The Kleene-Rosser Paradox, The Liars Paradox & A Fuzzy Logic Programming Paradox Imply SAT is (NOT) NP-complete

P = NP The Kleene-Rosser P arado x The Liar’ s P arado x & A F uzzy Logic Programm ing P arado x = ⇒ SA T is (NOT) NP-complete Rafee Ebrahim Kamouna Email: rafee102000@yahoo .com submitted to the A CM T ransactions on Computation Theory What is a Turing machine? Imeptuous Fire, Syntactico- Semantical! Ice and Desire, Computation wags on... [T uring ` a la “R ome o & Juliet”] Abstract After examining the P v ersus N P problem against the Kleene-Rosser parado x of the λ -calculus [9 4], it w as found that it r epresen ts a counte r- example to NP- completeness . W e pro v e that it con tradicts the pro o f of Co o k’s theorem. A logical formalization of the liar’s paradox leads to the same result. This formalization of the liar’s parado x in to a computable fo rm is a 2-v alued instance of a fuzzy logic pro gramming pa rado x disco v ered in the system of [90]. T hree pro o fs that sho w that SA T is (NOT) NP-complete ar e presen ted. The coun ter- example classes to NP-completeness are also coun ter- examples to F agin’s theorem [36] and the Immermann-V ardi t heorem [89,110], the fundamen tal results of descriptiv e complexit y . All these results sho w that ZF 6 C is inconsisten t. 1 1. In tro duction and the Kleene-Rosser P arado x: This pap er examines w ell-know n paradoxes against the fundamen tal question in complexit y theory , i.e. the P vs. NP problem. The Kleene-Rosser para do x of the inconsis tent λ -calculus disc ov ered in 1935 and a computable formalization of the liar’s paradox whic h is w ell-kno wn to happ en in natural languages. The liar’s paradox formalization happ ens to b e a 2-v alued sp ecial case of a more general m ulti-v alued one. The later being the case o f the fuzzy logic prog ramming parado x of the system in [90 ]. If the P vers us NP problem was eve r examined against an y of those paradoxes, it w ould hav e so o n b een disco v ered that it is a straigh tforward counte r- example to NP-completeness . Let L λ b e the language deﬁned b y the follow ing function when combined with itself, t hus k k : k = ( λ x. ¬ ( xx )) one then may deduce k k = ( λx. ¬ ( xx )) k = ¬ ( k k ) Ob viously , t he languag e L λ is decidable and in P . How ev er, it is ob vious L λ 6≤ p SA T , as how strings whic h are b oth “true” and “false” can be con verted to strings whic h are either “t r ue” or “false”. The coun ter-argument tha t a T uring machin e cannot diagonalize against itself leads to the fa ct that L λ w ould b e a coun ter- example to the the Ch urc h-T uring thesis instead of b eing a coun ter-example to NP-completeness. It is implausible to consider suc h a simply computable language as uncomputable. Also, writing L λ as a se ries of inﬁnite non-halting computations simply ignores t ha t it is programmably implemen ted and certainly halts. The follo wing pro of sho ws that this paradox results in NP-completeness undeﬁnabilit y when the languag e L λ is a ssumed to exist. Deﬁnition 1: Let LI AR Lang b e the class of all la nguages written in the LI AR logic system and F LP Lang b e the class of all languages written in the F LP logic system (deﬁned b elo w), then the class S y S B P D = { L λ : L λ ≡ The Kleene- Rosser par ado x, L λ ∈ LI AR Lang , L λ ∈ F LP Lang } . Deﬁnition 2: Le t M λ b e a progra m that chec ks for paradox es, i.e. a parado x recognizer. A computation M λ on w λ ∈ L λ prin ts “Y es” if w λ is a n instance of a parado x, i.e. w λ = “T rue” iﬀ w λ = “F alse” So: 1. M λ accepts w λ ∈ L λ iﬀ w λ is pa r ado xical, otherwise: 2. M λ rejects w λ ∈ L λ iﬀ w λ is satisﬁable. 2 Theorem 1.1: (Main Theorem) SA T is NOT NP-complete. The line of argumen tation of the original pro of of CNF SA T b eing NP-complete is a s fo llo ws as in [2 1 ] and quoted from [37]: “Let A b e a la nguage in N P accepted b y a non-deterministic T uring mac hine M : Fix a n input x . W e will create a 3CNF formula φ that will b e satisﬁable if and only if there is a prop er tableau for M and x .” Pro of: 1. Let M = M λ , A = L λ , x = w λ . 2. = ⇒ M λ accepts w λ . 3. SA T is NP-complete. 4. = ⇒ [ ∀ w λ ∈ L λ ∃ a prop er ta bleau fo r M λ and w λ ] ⇐ ⇒ [ φ is satisﬁable]. 5. = ⇒ φ is satisﬁable ⇐ ⇒ M λ accepts w λ . 6. But w λ is pa r ado xical, as a paradox . 7. = ⇒ φ is satisﬁable ⇐ ⇒ w λ is pa r ado xical. 8. φ is satisﬁable ⇐ ⇒ “F alse”. 9. φ is parado xical. 10. 6 ∃ φ : φ is satisﬁable. 11. SA T is ( NOT) NP-complete. Theorem 1.2: SA T is (NOT) NP-complete. Pro of: 1. SA T is NP-complete. 2. = ⇒ ∀ w ij ∈ L i ∃ f ( w ij ) = w SA T ∈ SA T . 3. Let w ij = w λj , then ∃ f ( w λ j ) = w S AT j . 4. w λj is “ true” iﬀ “fa lse” while w S AT j is either “true” or “false”. 5. 6 ∃ f : f ( w ij ) = w SA T ∀ w λ j . 6. SA T is (NOT) NP-complete. 3 Theorem 1.3: P = NP . Pro of: 1. SA T is (NOT) NP-complete. 2. = ⇒ NP-complete = ∅ . 3. = ⇒ P = NP . Th us, the Kleene-Rosser paradox kno wn as early as 1935 is suﬃcien t to ov erturn all NP-completeness results. Ho we ve r, other lo g ical languages may ha ve para- do xical b ehav ior as sho wn b elow. Note the misconception of a T uring mac hine cannot risk con tradiction is due to considering it as an enco ded in teger with no regard to its seman tics. Ob viously , no integer can form a par ado x. A paradox is an absolutely lo gic al situation whic h is related to language. Some may transform the Kleene-Rosse r paradox as an example of an inﬁnite lo op. Concealing this parado x into a ph ysical T uring machine that do es not halt w ould not eliminate it as it is in the language. The liar’s parado x whic h exists in natural langua ge can be easily formalized as below leading to the same ab ov e result. Ch urch’s λ -calculus is equiv alen t t o T uring machine s among other computational mo dels. The Syn tactico-Seman tical Bi-P olar Disorder T uring Mac hine P arado x Since the P ve rsus NP probnlem has all its ro ots in the mat hematics founda- tion crisis in the early XX th cen tury , an attempt to examine the reason b ehind these (negative) results introduce the “Syntactico-Seman tical Bi-P ola r Disorder” explained b elo w. The XX th cen tury most imp ortant results w ere re-organized a s b elo w: 1. Self-referen tia l S y S B P D : (a) Russell’s pa r a do x. (b) The Liar’s paradox. 2. G¨ odel Completeness/Inc ompleteness S y S B P D ; not e the relationship b e- t w een the pro of of his cele brat ed incompleteness theorem and the Liar’s parado x. 3. T uring Decidability/Unde cidability S y S B P D . 4 4. Finitenes s/Inﬁniteness S y S B P D : res ults in ﬁnite mo del theory that suc- ceed inﬁnitely and fail ﬁnitely . Most imp o rtan tly , G¨ odel’s completene ss theorem which is: (a) Positive : Completene ss/Incompleteness S y S B P D . (b) Ne gative : Finiteness/Inﬁniteness S y S B P D . All these S y S B P D ’s a re instances of the “Syn t actico-Seman t ical Prece- dence/Principalit y B i-P olar Disorder” . Note that G¨ odel completeness the- orem is considered a p ositiv e results in automated deduction an related areas while considered negativ e in ﬁnite mo del theory as it fails ﬁnitely . 1. Prec edence: syn tax deﬁnition precedes seman tics: [Syn tax < Seman tics] P re cdence . 2. Principalit y: during computation the input tak es v arious syn tactic fo rms where seman tics is principal o ve r syn tax in ev ery computation step: [Seman tics < Syn tax] P rin c ipality . 