔ ∃ f ∈ H ( f ≤ x ). The pro ofs of following t w o theorems are not difficult and are omitt ed. Theorem 4. A ne c essary c o ndition for P to c orr e ctly solve ( E , F ) is K er ( P ) c omplete for ( E , F ) . Let H ⊆ | Log E ( F ) | b e a complete set of strings for ( E , F ). W e define H to b e ir r educible for ( E , F ) as so on as no pro p er subset K ⊂ H ha pp ens to b e complete for ( E , F ). Theorem 5. L et | Log E ( F ) | b e irr e ducible a nd pr o gr ams P , Q b oth solve ( E , F ) . Then K er ( P ) = K er ( Q ) . 5 Indep enden ce of Decision Pro b lems W e first in tr o duce a notion of pairwise indep endence of strings relativ e to a reference set E . As next step, w e define a notion of in ternal indep endence of set E . Nex t w e define not io ns of in ternal and strong in ternal indep endence of a decision pro blem. 8 Mutual I ndep endence of Strings Let f , g b e any tw o strings in Σ ∞ ( E ) where E is a n y infinite recursiv e set o f w ords ov er alphab et Σ. According to definitions, f entangles g relative to E a s so on as, for all x ∈ E , x ≥ f ⇒ x ≥ g . W e agreed that f ⊒ E g means t ha t f en tangles g relative to E . Observ e that f fa ils to entangle g relative to E if and only if there ex- ists x ∈ E suc h that x con tains f and do es not contain g . If f 6⊒ E g a nd g 6⊒ E f then f and g are said mutual l y independent relativ e to E ; f and g are mutu all y dependent relativ e to E when they fail to b e m utually indepen- den t relative to E . If f , g are incompatible, then certainly f , g are m utually indep enden t relativ e to any E . Indep endence of a Recursiv e Set Our next step is to define the in ternal indep endence of a recursiv e set E . W e define E to b e inter nall y independent as so on as, give n any f , g ∈ Σ ∞ ( E ) o ne ha s f ⊒ E g if and only if f is par t of g , that is to sa y , if and only if f ≤ g . Indep endence of a Decision P r oblem Now w e are ready to in tro duce the simple internal indep endence of a decision problem ( E , F ). W e call ( E , F ) inter nall y independent as so on a s the strings in | Log E ( F ) | a r e mutually indep enden t take n tw o by t w o. Theorem 6. I f E is internal ly in dep endent then any de cision pr oblem ( E , F ) b ase d on E as r efer enc e set exhibits the simple i nternal indep endenc e pr op erty. Pr o of. Let E b e any infinite recursiv e set exhibiting the in ternal indep en- dence prop erty . Let ( E , F ) b e an y decision pro blem based on E as reference set. L et f , g b e any t w o strings in the reduced logog r a m | Log E ( F ) | of t he problem. (I) Assume f , g incompatible. Since f ∈ Σ ∞ ( E ), we hav e E ∩ E xp ( f ) 6 = ∅ . Let x ∈ E ∩ E xp ( f ). Then x is in E , x includes f and do es not include g . Analogously , one can find a y ∈ E whic h includes g and do es not include f . Th us, f , g are m utually indep enden t in E . (I I) Assume f , g compatible. By the minimalit y prop erty of the reduced logogram | Log E ( F ) | it cannot b e t ha t f ≥ g . By the in t ernal indep endence of the reference set E one has f ⊒ E g if a nd only if f ≥ g . Then, it also cannot b e the case that f ⊒ E g . As a consequence, there exists x ∈ E whic h includes f and do es not include g . Analogously , there exists y ∈ E whic h includes g and do es not include f . Thus, again w e ha ve that f , g are m utually indep enden t. W e conclude that problem ( E , F ) exhibits the simple in ternal indep en- dence pr o p ert y . 9 Strong Independence of a Decision Pr oblem Let us no w come t o t he strong in ternal indep endence o f a decision problem. W e kno w that ( E , F ) is in ternally indep endent as so o n as the strings in its reduced logog r a m | Log E ( F ) | are m utually indep enden t tak en tw o b y tw o. The simple in ternal indep endence of a decision problem ( E , F ) certainly is a form of in ternal indep endence of a decision pro blem, but w e ma y indeed ask f or more indep endence: W e ma y ask for indep endence of the elemen t s of the reduced ke rnel | Log E ( F ) | taken m b y m all m . The following no t io n o f in ternal indep endence of a decision pro blem captures this extreme form of in ternal indep endence of a problem. W e shall say that the decision problem ( E , F ) exhibits the prop erty of str ong inter nal independence if, for any c hoice of s distinct strings f 1 , .., f s in | Log E ( F ) | , the follo wing is true: F or eve ry i b etw een 1 and s there exists a w o rd x i ∈ E suc h that x i con tains f i and fails to con tain any of the remaining strings in { f 1 , .., f s } . It is left for the reader to sho w that strong internal indep endence of a decision problem implies simple in ternal indep endence. 6 Witness es and Wizards F rom Equation 8 we hav e Log E A 1 ∪ .. ∪ Log E A m ⊆ Log E ( A 1 ∪ .. ∪ A m ) (17) for closed A 1 , .., A m ⊆ E . No w replace m with α ( n ) and A i with F i : Log E F 1 ∪ .. ∪ Log E F α ( n ) ⊆ Log E ( F 1 ∪ .. ∪ F α ( n ) ) (18) The strings in Log E F 1 , .., Log E F α ( n ) are witnesses . The p ossible strings in Log E ( F 1 ∪ .. ∪ F α ( n ) ) − Log E F 1 ∪ .. ∪ Log E F α ( n ) (19) w e call “wizards” since they are so to sp eak able to p erceiv e t ha t a n input x shall b e in someone of the F i s but couldn’t sa y whic h. The p ossible existence of this t yp e of strings in the reduced logog ram | Log E ( F ) | of a decision pro b- lem ( E , F ) can b e demonstrated by examples. Wizards ha v e b een found to exist in the reduced lo gograms of follo wing pro blems (i) T o decide if a sym- metric lo opfree graph is connected, (ii) T o decide if a giv en p ositiv e in teger is comp osite (note, inciden tally , that P RI M E S is in P [8]). In a situation in whic h the t a rget set F is decomp osed according t o ∪ n F n = F , the witnesses are alw ay s there in the reduced logogram of set F relativ e to E . On the contrary , the wizards ma y b e missing. It p ertains to the structure of the computatio nal problem at hand whether the target set F has wizards. W e conclude this section pro ving a theorem: 10 Theorem 7. If F = ∪ n F n wher e the F i s ar e cylinders in E , then ∪ α ( n ) i =1 | Log E F i | is c omplete for ( E , F ) . Pr o of. Being a union of cylinders in E , F is a cylinder in E . Being cylinders in E , the F i s are endow ed with reduced logograms. This is to sa y that, for i = 1 , .., α ( n ) and an y x ∈ E , o ne has x ∈ F i if and only if there is g ∈ | Log E ( F i ) | suc h that x ≥ g . Since the target set ∪ n F n = F is itself a cylinder in E , Equation 18 holds. (I) Let f ∈ | Log E F 1 | ∪ .. ∪ | Log E F α ( n ) | a nd let x b e an input w ord of length n such that x ∈ E and x ≥ f . W e m ust pro v e x ∈ F . V ery o b viously w e ha v e f ∈ Log E F 1 ∪ .. ∪ Log E F α ( n ) . Since sequence F i has a cardinalit y function α ( n ), then, for input words x ∈ E of length n , equation ∪ n F n = F can b e rewritten F = F 1 ∪ .. ∪ F α ( n ) . Giv en F 1 ∪ .. ∪ F α ( n ) = F , f ∈ Log E ( F ) follows from Equation 18. Then x ∈ F follo ws fro m x ≥ f (taking into accoun t that F is a cylinder in E ). (I I) Let x ∈ E b e any input w ord o f length | x | = n . Assume x ∈ F . W e m ust pro v e that there exists f ∈ | Log E F 1 | ∪ .. ∪ | Log E F α ( n ) | suc h that x ≥ f . Since x ∈ F and F = F 1 ∪ .. ∪ F α ( n ) , there exists i , 1 ≤ i ≤ α ( n ), suc h that x ∈ F i . Since F i is a cylinder in E , the reduced logo g ram | Log E ( F i ) | exists. This implies that, if y ∈ E includes a string g ∈ | Log E ( F i ) | then certainly y ∈ F i . Con v ersely , if y ∈ F i then y includes at least a string g ∈ | Log E ( F i ) | . But this is just to say that | Log E ( F i ) | is a complete subset of | Log E ( F ) | for F i relativ e to E , whic h is to sa y , for problem ( E , F i ). Since | Log E ( F i ) | is complete fo r F i relativ e t o E , it follows f rom x ∈ F i that there exists a string f ∈ | Log E ( F i ) | such that x ≥ f . Then w e also hav e f ∈ | Log E F 1 | ∪ .. ∪ | Log E F α ( n ) | . W e hav e sho wn that, giv en any input w o rd x ∈ E suc h tha t | x | = n , o ne has x ∈ F if and only if there exists a string f in | Log E F 1 | ∪ .. ∪ | Log E F α ( n ) | suc h that f ≤ x . Th us, | Log E F 1 | ∪ .. ∪ | Log E F α ( n ) | is a complete subset of | Log E F | f or F relativ e to E . 7 Applicatio n to Bo ol e an F orm ulas The enco ding sche me that w e a do pt con verts C N F formulas into words o v er Σ = { 0 , 1 , 2 } . In what follows E = C N F , F = S AT . W e represen t clauses ov er x 1 , .., x n b y sequences of n co des fro m Σ. Co de 0 denotes absence of t he v a riable, co de 1 presence without min us, co de 2 presence with minus. E.g., clause x 1 ∨ x 3 ∨ − x 4 b ecomes 1012. A whole formula is enco ded as a sequence of clauses. W e define F nm = satisfiable form ulas with n v ariables and m clauses. 11 W e intro duce the sequence y 1 , y 2 , .. of solutions, and the corresp onding sequence F 1 , F 2 , .. or r ecursiv e subsets of F . Here the solutions y i are v a lue assignmen ts. The cardinality function is α ( n ) = 2 n . W e assume that F = S AT a s w ell as the regions F 1 , F 2 , .. are closed sets in E = C N F . Th us, all these sets are assumed to b e relative cylinders in E . These assumptions corresp ond to kno wn prop erties of S AT [5 ] [9]. Essen tially , our a pplication consists in inv estigating whether | Log E ( F ) | migh t p ossibly con t ain strings not already in some of the | Log E ( F i ) | . Before w e discuss the prop ositions that w e w ere able to derive, let us sp end a few w o rds on the lo g ogram of S AT . A string in | Log E ( F nm ) | is a prescription that a w ord in F nm ma y or may not b e conformant with. W e may represen t a string in | Log E ( F nm ) | as a word of length nm o v er { ♭ } ∪ Σ. Ex ample f o r n = m = 3: String ♭ ♭ 11 ♭ 2 ♭ 2 ♭ prescribes that first clause shall include x 3 , second shall include x 1 and − x 3 , thir d shall include − x 2 . Note that strings in | Log E ( F nm ) | only presc rib e either 1 or 2 as v alues (b y the minimalit y prop erty of reduced logogram). Theorem 8. Pr oblem ( C N F , S AT ) exhibits the str ong internal indep endenc e pr op erty. Pr o of. W e consider s distinct strings f 1 , .., f s in | Log E ( F nm ) | . Th us, regarded as a partial function, eac h f i will assign only v alues 1 or 2. W e m ust prov e that for eac h i = 1 , .., s there exists a string x i ∈ E nm = C N F nm suc h that x i includes f i and do es not include any of the remaining strings f 1 , .., f s . Let i b e any one of the indices 1 , .., s . Then D om ( f i ) ⊆ { 1 , .., nm } and, for a ll h ∈ D om ( f i ), we either ha v e f i ( h ) = 1 or f i ( h ) = 2. Let x i b e tha t word o f length nm ov er Σ = { 0 , 1 , 2 } suc h that for all h ∈ D om ( f i ) it holds tha t x ih = f i ( h ) while for h not in D om ( f i ) one ha s x ih = 0. Then certainly x i includes f i . Let f j b e an y one of the strings f 1 , .., f s b eing differen t fr o m f i . Th us, f j 6 = f i . W e m ust prov e that x i do es not include f j . (I) Assume D om ( f j ) = D om ( f i ). Since f i and f j are