3. (1) & (2) = ⇒ [Syn tax] <> [Semantics ], i.e. Bi-Polar Disorder. The question:“Are the XX th the o nly S y S B P D ’s” led to the disco very o f all recen t results. No w, w e hav e the Syn tactico-Semantical Bi-P olar Disorder T uring mac hine NP-completeness P aradox a s: SA T is NP-complete ⇐ ⇒ SA T is (NOT) NP-complete whic h is simply b ecause: w is paradoxi cal ⇐ ⇒ Maccept s w ⇐ ⇒ A ( w ) is satisfiable where A ( w ) [21]: A ( w ) = B ∧ C ∧ D ∧ E ∧ F ∧ G ∧ H ∧ I and b ecause P i s,t are prop ositional v aria bles in A ( w ) P i 1 1 , 1 ∧ P i 2 2 , 1 ∧ . . . P i n n, 1 is satisfiable iff w is paradoxical The reason for the paradox is that Co oks’s theorem is still tr ue despite all the results a b o v e and b elow of SA T b eing (NOT) NP-complete. Recen t results w ere obtained solely via logical syn tactico-seman tical pro ofs. On the other hand 5 Co ok’s pro of mixes the ph ysical w orld with the mental w orld. The formula A ( w ) in [21] consists of prop ositional sym b o ls whic h r efer to the ph ysical na t ure o f the T uring mac hine to prov e a pro p ert y of the set of strings it pro cesses. While the form ula is satisﬁable fr o m a ph ysical p oin t- of-view, it is no t alw ays the case from a logical p oin t-o f -view. It is clear that the pro of in [21] do es not mak e an accoun t of the meaning of the string w when there is a reference of a computation M on input w . A ( w ) is deriv ed f rom the mac hine ph ysical nature during the computatio n. An example o f tho se ph ysical facts is tha t if the mac hine ta p e head is at the lo cation k , then the next computation m ust b e either k + 1 or k − 1. T his - among many ot her similar thing - while b eing a true (ph ysical) prop ert y of the mac hine itself, it ma y not ha v e implications on the prop erties o f the lang ua ge b eing pro cessed. T his is the “Syn tactic-Semantic al Bi-P ola r Disorder T uring mac hine NP-completeness P aradox” whic h can b e stated more clearly as: “A logically satisﬁable form ula A ( w ) can alw a ys b e construct ed from the logically parado xical string w ” The source of this contradiction is w has no connection with phys ics, while A ( w ) do es ha v e. They b oth meet in the realm of “Syn tax” while they nev er do in the realm of “Seman t ics”, hence a syn tactico-semantic al paradox, whic h is an irreparable disorder o f computation and mathematics. It is possible for a seman tic pro of to o v erturn a syn tactic one, but not in this case when the pro of deriv es from the phy sical prop erties o f the non- detrministic T uring machine itself. Ob viously , no pro of (syn tactic or seman tic) can ov erturn an y physic al fact, e.g. that if the T uring machine head is at lo cation k , then the next computation step must b e either at lo cation k − 1 or at k + 1. This is a ph ysical fact. The prop ositional sym b ols constructed in the pro of are mostly deriv ed in this w ay . 2. The Liar’s Parado x: The fo llo wing theorem pro v es a formalizatio n of the Liar’s par a do x in a Prolog st yle programming la nguage. Th us, self-referen tial paradoxical languages can b e represen t ed in a programming language as w ell as in the ab ov e inconsisten t λ - calculus (recursion vs . self-reference). It is to b e noted tha t self-reference has b een remo v ed from ﬁrst-order logic delib er ately a priori in order to av oid suc h con tradictions. Ho w ev er, its elimination do es mean that those contradictions do not exist in the languages (elemen ts) of P and NP . Consider: P = { L 1 , L 2 , L 3 , . . . , L i , L j , L k , . . . } . Ob viously , L λ exists as some language in P as w ell as o t her pa rado xical la nguages lik e L ∈ LI AR b elo w. It is of no help to preclude them. Theorem 2.1: The Liar’s Parado x ≡ { English(John,F alse) } . Pro of: 6 1. The Liar’s P aradox ≡ { This sen tence is F alse } . 2. = ⇒ { This sen tence is F alse } ≡ { A = A is F alse } . 3. = ⇒ { English(John,F alse) } ≡ { A = A is F alse } . 4. = ⇒ { This sen tence is F alse } ≡ { English(John,F alse) } . 5. = ⇒ The Liar’s P aradox ≡ { English(John,F alse) } . The LI AR logic system has the same F LP [90 ] syn tax and seman tics but with truth constan ts restricted only to t wo v alues. Its formulas w ould lo ok like : P ( f ( t ) , f al se ), where f is a r ecursiv e f unction ov er the recursiv e t erm t . Simply , the Prolog at o m: English(John,F alse) w ould b e a statemen t that asserts its own falseho o d if and only if it is true, hence a para do x. The ﬁrst question to address is suc h statemen ts do exist or do not exist. The liar’s paradox do exist in natural language and is w ell-know n fo r more than t w o millennia. T o assume it is no t for- malizable in any computable form would nev er mean tha t it do es not exist. Suc h an assumption w ould not stand the test of time against a self-referen tial question suc h as P vs. NP whic h is itself a question in N P . The delib erate elimination of self-r eference t ha t may ha ve help ed the dev elopmen t of logic w ould hinder the progress of attacking this question. The reason is that in the dev elopmen t of a logical language, or a class of la nguages in a logic system, no suc h a question of whether a n inﬁnite class is equal/or no t to another inﬁnite class is addressed. F urther, in a logical la ng uage one is interes ted to remo v e any inconsistency a priori. In a ttac king P vs. N P , one cannot assume the Kleene-Rosser paradox ab ov e do es not exist nor ignore its implications. Lo g ical progra mming languages with paradox es can b e deve lop ed lik e formalizing the L ia r’s par a do x ab ov e which happ ens to b e a 2-v a lued instance of the multi-v alued fuzzy logic programming parado x b elow. 3. The F uzzy Logic Programming P arado x: There is a v ast literature with large n umber of results in b oth “Mathematical F uzzy Logic” and “F uzzy Logic Pro g ramming”, (see the references). Mathemati- cal fuzzy lo gic systems w ere dev elop ed b y Ha jek [50 - 87], Estev a [26,29-3 5], Go do [37-49] and others. Systems lik e BL, Luk asiewicz, G¨ odel and Product logics ha v e been formulated with v arious rigorous prop erties and ha v e become stan- dard. F uzzy logic progr a mming and p ossibilistic logic prog ramming systems in the works of Go do and Alsinet et al. [1-18], V o jt a s et al. [111-115 ] we re dev el- op ed with lar ge num b er of soundness and completeness results with interes ting prop erties. V ariations as the multi-adjoin t lo g ic pro gramming w as dev elop ed by 7 Medina et al. [98-101 ]. The hu ge n um b er of results is clear and of course this is not an exhaustiv e listing. The ﬁrst use of truth constants in the language syn tax ﬁrst app eared in P a v elk a ’s logic [106 ] as early as 1979. Before tha t, truth w as express ed only in the la nguage seman tics as in Luk asiewicz and Kleene many-v alued logics. Pa ve lk a ex tended Luk asiewicz lo g ic with rational truth constan ts. No v ak [102-1 0 5], in his weigh ted inference sys tems dev elop ed a syn tax of pa irs: (fo rm ula, truth v alue). Expansions of other logics with truth constan ts in Estev a et al. 2000, and recen tly in Estev a et a l. 2006 [23-25 ,2 7], and Savic ky et al. 2006 [10 8 ]. In 2007, truth constants app eared in Estev a et a l. [28]. The w ork of Straccia et al. [19,20,96,1 07,109] in fuzzy description logics emplo ye d truth constan ts as w ell. So, the idea of having a truth constant in the language syn tax is w ell-established. A coun ter-example to the NP-comple teness prop erty written in F LP [90] lan- guage is presen ted. A class of inﬁnite n um b er of languages is c haracterized - S y S B P D : t he syn tactico- semantical bi-p o lar disorder class including all para- do xical langua g es of F LP as w ell as t ha t of the liar’s paradox and λ -calculus. Eac h elemen t in this class constitutes a coun ter-example as w ell. A o ne- step com- putation L is in tro duced to motiv ate t he presen tation. Theorem 3.1 establishes the paradox a nd theorem 4.1 sho ws that L is de cidable and L ∈ P . Theorem 4.2 establishes the counter-example and sho ws tha t SA T is NOT NP-complete using the same pro of o f the Kleene-Rosser paradox. First, w e recall the fact the syn tax of F LP is absolutely classical. All the well- formed f orm ulas of F LP are well-formed fo r m ulas o f classical logic. Ho w ev er, F LP uses non- classical seman tics for the same classical syn tax. First, the classi- cal deﬁnition of an Herbrand interpretation and a n Herbrand mo del are recalled. Second, it is sho wn that if truth constan ts are a llo w ed in the language syn tax in the sens e of [90 ], then ev ery Herbrand in terpretatio n of any F LP language is a mo del iﬀ it is not a mo del , when the case of F LP collapses to classical logic, i.e. µ = “0 ” o r µ = “1” ; the F LP paradox is the liar’s paradox. This is the “ Syn tactico-Seman tical Bi-P olar Disorder F LP Parado x”. All LI AR w ell- formed formulas are F LP w ell-formed fo rm ulas. This is wh y refuting the F LP parado x necessitates refuting its sp ecial 2- v alued version whic h is the liar’s pa r a- do x. It is not easy to refute the lia r’s paradox nor to sho w that it is imp ossible to b e formalized in a programming language resulting in the ab o v e discussed consequenc es. Deﬁnition 3.1: Let L b e a languag e ov er an alphab et Σ containing a t least one constan t sym b ol. The set U L of all ground terms constructed fro m functions and constan ts in L is called the Herbrand unive rse of L . The set B L of all ground atomic form ulas ov er L is called the Herbrand base of L . 8 Deﬁnition 3.2: The Herbr and interpr etation I L for a language L is a structure I L ≡ whose domain of discourse is U L where: 1. ∀ c ∈ L : c is a constan t: I c ( c ) = c . 2. ∀ f ∈ L : f is a function sym b ol o f arity n , and t 1 , t 2 , . . . , t n are terms: I f ( f )( t 1 , t 2 , . . . , t n ) = f ( I ( t 1 ) , . . . , I ( t n )) 3. ∀ p ∈ L : p is a predicate o f arity n : I p ( p ) : B L → { 0 , 1 } Deﬁnition 3.3: The Herbr and interpr etation I L for a lang uage L is a mo del iﬀ I L : B L → { 1 } ∧ B L 6→ { 0 } . Let L b e the classical logic program consisting of the single (ground) fact: p ( c 1 , c 2 , . . . , c n ) ← and let c n = µ ∈ C ⊆ [0 , 1] b e a trut h constan t. If I L is an Herbrand inte rpre- tation fo r L , then I L is a mo del iﬀ it is not a mo del . I L in terprets the predicate sym b ol p (classically) as a relation b et w een the domains fr o m whic h the n-tuple ( c 1 , c 2 , . . . , c n ) is extracted. The last mem b er of the tuple c n is a real num b er in a coun table C ⊆ [0 , 1]. Whe n constan t sym b ols are interprete d in classical seman tics, it banishes an argument of a predicate to b e the t r ut h constant of the same predicate. F LP non-classical semantics enforces an arg umen t of a predi- cate to b e a truth constan t of the same predicate. Seman tics of formal languages are enforced in the same wa y as in natural languag es. Since the string “ma in” o v er the La tin alphab et is in terpreted diﬀeren tly in English and F renc h (the w ord “main” in F renc h means “hand”). Obviously , O xf or d ( main ) 6 = Laro u sse ( main ) I L C lassical ( p ) 6≡ I L F LP ( p ) Neither the English p eople ma y ask the F renc h to f ollo w Oxford dictionary , nor the F renc h ma y ask the English to follow Laro usse. F orbidding arguments of a predicate to b e the tr uth constan t of the same predicate is equally una ccept able . Moreo v er, in the case of the P vs. NP question, the en tire scien tiﬁc communit y is pre-o ccupied with ANY set of strings (a langua ge) that may separate the t w o 9 classes. Usually , a set of strings in N P and not in P , hence the question is settled. Let alone the self-referen tial nature of the question, i.e. P vs. NP is a question in NP . So, if X is the decision pro blem X ≡ P =? NP , then X ∈ NP . But classes are (forbidden) to b e elemen ts, so suc h an argumen t is a meta- mathematical/philosophical one ( X is not a v a lid mathematical ob ject). Just consider an analogy of the question: x ? = y , x ∈ N , y ∈ R . Obviously , this later question is a n ill- p osed one. F or the ab ov e considerations, the author is not deterred to enforce suc h seman tics on the same syn tax of classical log ic, then examine the consequences . F orbidding suc h seman tics w on’t help b ecause b oth classes contain inﬁnite n um b er of lan- guages. An y metho d to for bid suc h seman tics can ob viously b e eliminated with a coun ter-part to enforce whatev er semantics t o examine its implications to this long outstanding question. In o ther w ords, a coun ter-argumen t against F LP non- classical seman tics should prov e that suc h la nguages don’t exist at all. The fact that it leads to paradoxical and inconsisten t computations nev er means that these computations are wrong or meaningless. The attac hed tw o meta-in terpreters w ork quite w ell meaningfully from a practical engineering p oin t-of- view. The reason for this is that in a lo gic programming system, t he user is interes ted in answ er substitutions rather than logical consequences a s in automat ic theorem pro ving. Cantor’s set theory has its famous paradox es, one can nev er argue it is wrong, t ho ugh initially it w as con trov ersial. The follow ing theorem pro ve s that lang uages written in F LP can ha v e in terpretations consisting of paradoxical structures. Theorem 3.1: Let L b e the classical logic prog ram consisting of the single (ground) fact: p ( c 1 , c 2 , . . . , c n ) ← and let c n = µ ∈ C ⊆ [0 , 1] b e a truth constant. If I L is a n Herbrand interpreta- tion for L , then I L is a mo del iﬀ it is no t a mo d el . Pro of: 1. I L ≡ ≡ . 2. ⇒ I c ( c 1 ) = c 1 . 3. ⇒ I c ( c 2 ) = c 2 . 4. · · · 5. · · · 6. · · · 10 7. ⇒ I c n − 1 = c n − 1 . 8. ⇒ I c ( µ ) = µ ∈ [0 , 1]. 9. ⇒ I p ∈ { 0 , 1 } . 10. ⇒ I L ≡ . 11. ⇒ I L ≡ 12. ⇒ ∀ I c ∈ ]0 , 1[ , I L is a mo del iﬀ it is not a mo del 4. The F LP Counter-Examp le t o NP -completeness: Consider an F LP program (when F LP is mentioned in this pap er, it is mean t as deﬁned in [9 0]). The deﬁnition of a fuzzy atom in F LP is: p ( t 1 , t 2 , . . . , t n , µ ) Where µ ∈ [0 , 1] is the truth constan t. This atom is a classical one despite the w eigh t a ttac hed to it. Consider the F LP program consisting of one f act: Age-Ab out-21( Jo hn,0.9) ← The syn tax of this program constitutes a well-formed formula of classical logic programming. Consider the goal: ← Age- Ab out-21(John, µ ) . This goal succeeds with t w o con tradictory truth v alues, namely “1” and “ 0 .9”. In computation theory terms, this logic prog ram is a T uring mac hine M that co des the input string “ J ohn ” with b oth “Y es” and “No” at the same time. One for the truth v alue µ = “1 ” and the o ther for µ = “0 . 