differen t, there is k ∈ D om ( f i ) suc h that f i ( k ) 6 = f j ( k ). But x ik = f i ( k ), then x ik 6 = f j ( k ). Then x i do es not include f j . (I I) L et Assume D om ( f j ) 6 = D om ( f i ). Then either there is a ∈ D om ( f j ) suc h that a 6∈ D o m ( f i ) or there exists b ∈ D om ( f i ) suc h that b 6∈ D om ( f j ). Assume that a exists. Then x i do es no t include f j since x ia = 0 while f j ( a ) / ∈ 0, hence x ia 6 = f j ( a ). Analo gously , x i do es not include f j in case b exists. Theorem 9. The r e duc e d lo g o gr am | Log C N F ( S AT ) | do es not c ontain w iz- ar ds. 12 Pr o of. W e m ust pro v e: Log E F nm 1 ∪ .. ∪ Log E F nm α ( n ) = Log E ( F nm ) (20) where α ( n ) is the car dina lity f unction of sequence F 1 , F 2 , .. . Here F i is the range o f the v alue assignmen t y i (set of form ulas in F that are satisfied b y y i ) and is a cylinder in E . Since Equation 18 holds, w e j ust ha v e to prov e t ha t the righ t-hand side of Equation 20 do es not con tain wizards. W e actually will pro v e: | Log E F nm 1 | ∪ .. ∪ | Log E F nm α ( n ) | = | Log E ( F nm ) | (21) whic h is eviden tly equiv alen t to Equation 20. W e write K nm for | Log E ( F nm ) | and, for ev ery intege r i = 1 , .., α ( n ), we write K nm i = | Log E ( F nm i ) | . W e m ust pro v e K nm = K nm 1 ∪ .. ∪ K nm α ( n ) . First o f all, note that the set of all witnesses K nm 1 ∪ .. ∪ K nm α ( n ) is complete for the target set F = S AT relativ e to reference set E = C N F b y Theorem 7. This implies that, if x ∈ F nm , then x includes a string f ∈ K nm 1 ∪ .. ∪ K nm α ( n ) . Let h ∈ K nm . Since K nm is included in Σ ∞ ( E ), w e hav e h ∈ Σ ∞ ( E ). Then there is an x ∈ E nm suc h that x ≥ h . O n the other side, if x is in E nm and includes string h , then x ∈ F , hence, since K nm 1 ∪ .. ∪ K nm α ( n ) is complete for F relative to E , there shall exist a string k ∈ K nm 1 ∪ .. ∪ K nm α ( n ) suc h that x ≥ k (a nd h, k shall ha v e to b e compatible to one another). W e then set h → k to mean that (i) k is a mem b er of K nm 1 ∪ .. ∪ K nm α ( n ) , (ii) there exists x ∈ E nm suc h that b oth h ≤ x and k ≤ x . (Th us, h → k implies that h and k a r e compatible.) Besides, we in tro duce the set U ( h ) = { k | h → k } of those witnesses (mem b ers of set K nm 1 ∪ .. ∪ K nm α ( n ) ) that ar e r elat ed to h . No w, b y w ay of contradiction, w e assume t hat h do es not b elong to U ( h ). W e t hen hav e that the elemen ts in the set { h } ∪ U ( h ) ar e all distinct. By the strong in ternal indep endence o f S AT , in corr esp o ndence to eac h string f ∈ { h } ∪ U ( h ) there exists a w ord x ∈ E nm suc h that f ≤ x and for no g ∈ { h } ∪ U ( h ) b eing distinct f rom f one has g ≤ x . Let x ∈ E nm b e suc h that x ≥ h and for no g ∈ U ( h ) one has the inclusion g ≤ x . W ord x is in F = S AT since x includes h whic h is an elemen t of | Log E ( F nm ) | . Besides, x do es not con tain any elemen t from U ( h ). But that in turn means that x do es not con tain an y strings f rom the witset K nm 1 ∪ .. ∪ K nm α ( n ) . (Should x include a string k from K nm 1 ∪ .. ∪ K nm α ( n ) that w o uld mean that b oth x ≥ h, x ≥ k w ould hold, hence k w ould b e related with h whic h would imply k ∈ U ( h ).) This is absurd, since K nm 1 ∪ .. ∪ K nm α ( n ) is complete for S AT relat ive to C N F . W e conclude that h is a mem b er of U ( h ), and hence is in K nm 1 ∪ .. ∪ K nm α ( n ) . 13 Since we already kno w that K nm 1 ∪ .. ∪ K nm α ( n ) is a subset of K nm , we conclude that K nm = K nm 1 ∪ .. ∪ K nm α ( n ) . Th us, S AT has no wizards. Theorem 10. The r e duc e d lo go gr am | Log C N F ( S AT ) | is irr e ducible. Pr o of. Let g ∈ | Log E ( F nm ) | . By Theorem 9 w e know t hat g mu st b e a witness. Th us, g is a string con vey ing the sp ecification of exactly one v alue assignmen t. Besides, g is minimal (no prop er restriction of g is a sufficien t condition for ev ent x ∈ F ). These t w o facts mak e it a straigh tforw ard task to sp ecify the general shap e that string g shall exhibit. First of all, D om ( g ) shall ha v e to b e a set of exactly m n um b ers ta k en from { 1 , .., nm } . The first of these num b ers is to b e ta k en from the first blo ck { 1 , .., n } (where the first clause is allo cated), the second is to b e from the second blo c k { n + 1 , .., 2 n } ,.., the m th is from the m th blo c k { n ( m − 1) + 1 , .., nm } (where the last clause is allo cated). Thus , there are nm p ossible determinations for D om ( g ). W e kno w that regarded as a prescription, g can only prescrib e the tw o v a lues 1 and 2. (T o help in tuition, string g can b e though t of as a sequence of flats ♭♭..♭ of length nm in which some of the flats (as man y as m ) hav e b een replaced with 1 s or 2 s .) With an y g that satisfies the ab ov e requiremen ts w e asso ciate a form ula γ ( g ) as follo ws. W e note that, regarded as a prescription, g prescrib es the presence o f exactly one literal in eac h clause of a form ula x consisting of m clauses: W e then state that the i th clause of γ ( g ) shall consist of exactly the single literal that g prescrib es to the i th clause of x . Eviden tly , γ ( g ) is satisfiable and g ≤ γ ( g ). W e claim that γ ( g ) do es no t include mem b ers of | Log E ( F nm ) | o t her than g . Indeed, the strings in | Log E ( F nm ) | nev er prescrib e 0 as v a lue, and g is the la rgest string b eing included in the co dew ord of γ ( g ) which do es not prescribe 0 as v alue. Thus , the only strings that do not prescrib e 0 a s v a lue and happ en to b e included in the co dew ord of γ ( g ) are exactly string g itself and the prop er restrictions of string g . Since g is minimal, all of its prop er restrictions are not mem b ers of | Log E ( F nm ) | . Th us g is the only string b eing included in t he co dew ord of γ ( g ) to b e found in | Log E ( F nm ) | . Hence, | Log E ( F nm ) | − { g } is not complete for F nm relativ e to E nm . 8 SA T as Searc h Prob lem The searc h vers ion of a decision problem consists in obta ining solutions for a given instance x . Thus, with an y NP pro blem ( E , F ) we asso ciate the follo wing searc h problem: Giv en x find a solution y for x or stat e that no suc h y exists. 14 It is kno wn that, by self-reducibilit y o f S AT , if w e had a p olynomial algorithm for S AT , then w e w ould also hav e a p o lynomial algor it hm for the searc h problem asso ciated with S AT [9]. The results of previous sections sho w that w e can sa y more: It is imp ossible to solve S AT without at the same time solving the searc h problem asso ciat ed with S AT . These remarks suggest that w e may wish to fo cus on the search problem asso ciated with S AT . This is what w e do in this section. Giv en any NP problem ( E , F ), w e introduce the cov er o f the target set F asso ciated with | Log E ( F ) | to b e the family of sets D E ( F ) = { E xp E ( g ) ⊆ F : g ∈ | Log E ( F ) |} . (22) Its members a re the char ts or else r eg ions of the co v er. The cov er that is asso ciated with the k ernel of a program P solving ( E , F ) is then F P ( E , F ) = { E xp E ( g ) ⊆ F : g ∈ K er ( P ) } . (23) Both D E ( F ) and F P ( E , F ) a r e f amilies of subsets of the tar g et set F whose union is F , with F P ( E , F ) b eing a subfamily of D E ( F ). F or S AT we ha ve the f ollo wing situation: F P ( E , F ) = D E ( F ) b