9”, and vice v ersa. In o ther w ords, if the T uring mac hine halts in the q accept state, its tap e sym b ols imply that it is in the q r ej e ct state. On the other hand, if it halts in t he q r ej e ct state, it s tap e sym b ols imply that it is in the q accept state. This is the S y S B P D “ Syn tactico-Seman tical Bi-P olar Disorder” paradox . Since t he ato m in F LP is a classical o ne despite the w eigh t atta c hed to it, it is b oth classical and fuzzy . So, the S y S B P D para do x is due to the fact that:“ p is fuzzy iﬀ p is not fuzzy”, or “ p is t w o- v alued iﬀ p is man y-v a lued” where p is an atom of F LP . T he syn tax/seman tics dic hotomy is bi-p olarity , and the parado x is the undesirable disorder. Theorem 4.1: Let L b e the language deﬁned by the ab ov e progra m, then L is de cid a ble and L ∈ P . Pro of: 11 As in [21], t M ( w ) denotes the n umber of steps in t he computat io n of M on input w , and T M ( n ) the w orst case run time of M : T M ( n ) = max { t M ( w ) | w ∈ Σ n } where Σ n is the set of all strings o v er Σ of length n . Let M b e the T uring machine asso ciated with the one step computation deﬁned ab ov e, clearly: 1. T M ( n ) = m ∈ N. 2. ⇒ t M ( w ) 6 = ∞ . 3. ⇒ L is decidable. 4. T M ( n ) = m ∈ N ⇒ L ∈ P . The computation M on an y input to the a b o v e program c ertainly halts and L ∈ P . Theorem 4.2: SA T is NOT NP-complete. Let L o v er an alphab et Σ b e the language deﬁned by the F LP pr o gram ab ov e, then L can NEVER b e reduced to SA T , hence SA T is not NP-complete. The same pro of of theorem 1.1 a pplies. Pro of: 1. SA T is NP-complete. 2. ⇒ L ≤ p SA T . 3. ⇒ ∃ f : ∀ x ∈ L ⇔ f ( x ) ∈ SA T , [22]. 4. x ∈ L ⇒ ∀ x, x is b oth ac c epte d AND r eje cte d b y M . 5. y ∈ SA T ⇒ ∀ y , y is either ac c epte d OR r eje cte d b y M . 6. ⇒ 6 ∃ f : f ( x ) = y . 7. ⇒ L can NEVER b e reduced to SA T . 8. ⇒ SA T is NOT NP-complete. 5. Wh y SA T is NOT NP-complete Let L b e the following Prolog progr a m consisting of the single-fact: Age − About ( John , 0 . 9 ) ← Running this pro g ram with any ground goal MUST generate contradictory truth v alues: 12 1. Seman tic:“1” , or “0”. 2. Syn tactic:“0.9 ”. There are only tw o p ossibilities that ha ve no third: 1. L has an a sso ciated T uring mac hine M , or: 2. L do es not ha v e one; the coun ter-argumen t that it is imp ossible for a T uring mac hine to diagonalize against itself. Case (1): L has an asso ciated T uring mac hine M 1. SA T Decision Problem: The SA T decision function is R ( F , x ), assigning a truth v a lue x for a Bo olean form ula F . (a) Input: Bo olean F orm ula. (b) Output: “1 ” or “0”. 2. L ∈ LI AR and L ∈ F LP Decision Problems: Both decision problems fo r m a relatio n that is not a function R ( F , x, y ), assigning tw o distinct truth v alues, x (AND) y for an F LP , or LI AR form ula F . One truth v a lue is syn tactic “0.9” written ab o v e in the prog ram. The other is seman tic. Any basic kno wledge of logic is suﬃcien t to view b oth con tradictory truth v alues. (a) Input: F LP or LI AR F ormula. (b) Output: i. “ 1”, F LP seman tical truth-v alue; AND (no t or): ii. “ 0.9”, syn tactical t ruth-v alue; o r “0 ” in the case of the 2 -v alued F LP , i.e. L ∈ LI AR . Those tw o v alues are not only irr econcilable, but also irreducible into a (single) truth v alue. No w, the problem is ho w to write the reduction function f to reduce L to SA T : L ≤ p SAT This is a counter-examp le argumen t that can b e refuted by exp erimen t. If the reader gets angry at this, neither Einstein nor Popper ( i.e. “ T estabilit y”) would. Who ﬁnds himself angry should presen t the reduction function f reducing L to 13 SA T as a refutation to this counte r- example. Ob viously , this counter-example is just a mem b er o f the S y S B P D class ( λ -calculus, F LP , LI AR and p o t entially more) of inﬁnite num b er of languages ha ving the same prop ert y . Case (2): L do es not hav e an asso ciated T uring machine , but L is computable on the von Neumann machine . Then, this is a counte r- example to the Ch urch-T uring thesis. The situation b ecomes: L ∈ P ⇐ ⇒ L 6∈ P 1. L ∈ P , as it is a one-step computation. 2. L 6∈ P , the class P is deﬁned only on T uring mac hines. The claim that L do es not ha ve an asso ciated T uring mac hine should b e prov ed (ph ysically) b y building t he ma chine and demonstrating its incapacit y compared to the von Neumann mac hine, i.e. L is not T uring-computable. The sk eptic should make a public demonstration of a T uring mac hine that he claims to b e capable of computing ev erything in history except the ab ov e example. In other w ords, Prolog is (NOT) programmable on T uring mac hines. Ob viously , the ab ov e program can b e written in all Prolog v ersions. Th us, he ha s to prov e (exp erimen- tally) that PROgramming in LO G ic is imp ossible. A mathematical pro of that a T uring mac hine cannot compute the ab ov e program is irrelev ant to the phys ical phenomenon of computation. It w ould b e certainly intere sting for ev eryb o dy to see this mac hine in public. Of course, not only for the scien t iﬁc comm unity , but for t he whole w orld. Then something m ust b e wrong somewhere. If the T uring mac hine deﬁnition as a tuple in [22] < Σ , Γ , Q, δ > , then the coun ter-a r gumen t that it is imp ossible for a T uring mac hine to diagonalize against itself deﬁnitely a ssumes that the transition δ function may not b e a log ical one. The T uring SySBPD mac hine in tro duced b elo w emphasizes computable lo gical functions by assigning log ical prop erties to δ , i.e. < Σ , Γ , Q, p ( δ, µ ) > . It is eas y to see that if a T uring mac hine cannot risk con tradiction (as claimed ab ov e), then the T uring SySBPD may . Ho w ev er, b oth mac hines a r e equiv a lent with T uring SySBPD emphasis of p ossible logical con tradiction. 6. Another P r o of: This pro o f is entirely indep enden t of T uring mac hines. It is easy to see the p ossibilit y of suc h an approac h since the SA T problem (as w ell a s F LP ) are logical problems that exis t indep enden t of complexit y theory . First, the SA T and F LP decision problems ar e deﬁned, then follo we d b y the pro o f . 14 Deﬁnition 6.1: The SA T Decision Problem. Let F be a SA T formula, the SA T computation on F assigns a function h to F : h ( F ) ∈ { 0 , 1 } , t hus h is a pair. Either h = ( F , 0) or h = ( F , 1). In ot her w ords, input string are co ded either “Y es” or “ No”. Deﬁnition 6.2: The F LP Decision Problem. Let G b e an F LP form ula, the F LP computation o n G assigns a r elation r to G : r ( G ) is a triple r ( G ) = ( G, x, y ) , x ∈ [0 , 1] , y ∈ { 0 , 1 } b oth x, y are non-empt y , x = y only when x, y ∈ { 0 , 1 } , otherwise x 6 = y ; where: 1. x: sy ntactic F LP truth v alue. 2. y: se mantic F LP truth v alue. In this case, input strings are co ded b oth “Y es” and “No”. Theorem 6.1: SA T is (NOT) NP-complete. Pro of: 1. SA T is NP-complete. 2. = ⇒ ∀ L written in F LP , L ∈ P , SA T is NP-complete = ⇒ L ≤ p SA T . 3. L ≤ p SA T = ⇒ r is not a relation, but a function, i.e. when the triple mus t b ecome a pair. 4. r is a relat io n = ⇒ L 6≤ p SA T , con tra-p ositive of 3. 5. r is a relat io n, b y Deﬁnition 6.2. 6. = ⇒ SA T s (NOT) NP-complete. It is easy to see that infnite-v alued F LP is not necessary for the ab o v e res ult and it can b e arriv ed a t via o nly 3 -v alued F LP a s w ell as 2 - v alued F LP , i.e. the system LI AR . The fo llowing section presen ts example progra ms to demonstrate the inv alidity of the counte r- argumen t of the imp ossibilit y to write suc h t yp e of programs. 7. The S y S B P D Class of Counter-Examp les The language L ab o ve constitutes a coun ter- example for the NP-completenes s prop ert y . In fact, there is not only one suc h language but an inﬁnite class of languages, recalling examples in [90] in the contex t of this pap er: Example 7.1 [90]: 15 Mature-Studen t(x, µ ) ← St udent(x),Age-About- 21(x, µ ) Age-Ab out-21( Jo hn,0.9) ← Age-Ab out-21( Peter,0.4) ← Studen t(John) ← Studen t(P eter) ← Here, w e ha v e three predicate sym b ols, namely , Studen t, Mature-Studen t and Age-Ab out-21. The n- ary predicate sym b ol b ecomes an n-a ry+1 if t he pr edicate is a fuzzy one. This is to allo w fo r the µ indicating the membership v alue. Ob viously , Mature-Studen t and Age-Ab out-21 ar e fuzzy predicates. No w, w e consider the go al ← Mature-Student(John, µ ). This will unify the head of the ﬁrst rule with uniﬁcation (x = John, µ = µ ). Th us, resulting into tw o subgoals, the ﬁrst Studen t(John) whic h succeeds. The other subgoal is Age-Ab out- 21(John, µ ) whic h succeeds with the v alue µ = 0 . 