y Theo- rem 10 and the strings in | Log E ( F ) | are all witnesses b y Theorem 9. Thus an y of these strings, call it g , has an asso ciated relativized cylinder E xp E ( g ) b eing fully included in only o ne of the regions F i s . Since for E = C N F , F = S AT , E xp E ( g ) is actually a n in tersection o f t w o absolute cylinder sets E xp ( g ) and E , then E xp E ( g ) itself is an absolute cylinder. In general, E xp E ( g ) will interse ct certain ot her regions F h , F k ,.. , but t here exists only one region F j whic h completely includes E xp E ( g ). Besides, ev ery region F i shall ha v e to include at least one suc h elemen tary relativized cylinder E xp E ( g ). As a consequen ce, the cardinalit y of t he co v er D E ( F ) cannot b e smaller than that o f the family of sets { F n i : i = 1 , .., 2 n } , hence it is exp onen tial. Remarks on the Time Complexity of SA T In the rest of this section w e mak e remarks on the time complexit y of SA T in the ligh t of Theorems 8 , 9, 10. W e will b e less formal than in previous sections. Our remarks consist of tw o pa rts: P art One It f o llo ws from Theorem 10 that there is a unique subfamily F of D E ( F ) suc h that F = S F , namely D E ( F ) itself. As a consequence, fo r an y prop er subset F ⊂ D E ( F ) one has F 6 = S F . W e then hav e that it cannot b e t ha t F P ( E , F ) is a prop er subfamily of the full co v er D E ( F ), otherwise we w ould hav e F 6 = S F P ( E , F ), and then P 15 could not b e correct as a pr o gram. In particular, since D E ( F ) is exponential, F P ( E , F ) is not allo w ed to b e a p olynomial subfamily of D E ( F ) ): No searc h algorithm for S AT can only searc h a p olynomial family of sets. P art Two It remains for us to discuss the p o ssibilit y that one single algo - rithm can solve the full searc h problem fo r x by directly searc hing the full exp o nen t ial family D E ( F ) in p olynomial time. Ho w ev er this can scarcely b e the case due t o complete absence of an y form of dep endence among subsets in the reduced logo g ram | Log E ( F ) | for E = C N F , F = S AT . By this lac k of in ternal dep endence, any computation of a program P solving ( C N F , S AT ) is suc h that the result o f any computation step do es not change the results that are left p ossible fo r the subsequen t steps. In the rest of this part we mak e a few informal remarks on how this lack of dep endence comes into pla y . W e take a general purp ose program mac hine M as computation mo del. (That M is a program mac hine means tha t the pro cess carried out b y M is determined by a running program.) W e assume tha t only one program is running at any momen t of time within M . W e ke ep mac hine M fixed while w e consider an infinite set of programs solving S AT (a ctually the set of a ll programs that run on M and solv e S AT ). W e emphasize that the hardw are is kept fixed while differen t programs all running on that hardware are compared. Let B ( x, m ) b e a program whic h for a n y giv en input x of size n and ev ery in teger m b etw een 1 and 2 n will decide if x ha s solutions in the range b et w een y 1 and y m . T a k e T im e B ( x, m ) b e the num b er of time units that B uses on inputs x, m . W e will ma ke remarks that con v ey evidence f or following stat ement: If for an y x and m < 2 n w e ha ve T im e B ( x, m ) = T ime B ( x, m + 1), then w e ma y r eplace B with a new pro gram C running on M and suc h that T ime C ( x, m ) < T ime C ( x, m + 1) = T ime B ( x, m + 1). Indeed, under the ab ov e hy p o theses on M , w e can sp eak of the the class of all pro grams B , C ,.. that solv e S AT on machine M , and we can in tro duce a most efficien t progra m A in this class. W e understand that A is a most efficien t program as so on as T ime A ( x, 2 n ) ≤ T ime C ( x, 2 n ) f o r any other program C o n a ny input w ord x . It is sufficien t for us to giv e a hint fo r T ime A ( x, 1) < T ime A ( x, 2). Our hin t is the fo llo wing. Since, b y Theorem 9, w e ha v e | Log E ( F 1 ∪ F 2 ) | = | Log E ( F 1 ) | ∪ | Log E ( F 2 ) | and | Log E ( F 1 ) | ∩ | Log E ( F 2 ) | = ∅ , a computation that implemen ts the collection of tests in | Log E ( F 1 ∪ F 2 ) | consists of tw o distinct computations, one implemen ting collection | Log E ( F 1 ) | and the other imple- men ting collection | Log E ( F 2 ) | . Th us, computation A ( x, 1) b eing a prop er 16 prefix of computation A ( x, 2) is compatible with assumed optimality of A , whence T ime A ( x, 1) < T ime A ( x, 2). 9 On Ascribing Kno wledg e to Programs Our theory has ro ots in the b o dy of formalized concepts referred to as Scott’s theory of computation [10]. Th us, the reduced logogram | Log E ( F ) | asso ci- ated with problem ( E , F ) is a n inf ormation sy stem [6], [7] (how ev er, the v ery imp ortant relation is not entailmen t but entangleme n t). Eve n more relev an t are the relationships with the “ dynamical” part of Scott’s theory , the one regarding computations as sequenc es o f steps through whic h the r un- ning program’s kno wledge increases [3]. W e also, in this latter resp ect, used concepts from the mo del t heoretic analysis of program kno wledge [11 ]. In this section we briefly review relationships of the ab ov e theory with formalisms that ascrib e kno wledge to a running program. In Scott’s theory the computations that pro gram P do es are functionally equiv alen t to sequences of tok ens (or tests) b eing consisten t with the input string x . In our dev elopmen ts, the “tests” or “tokens ” are iden tified with the strings in K er ( P ). In Scott’s theory , the state of knowle dge of a running program P consists of a pile o f asser tions . These are consisten t (indeed, they are prop ositions that are true of one a nd the same ob j ect x ). As so on as the pile b ecomes a decisiv e one, the progr am mak es its decision and stops. Our addition is: The “ assertions” are of the fo rm x ∈ E xp ( g ) or else x 6∈ E xp ( f ) where f , g ∈ K er ( P ). Searc hing x for a string g a moun ts to same as asking if x happ ens to b elong in the a bsolute elemen tary cylinder E xp ( g ) a sso ciated with g . W e th us arriv e at the conclusion that all that P can p ossibly do to make a decision consists in asking questions of this form. Thus , the computatio ns that P p erforms ar e just sequenc es of tests in disg uise . Note that P has not got to ask whether x is in E xp E ( g ) since P already know s that x is in E . (This is an imp orta nt p o in t since asking if x is in E xp E ( g ) w ould b e more computationally exp ensiv e.) In this theory , information regarding x is acquired b y P in lumps. The acquisition of a piece of information o ccurs at the momen t when the exe- cution of a sequenc e of tests is completed ( i.e., when the computation that implemen ts that sequence of tests is completed). W e ma y well think of a piece of info r mation as b eing a piece of pa p er carrying a written no te suc h as “ x is in E xp ( g )” or “ x fails to b e in E xp ( g ).” These notes stac k one up on the other until the pile b ecomes a decisiv e o ne: This is the case when the data that w a s gat hered entails one of the ev ents x ∈ F or else x ∈ E − F . 17 Note that loading a n input x in memory do es not imply computations, hence no tests are made o n x while loading, hence no know ledge is acquired ab out x . After lo ading x , the pile o f assertions that represen ts progr am’s kno wledge is empt y . 