9 f or John. It is ob vious that the predicate Mature-Studen t leads to the same S y S B P D paradox as the Age-Ab out- 21 did ab ov e. Example 7.2 [90]: P oten tial- Customer(x, µ 1 ) ← Customer(x), µ 1 ≥ 0 . 7 T op-P otential-Customer(x , µ 2 ) ← Customer(x), µ 2 ≥ 0 . 9 Go o d- Credit-Customer(x, µ 3 ) ← Balance-leve l(x,y , µ 3 ) , µ 3 ≥ 0 . 7 Customer(John) ← Balance-Lev el(John,400,0.7) ← Customer(Ric hard) ← Balance-Lev el(Ric hard,500,0 .8) ← Consider the g oal ← Go o d-Credit-Customer(Ric hard, µ ) It is o b vious that the predicate Go o d- Credit-Customer leads to the same S y S B P D parado x as the Age-Ab out-2 1 did ab ov e. Example 7.3 [90]: R1: p ( x, y , µ p 1 ) ← q ( x, µ q 1 ) , r ( y , µ r ) R2: p ( x, y , µ p 2 ) ← q ( x, µ q 2 ) , s ( y , µ s ) R3: q ( m, 0 . 3) ← R4: r ( x, µ r ) ← t ( x, µ t ) R5: s ( n, 1) ← R6: t ( n, 0 . 4) ← Consider the fuzzy goal ← p ( m, n, 0 . 3) whic h uniﬁes with the ﬁrst fuzzy rule giving the t wo fuzzy sub-go a ls, where the success of eac h leads to the S y S B P D parado x: 1. ← q ( m, µ q 1 ) , µ q 1 ≥ 0 . 3 , 16 2. ← r ( n, µ r ) , µ r ≥ 0 . 3 . The fuzzy subgoal (1) uniﬁes with R3 and succeeds while the second fuzzy subgoal uniﬁes with R4 and results with another t wo fuzzy subgoals with the second b eing µ r ≥ 0 . 3 resulting in the g oal ← ( t, 0 . 3) whic h succeeds when unifying with R6. As a result, the original goal ← p ( m, n, 0 . 3) succeeds as far as matc hing with rule R1 is considered. When matching with rule R2, t w o fuzzy subgoals are generated, they are (where the success of eac h - again - leads to the S y S B P D pa r a do x - and this situatio n recurs): 1. ← q ( m, µ q 2 ) , µ q 2 ≥ 0 . 3 , 2. ← s ( n, µ s ) , µ s ≥ 0 . 3 . The ﬁr st succes sfully match es with R3 and the second as w ell with R5. So, the original fuzzy goal succeeds in this case. No w consider the fuzzy go al ← p ( m, n, 0 . 2 ) when matc hing with R1, t w o fuzzy subgoals are generated, namely: 1. ← q ( m, µ q 1 ) , µ q 1 ≥ 0 . 2 , 2. ← r ( n, µ r ) , µ r ≥ 0 . 2 . The ﬁrst fuzzy subgoal of (1) ← q ( m, µ q 1 ) uniﬁes with R 3 giving µ q = 0 . 3 and a s a result the second fuzzy subgoal µ q ≥ 0 . 2 succeed s. F or the second fuzzy subgoal ← r ( n, µ r ) , µ r ≥ 0 . 2, w e ha ve o nly rule R4 whic h uniﬁes successfully resulting in the goal ← ( t, 0 . 2) which succeeds when unifying with R6. As a result, the original fuzzy goal ← p ( m, n, 0 . 2) succeeds. When matc hing with R 2, tw o fuzzy subgoals are generated, namely: 1. ← q ( m, µ q ) , µ q ≥ 0 . 2 , 2. ← s ( n, µ s ) , µ s ≥ 0 . 2 . The ﬁrst subgoal matc hes with R3 and succeeds. T he second fuzzy subgoal matc hes with R5 and succeeds . No w consider a fuzzy goal with a v ariable µ , i.e. ← p ( m, n, µ ), matc hing with R1, we get: 1. ← q ( m, µ q ) , µ q ≥ µ , 2. ← r ( n, µ r ) , µ r ≥ µ . The ﬁrst matche s with R3 a nd µ q = 0 . 3, thus solving µ ≤ 0 . 3. T he second will unify with rule R4 then rule R6 returning µ ≤ 0 . 4. The or ig inal goal succeeds with ( µ ≤ 0 . 3 ) ∧ ( µ ≤ 0 . 4). Th us µ ≤ 0 . 3 . When matchin g with rule R2, t w o fuzzy subgoals are generated: 17 1. ← q ( m, µ q ) , µ q ≥ µ , 2. ← s ( n, µ s ) , µ s ≥ µ . The ﬁrst matc hes with R3 giving µ ≤ 0 . 3. The second matc hes with R5 giving µ ≤ 1. The original goal succeeds with [( µ ≤ 0 . 3 ) ∧ ( µ ≤ 1 ) ] ∨ [( µ ≤ 0 . 3 ) ∧ ( µ ≤ 0 . 4)]. Th us, µ ≤ 0 . 3. Th us, the S y S B P D par ado x is generated and re- g enerated in this simple prog ram. 8. SySBPD Implemen ted: 8.1 An F LP Meta-Interpreter: Sun- Unix (IC-Prolog) In this section, a meta-inte rpreter is presen ted to the S y S B P D class. The meta- in terpreter is implemen ted in IC-Prolog . Giv en the rule: ← < q ( x ) , µ q 1 > . It can b e read declarativ ely or pro cedurally: 1. The declarativ e reading states that: f o r a certain v alue o f the v ar iable x , p 1 should b e true t o a lev el µ p 1 ≥ µ q 1 . 2. The pro cedural reading states that: for a fuzzy g oal ← to succeed, the fuzzy subgoal ← < q ( m ) , 0 . 3 > m ust succeed. F urther, for the fuzzy goal ← , the f uzzy sub-go al q ( m, 0 . 4) m ust succeed. So, as fa r as execution is concerned, b oth v alues of µ are instan tiated in the fuzzy rule with the same constan t lev el in the go al and then attempt succeeding the fuzzy sub-goal. Then, using the meta-interpreter, the rule is rewritten as follows: R 1 : ← < q 1 ( x ) , µ q 1 > as R 1 ′ : p 1( X , M p 1) : − q ( X , M p 1) . No w, consider the fuzzy g oal ← , where V is a v ariable. Now , the s ystem is q ueried to what maxim um lev el this fuzzy goal can b e satis- ﬁed. T his is done via the meta-in terpreter predicate solv e ( A ) whic h b ecomes ← sol v e ( p 1 ( m, V )) . The system predicates functor and ar g are used. When rewriting the fuzzy logic programs in IC-Prolog or standard Prolog , care should b e taken as the semantics asso ciated with fuzzy logic programs are diﬀer- en t than that of standard Prolog. F or instance, give n the fact < q ( m ) , 0 . 3 > ← , in fuzzy logic prog ramming, it is considered as a fuzzy fact. q is said to b e true to a lev el µ where 0 < µ ≤ 0 . 3 . In standard Prolo g, the go a l ← q ( m, 0 . 25 ) would return the answ er “No”. So, to write a fuzzy fact in Prolo g , it should b e written as: q ( m, M q ) : − ( M q ≤ 0 . 3) , ( M q > 0) 18 During execution within the Prolog mo del, the answ ers conform t o the give n seman tics. No w, the extended rules are extended with a factor f ∈ [0 , 1] doubting the rule: ← (0 . 9) − < q ( x ) , µ q > . F or the g o al ← to succeed, the fuzzy goal ← q ( x, µ q ) mus t succeed at least with the v alue 0 . 3 / 0 . 9. T o do this in standard Prolog, t he fuzzy fact and the fuzzy r ule ar e rewritten as follows : p 1 ( X , M p 1) : − q ( X , M p 1) . q ( m, M q ) : − ( M q ≤ 0 . 3 / 0 . 9) , ( M q > 0) . whic h will lead to the in tended meaning. No w, if the predicate q happ ens to b e in the b o dy of t w o fuzzy rules with diﬀeren t f factors, a diﬀeren t rewriting of the facts is required. F or instance, one obtains the follo wing tw o rules and t w o facts: R 1 : ← (0 . 9) − < q ( x ) , µ q > R 2 : ← (0 . 7) − < q ( x ) , µ q >, < s ( Y ) , µ s > . F act 1 : < q ( m ) , 0 . 3 > ← F act 2 : < s ( n ) , 0 . 4 > ← If this f uzzy logic program is rewritten in Prolog, one g ets: R 1 ′ : p 1( X , M p 1) : − q ( X , M p 1) . R 2 ′ : p 2( X , Y , M p 2) : − q ( X , M p 2) , s ( Y , M p 2) . and the t w o fuzzy facts: F act 1 ′ : q ( m, M q ) : − ( M q ≤ 0 . 3 / 0 . 9) , ( M q > 0) F act 2 ′ : s ( n, M s ) : − ( M s ≤ 0 . 4 / 0 . 7) , ( M s > 0) . If a fuzzy goal matc hes with R 1 ′ , then F act 1 ′ , this w ould b e ﬁne. But if a fuzzy g oal matc hes with R 2 ′ , the q fuzzy subgoal m ust ha v e f = 0 . 7 not 0 . 9. Th us, g iv en the same predicate o ccurring in the bo dy of t w o fuzzy rules with diﬀeren t f factors, it should b e renamed when rewriting. As a result, the predicate q is renamed in R 2 to h , and one obtains t w o fuzzy facts F act 1 ′ and F act 2 ′′ corresp onding to F act 1 in the origina l program: R 1 ′ : p 1 ( X , M p 1) : − q ( X , M p 1) . R 2 ′ : p 2 ( X , Y , M p 2 ) : − h ( X , M p 2) , s ( Y , M p 2) . F act 1 ′ : q ( m, M q ) : − ( M q ≤ 0 . 3 / 0 . 9) , ( M q > 0) . F act 1 ′′ : h ( m, M h ) : − ( M h ≤ 0 . 3 / 0 . 7) , ( M h > 0 ) . F act 2 ′ : s ( n, M s ) : − ( M s ≤ 0 . 4 / 0 . 7) , ( M s > 0) . In the following, a co de listing for the meta-in terpreter is presen ted and a rewrit- ten fuzzy log ic progra m in IC-Prolog that w as tested with the results exp ected from the seman tics for f uzzy logic programming. The [0 , 1 ] in terv al has b een assumed a s [0 ,1 00], i.e. one hundre d incremen ts. p1(X,Mp1):- q(X,Mp1). p2(X,Y,Mp2):-q(X,Mp2),s(Y,Mp2). 19 p3(X,Y,Z,Mp3):- s(Y,Mp3),t(Z,Mp3),q(X,Mp3). q(m,Mp2):-(Mp2= < 4/9),(Mp2 > 0). s(n,Mpr):-(Mpr= < 3/7),(Mpr > 0). t(l,Mpr):-(Mpr= < 1/2),(Mpr > 0 ) . solv e(A,0). solv e(A,X) :- X > 0 , functor(A,F ,N) ,F =A,arg(N,A,H),v ar(H),a rg(N,A,X),A,!. solv e(A,X) :- X > 0, Z is X - 1, solve (A,Z). solv e(A) :- solv e(A,100). n t(A):-solve (A),functor(A,F ,N) ,a rg(N,A,H),Y is 100 -H,write(Y). The solv e predicate ﬁnds the threshold if the goal contained v a r iables. F or a negated g oal not con taining v a riables the built-in not predicate w ould pro duce the right answ er. If the negated goa l con tained v ariables, the nt predicate ab ov e giv es the threshold. 8.2 Meta-In terpreter: PC:Win-Prolog The following a re three clauses whic h f orm the progr am in question. The meta- in terpreter will run in conjunction with this prog ram. This program can be c hanged a nd edited eac h run while the meta-inte rpreter is re-usable across diﬀer- en t programs. p1(X,Mp1):- q(X,Mp1). p2(X,Y,Mp2):-q(X,Mp2),s(Y,Mp2). p3(X,Y,Z,Mp3):- s(Y,Mp3),t(Z,Mp3),q(X,Mp3). The following are clauses to establish the allow able ranges for truth v alues in a Prolog syntax. q(m,Mp2):-(Mp2= < 4/9),(Mp2 > 0). s(n,Mpr):-(Mpr= < 3/7),(Mpr > 0). t(l,Mpr):-(Mpr= < 1/2),(Mpr > 0 ) . Here starts the meta-interprete r: three diﬀerent predicates: solv e(A) uniary predicate, solv e(A,X) binary predicate and n t(A) Base predicate to pass t he v alue of zero lev el without at t empting recursiv e calls solv e(A,0). Base predicate to pass v alues greater than zero solv e(A,X) :- X > 0 , functor(A,F ,N) ,F =A,arg(N,A,H),v ar(H),a rg(N,A,X),A,!. Recursiv e calls to determine the exact levels solv e(A,X) :- X > 0, Z is X - 1, solve (A,Z). Initial run of the goa l uniﬁes with this clause head solv e(A) :- solv e(A,100). T o pro duce results for a negated goal: n t(A):-solve (A),functor(A,F ,N) ,a rg(N,A,H),Y is 100 -H,write(Y). 9. On t he a voida bility of t he Complexity Class:“SySBPD”: 20 Ov erall, the SySBPD la ng uages w ould constitute a r e duction obstruction . O b vi- ously , reduction is of cen tra l impo rtance in computability and complexit y the- ories. T his new complexit y class SySB PD w ould ov erlap virtually ev ery com- plexit y class. Its eﬀect is not conﬁned to obstructing reduction only . It w ould propagate to many results of descriptiv e complexit y . F ag in’s theorem [36] as w ell as the Immermann-V ardi theorem [8 9,110] are examined after t he disco v ery of this class. How ev er, the second F LP paradox (app earing when considering the cardinality of the v alid formulas of the underlying paradoxic al system) has far-reac hing implications in mathematics outside complexity theory . It turns out that this pa r ado x *pro v es* the existence of a transﬁnite cardinal, hence the “Con- tin uum Hyp othesis” & the “Axiom of Choice” are fa lse and ZF C is inconsisten t [92]. Clearly , this inconsistency result aﬀects all of mathematics a nd mathemat- ical disciplines: ph ysics, computer science, etc, a part from inconsistency results due to NP-completeness a nd descriptiv e complexit y . In f uzzy lo g ic applicatio ns, clearly the para doxical feature of F LP is undesirable. P erhaps tha t system w as nev er a do pted, unles s f r o m a practical p oin t-of - view. Theoretically , it is certainly paradoxic al. Practically , the meta-inte rpreter pre- sen ted ab o ve could b e used in “fuzzy expert systems” without an y problems. Moreo v er, f uzzy logic programming can compute ev en more computable func- tions. How ev er, its theoretical para do x is av oidable, see [91] and other F LP systems in the references b elow . Nev ertheless, this paradoxical class of languages is ( unav o idable) in complexit y theory . The reason is t ha t complexit y theory is a theory that studies the computational complexit y of classes of inﬁnite num b er o f languages. So, ev en if F LP is ignored, it do es not mean that it do es not exist. Moreo v er, it has b een demonstrated that the 2- v alued F LP parado x is precisely the liar’s parado x whic h is inevitable in natural languag es. Perhaps mo r e inter- estingly , the paradoxical F LP relation o ccurs in nature. It reconceptualizes the relation b etw een space and time ma king a quan tum theory of g ra vit y p ossible, the long outstanding question of t heoretical ph ysics [93]. As suc h, a substan tial class of paradox ical languages do es exist within the robust class P . One has the four new computat io nal complexit y classes: 1. P S y s = { L : L ∈ P ∩ S y S B P D } 2. P N onS y s = { L : L ∈ P , L 6∈ S y S B P D } 3. N P S y s = { L : L ∈ N P ∩ S y S B P D } 4. N P N onS y s = { L : L ∈ N P , L 6∈ S y S B P D } Related to the conv en tional P and NP as follo ws: 1. P = P S y s ∪ P N onS y s , P S y s ∩ P N onS y s = ∅ 21 2. NP = N P S y s ∪ N P N onS y s , N P S y s ∩ N P N onS y s = ∅ The NP-completeness prop ert y f o r SA T could b e r evised in the ligh t of the disco v- ery o f the new class as t he la nguage whic h is complete to the new computat io nal complexit y class NP N onS y s : Empirical Observ ation: SA T is NP N onS y s -complete. ∀ L ∈ N P N onS y s L ≤ p SA T = ⇒ SA T is NP N onS y s -complete. Other complete languages for other classes should b e appropriately mo diﬁed to exclude an y language in the S y S B P D class, as it cannot b e reduced to suc h a complete language. F o r instance HP is (NOT) c.e.-complete with similar con- siderations. Answ ering an y of the fo llo wing questions, answ ers the P =? NP question: 1. P S y s =? NP S y s . 2. P N onS y s =? NP N onS y s ; the o ld question. Observ at ion: SA T ∈ P N onS y s = ⇒ P N onS y s = NP N onS y s . 10. Descriptiv e Complexit y: F undamen tal results of descriptiv e complexit y mus t b e examine d against the S y S B P D computational parado xes. Similar argumen ts as to NP-completeness can b e demonstrated as b elow. 10.1 F agin’s theorem [36]: NP = SO ∃ . Theorem 10.1: NP 6 = SO ∃ . Pro of: Let L b e a one step paradox ical F LP computatio n a s ab ov e. 1. L ∈ P . 2. L ∈ NP . 3. L 6∈ SO ∃ , L ∈ SO ∃ ⇐ ⇒6 ∃ p ( t 1 , t 2 , . . . , t n , µ ) ∈ L . 4. NP 6 = SO ∃ Observ at ion: N P N onS y s = SO ∃ − FLP By the nota tion SO ∃ − FLP , it is mean t that atoms of the for m p ( t 1 , t 2 , . . . , t n , µ ) , 22 µ ∈ [0 , 1] are forbidden, i.e. only purely classical atoms. This observ at io n is a restatemen t of the old result excluding parado xical S y S B P D languages. NP S y s =? SO ∃ remains an op en question. Immermann-V a rdi theorem [89,110]: P = FO+LF P . Theorem 10.2: P 6 = FO+ LFP . Pro of: Let L b e a one step paradox ical F LP computatio n a s ab ov e. 1. L ∈ P . 2. L 6∈ F O+LF P , L ∈ F O+LFP ⇐ ⇒6 ∃ p ( t 1 , t 2 , . . . , t n , µ ) ∈ L . 