10 Concl u sions W e adv o cated strings (with sp ecial meaning for the term) as a fundamen- tal notion fo r studies of computation. So to sp eak, strings are needed to express the notions of in ternal and strong in ternal indep endence of a deci- sion problem that underly our t heory of decision problems. W e we re led to form ulate strings to b ecome able to deriv e the v ery basic notion of internal indep endence of a decision problem. St r ing s seem to b e useful since they ar e absolutely elemen t ary . Note tha t they are already at w ork in Computability . The “restrictions” that are often used in the study of circuit complexity a re finite Bo olean v ersions of the strings [9]. Strings are not made of consecutiv e letters. A string can b e in t ersp ersed in a w ord: By canceling zero or more letters in a w ord x , and b y leaving blanks in places of letters, w e get a string f whic h is a substring o f the origina l word x . In a string, o ne has information asso ciated with spaces b et w een letters (and hence with p ossible multiple p erio dicit y with whic h letters ma y o ccur). As so on as w e hav e the strings, w e a re able to define the k ernel K er ( P ) o f a decision pro gram P , a set of strings whic h capture structural features of b oth program P and the decision problem ( E , F ) that P solv es. K er ( P ) is a subset of the reduced logo gram | Log E ( F ) | of target set F in base E . The reduced logog r a m consists of substrings of the w ords in F whic h exhibit the following prop erty: If a w ord in E includes one of these substrings then it b elongs to F . W e may think of the strings in | Log E ( F ) | as kind of genes of the w ords in F . (In early notes the logogram w as the j innee or g enie of problem ( E , F ).) The idea clearly comes from biology , where it is kno wn that certain o ccurrences at giv en in terv a ls of certain letters within DNA sequences con v ey structural information, and yield observ able c ha racters in the macroscopic dev elopment of the structures. Our application to S AT uses a structural prop ert y of t hat problem that seems to hav e escap ed atten tion so far. W e called it “strong internal inde- p endence.” The orem 8 sho ws that S AT exhibits the strong in ternal inde- p endence pro p ert y . Theorem 9 show s that, by tha t prop erty , S AT cannot ha v e collectiv e certificates in its reduced logogram. As a consequence, all the programs that solve S AT ha ve same k ernel (Theorem 10). The remarks in Section 8 suggest ho w Theorems 8, 9, 10 can p ossibly b e 18 used to put SA T under scrutin y . Our ultimate concern in this pap er has b een to set fort h our dev elopmen ts as a p ossible new tec hnique to attac k decision problems, where “tec hnique” is here used in the sense tha t Hemaspaandra and Ogihara ga v e to this term in the preface of their “Companion.” 11 Ac kno wledge men t s In the dev elopmen t of this researc h I receiv ed advice f r o m Proff. F abrizio Luccio, Joha n Hastad, Giancarlo Mauri, and Claudio Pro cesi. These results w o uld not ha v e b een a chiev ed without that help. References [1] Kirousis K and Kolaitis P . The complexit y of minimal satisfiability prob- lems . Information a nd Computation, 187(2003) , 20-39. [2] Odifreddi P . Classical Recursion Theory . North-Holland, 1989 [3] Gierz G , Hofmann K, K eimel K, Lawson J D , Mislo v e M W, Scott D. Con tin uous Lattices and Domains . Cam bridge Univ ersit y Press [4] Birkhoff G. Lattice theory . AMS V olume 25 [5] Balcazar J, Diaz J, Gabarro J. Structural Complexit y I I . 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