3. P 6 = F O+LFP Observ at ion: P N onS y s = [FO - FLP]+LFP . The question P S y s =? F O+LFP remains op en. Similar a rgumen ts hold for o ther descriptiv e complex ity results o v er the computational complexit y hierarc hy . Theorem 10.3: ZF C is inconsisten t. Pro of: [Co ok’s Theorem [21] V Theorems 1.1, 4.2, 6 .1] W [F agin’s Theorem V Theorem 10.1] W [Immermann-V ardi Theorem V Theorem 10.2] = ⇒ ZF C is inconsisten t SySBPD Impli cations The P v ersus NP Problem The problem certainly su rvive s the S y S B P D class o f coun ter-examples to the NP-completeness prop erty . Ho w ev er, a p olynomial-time algorithm for SA T no longer implies P = NP . Nor t he non-existence of suc h an alg orithm w ould imply P 6 = N P . In its basic informal deﬁnition:“Whether easy recognition of a solutio n implies easy ﬁnding one”, t he problems surviv es a s it alwa ys had b een. How ev er, the precise deﬁnition of the class P is divided into t w o (disjoin t) classes P S y S B P D and P N onS y S B P D , written simply as P S y S and P N onS y S : 23 P = { L | L = L ( M ) f o r some T uring mac hine M whic h runs in p olynomial time } P = P S y S ∪ P N on S y S P S y S = { L | L = L [ M ] for some T uring machin e M whic h runs in p o lynomial time } , where L [ M ] denotes M accepts L iﬀ M rejects it. P N onS y S = { L | L = L ( M ) for some T uring mac hine M whic h runs in p olynomial time } , where L ( M ) denotes M accepts L and strictly do es no t reject it. The S y S B P D could hav e members across the entire arithmetic hierarc h y . Since the fuzzy logic programs [90] are classical, they hav e the complete T uring hierar- c h y computat io nal capability . The usual hierarc hy nicely presen ted in [87] MUST b e augmen ted with the class S y S B P D , resulting in lots o f class separation ques- tions. It is ob vious that the counter-example to t he NP-completeness prop erty is also a coun ter-example t o c.e.-completeness. The same pro of ab o ve sho wing SA T not to b e NP-complete can b e used to pro v e tha t HP is NOT c.e.-complete. Ho w ev er, it is undecidable. The new complex ity hierarc hy - incorp o rating the S y S B P D class describ es computable languag es on the T uring SySBPD ma- c hine. While the T uring machine had only tw o halting states: q accept and q r ej e ct , the T uring SySBPD mac hine is a T uring mac hine that has the fo llo wing halting states: 1. q accept : M halt s in q accept and only q accept , i.e. no paradoxical halting. 2. q r ej e ct : M halts in q r ej e ct and only q r ej e ct , i.e. no paradoxical halting. 3. q S y S B P D . M halts in the state q S y S B P D when it halts in q accept iﬀ it halts in q r ej e ct , i.e. M halts p ar adox i c al ly . The a b o v e (review ed) deﬁnition of the class P is on the T uring SySBPD ma- c hine. 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European Conference on Artiﬁcial In telligence 98, Univ ersit y of Brig h ton, 1998, 7 pages 115. P . V o jt´ as. L. P aulak:“Soundness and completeness of non-classical ex- tended SLD- resolution”, in Pro c. ELP’96 Extended lo g ic programming, Leipzig, ed. R. D yc khoﬀ et al., Lecture Notes in Comp. Sci. 1050 Springer V erlag, 1996, 289-301. 116. P . V o jt ´ as, M. V omlelo v´ a:“T ransfor mat ion of deductiv e a nd inductiv e tasks b et w een mo dels of logic pro gramming with imp erfect inf o rmation”, In Proc. IPMU 2004, B. Bouc hon- Meunier et al. eds. Editrice Univ ersita La Sapienza, Roma, 2 004, 839-846 33 A Spatio-T emp or a l Bi-P olar Disorder Quan tum Theory of Gra vit y A F uzzy Logic Programm ing Reconciliation S y S B P D ⇐ ⇒ S p T B P D Rafee Ebrahim Kamouna What is Gravity? Imeptuous Fire, Space-Tempo ral! Ice and Desire, The Universe wag s on.. . [Einstein ` a la “R ome o & Juliet”] What is a Turing machine? Imeptuous Fire, Syntactico- Semantical! Ice and Desire, F uzzy L o gic Pr o gr ammin g goes on... [Einstein me ets T uring] Abstract A theory of quan tum gravit y founded on fuzzy logic programming F LP [1] is presen ted. The connection b etw een space and time of general relativit y is re- examined from a logical p oin t-of- view. A one-to-one corresp ondence b etw een the space/time dic ho tom y and syn tax/seman tics of logic w as disco v ered. The Syn tactico-Seman tical Bi-P olar Disorder nature of F LP ( S y S B P D ) naturally expresse s the space/time relationship as w ell a s unifying it with quan tum me- c hanics particle/an ti-particle dic hoto m y . The Spatio-T emp oral Bi-P olar D isor- der ( S pT B P D ) theory mak es new predictions that can b e tested by exp erimen t, form ulates new hy p ot heses as w ell as shedding ligh t on previously unexplained observ ed phenomena, e.g. “CP violatio n” and the 720 degrees instead o f 360 for an electron t o return to its state. In tro duction: Einstein’s general relativit y is the most accepted theory of gravit y conﬁrmed by exp eriments and observ ations. It is mathematically expressed as tensor equations whose solution is Loren tzian manifolds of curved spacetime (R iemannian/ Pseudo- Riemannian space). Dirac’s eq uatio n is the experimen tally-ve riﬁed relativistic 34 quan tum mec hanics theory that succes sfully uniﬁed quan tum mec hanics and sp e- cial relat ivity (ﬂat spacetime - Mink ows ki space) whose solution is a w av e func- tion. Establishing a theory of quan tum gravit y remains (undoubtedly) as the theoretical ph ysics outstanding problem fo r decades [3]. The standard mo del of particle physic s uniﬁed all nature fundamen tal forces except g ra vit y . Ha ving a uniﬁed theory of all f undamental fo rces of nature is obv iously a goa l lo ngtime sough t a f ter. Einstein had famously spent quite a long time in searc h for a “Uniﬁed Field Theory”. This pap er presen ts a f uzzy lo gic p r o gr amm ing (FLP) reconciliation of the theory of g eneral relativity (Einstein’s ﬁeld equations) and relativistic quan tum mechanic s (Dirac’s equation). This problem can b e formu- lated as: “If general relativit y regards gra vit y as spacetime and quantum mechan- ics provide s a wa ve function (Ψ - Dira c’s equation) ev olution in time, it seems imp ossible for those t wo theories to b e (mathematically) uniﬁed. A t t empts in- clude unsuccessful p erturbativ e quantum gravit y , string theories culminating in M-Theory but with neither exp erimen tal results nor observ ations [3]. It w as found that the language of “F uzzy Logic Programming F LP ” [1] can natur al ly do the job. This is done via re-examination of the logical relationship b et w een space and time. The discov ery of a one-to- one corresp ondence b etw een the space/time dic hotomy and that of syn tax/seman tics made F LP a naturally app ealing candidate for this intractable reconciliation; S y S B P D vs. S pT B P D . Philosophical F oundation: E = mc 2 implies that E and m are diﬀeren t (manifestations) “essences” of the same “exis tence”. The “Principalit y” of the existence o v er the essence should b e ob vious. There can b e no “Principalit y” for E o v er m nor vice v ersa. This can b e called “Energy/Mass” Principalit y Bi-P olar Disorder; to render the term c onn o tating and its meaning connotated to! More imp orta n tly , the lessons of general relat ivity dictate t ha t if t he Earth gets in to the ev en t horizon of a black hole, space and time w o uld sw ap po si- tions. This implies that space and time are diﬀerent manifestations “essences” of the same “existence ”. It should b e self-eviden t that there is no “Principalit y” out w eigher for neither space nor time o v er one another. This is the “Space- time/Timespace Principality Bi- P olar D isorder”. S pT B P D exploits R E , m ( E , m ) vs. R S p T ( space, time ), where R E , m ( E , m ) means energy a nd mass can sw ap p o- sitions in sp ecial relativit y and R S p T ( space, time ) means space and time swap p ositions in general relativit y . It is easy to see that space and time are a lwa ys sw a pping p o sitions but only com- pletely within a blac k ho le. Solutions to Einstein’s ﬁeld equations a r e spacetimes whic h are Lorentz ian manifolds. The tangen t v ector at an y p o int in the ma nif o ld 35 is classiﬁed as spacelik e (sw apping p ositions spacewise ) or timelike (sw apping p o- sitions t imewise) according to the negative /p ositiv e v alue of the manifold’s metric [4]. If ( M , g ) is a Lo ren tzian manifold (so g is the metric on the ma nif o ld M ) then the tangent vec tor s at each p o in t in the manifold can b e classed into three diﬀeren t types. A tangen t v ector X is: 1. timelik e if g ( X , X ) > 0 2. n ull if g ( X , X ) = 0 3. spaceli ke if g ( X , X ) < 0. General relativit y [5] is understo o d as spacetime tells matter how to mov e, then in S pT B P D so should timespace. And if matter tells spacetime how to curve in general relativit y , then in S pT B P D it should tell timespace to o. The diﬀerence b et w een spacetime and timespace: 1. F or a single observ er at one p oint, they are iden tical. 2. F or tw o observ ers A, B at tw o diﬀeren t lo cations X , Y , w e hav e: Spacetime(A,X) = Timespace(B,Y) or Timespace(A,X) = Spacetime(B,Y) That is to say , they are recipro cal. This is a corolla r y of the Space/Time Prin- cipalit y Bi-Polar Disorder. So, S pT B P D Quantum Theory of Grav ity regards gra vity as r ecipro cal spacetime/timespace and quan tum mec hanics as recipro cal w a ve functions Ψ-(particle)/Ψ B P D -an ti- pa rticle. S pT B P D regards spacetime ge- ometry as giv en b y the Einstein F ield Equations is a result of a fermion spin-lik e angular mo t io n (in a Hilb ert space) of ﬂat spacetime and ﬂat timespace. This w ould justify the dynamic spacetime geometry . This philosophical in t erpreta- tion of spacetime geometry could extend ( p oten tially r econciling) v on Neumann mathematical f o undations of quantum mec hanics to general relativit y . This is a new hy p ot hesis. Another one is quantum in terpretation of the Big Bang as well as the expansion of the univ erse. This is due t o the particle/an ti-pa r t icle view of spacetime/times pace. Spacetime (in-order) is iden tical to timespace (disor- der). The diﬀerence is a matter of state. The Pauli exculsion principle could b e extended from quan tum mec hanics t o gravit y in S pT B P D . Mathematical F orm ulation: S y S B P D vs. S pT B P D The followin g equations relate tw o solutions of Einstein’s equations with a nother t w o of D irac’s. Einste in’s solutions are t w o B P D - conformally related Lo ren tzian 36 manifolds (as deﬁned b elow ). Bot h are related by a S y S B P D F LP predic ate Gr av ity . Dirac’s tw o solutions are tw o w a ve functions, for particles and anti- particles. Both are relat ed by another S y S B P D F LP predicate Quantum . The t w o manifolds with b oth wa v e functions a r e related b y κ , the Univ erse Bi-P olar Disorder Constan t, the prediction of S pT B P D . 1. E q ual ( µ Ψ , | Ψ | − | Ψ B P D | ) ← Quantum (Ψ , Ψ B P D , µ Ψ ) 2. E q ual ( µ g r av ity , κ.µ Ψ ) ← Gr av ity ( Lor entz , Lor entz B P D , µ g r av ity ). Where Gr av ity and Quantum are tertiary F LP predicates as in [1], and E q ual is a binary non-f uzzy predicate whose meaning is obvious . Lor entz a nd Lor entz B P D are any tw o B P D -confo rmally related Loren t zian solutions of Einstein’s equa- tions. Let g b e t he Lor entz ian manifold metric and ˆ g Lor entz B P D manifold metric, they are conformally relat ed if ˆ g = Ω 2 g (standard deﬁnition [3]) and B P D -conformally related if ˆ g = − Ω 2 g (new deﬁnition). So, µ g r av ity = Ω 2 . F rom [3], and fo r the pap er to b e self-con tained: “Tw o metrics g and ˆ g are conformally related if ˆ g = Ω 2 g for some real function Ω called the conformal factor . Loo king at the deﬁnitions of whic h tangen t v ectors are timelik e, null and spacelik e w e see they remain unc hanged if w e use g or ˆ g . As an example supp ose X is a timelik e tangen t v ector with respect to the g metric. This means t hat g ( X , X ) > 0. W e then hav e that ˆ g ( X , X ) = Ω 2 g ( X , X ) > 0, so X is a timelike tangen t v ector with resp ect to the ˆ g to o. It follows from t his that the causal structure of a Lorentzian manifold is unaﬀected b y a conformal transformation.” A solution to Dira c’s equation is the w av e function Ψ associated with the quan tum system particles and Ψ B P D is the w av e f unction asso ciated with the corresp onding an ti-part icles. It is to b e noted that the ab ov e F LP rules/equations a r e not equiv alen t to their algebraic coun terpart: Ω 2 = κ .µ Ψ = κ . | Ψ − Ψ B P D | The in terpretation of the predicate Gr av ity ( Lor entz , Lor entz B P D , µ g r av ity ) is t hat for t wo B P D -confo rmally r elated spacetimes, for them to b e i n -or der they hav e to b e in disor der . Non-fuzzy Gr av ity implies iden tical manifolds while fuzzy Gr av ity a dmits diﬀeren t ma nif o lds. So, “ p is fuzzy iﬀ p is no t fuzzy” reads dy- namic p erp etual oscillations (grav itatio nal wa ve s) of spacetime. These w av es are con tin uous and p erp etual and obvious ly m uc h easier to phra se logically . They con tin ue like this perp etually as the lessons of general relativit y dictate a dynamic geometry o f spacetime. Space and time lose order the more the sp eed approac hes 37 sp eed of ligh t when they sw a p p ositions. S y S B P D is expres sed as “ p is f uzzy iﬀ p is fuzzy”, where p as in [1]: p ( t 1 , t 2 , . . . , t n , µ ). Or, “ p is an atom of classical logic iﬀ it is not an a t om of classical log ic”. This is the S pT B P D / S y S B P D space/time vs. syn tax/seman tics mathematically represen ting the dynamic nature of space- time a s w ell as unifying it with quan tum mec hanics. κ is t he “Bi- P olar Disorder Univ ersal Constant” whic h can b e observ ed b y exp erimen ts relating gr avitational waves to quantum ones. Where the F LP equations ab ov e predict a Bi-Polar D isorder dic hotomy r a ther than the usual symmetry interpretation. S pT B P naturally explains “CP viola- tion” a s w ell as the 72 0 degrees for a n electron to return to its state rather than 360 degrees ( either in-or der state or disor der , thus Bi-Polar Disorder). This explanation cannot b e prov ided b y the algebraic equation whic h only gives the mathematical prediction. T he meaning of this form ula that the tw o Lorentz manifolds (perp etually) o scillate betw een tw o states. Once they are iden tical (classical logic progra mming), the other with the deﬁcit µ g r av ity (FLP). It is im- p ossible to form ulate this sort of oscillation as a w a v e against time; as usual in ph ysics. This new lo gic al form ulatio n resolv es the problem. Lorentz and Lor entz B P D metrics describ e t w o curv ed spacetimes (spacetime and timespace) in B P D , t h us (p oten tially) explaining ripples in spacetime geometry . The ﬁnal p ar adox is that S y S B P D is highly undesirable for computer science w ouldn’t b e at a ll f o r phy sics. Discussion & Conclusion: The follo wing questions are addressed: 1. Can S pT B P D b e tested by exp erimen t? 2. Do es it mak e new predictions? 3. Do es it generate new hypotheses? 4. Do es S pT B P D provide new explanations for strange o bserv ations? 5. S y S B P D implications to ph ysics is S pT B P D compared to la te a w aken ing to G o del’s Incompletenes s Theorem [19 3 0-2002!!!]; in [2]. 6. Is it a prop osal f or ﬁnal theory? No theory is considered to b e ap o dictically true unless supp orted by exp erimen- tal results a nd observ a tions. S pT B P D is f o unded on general relativity and rel- ativistic quan tum mec hanics. So it is o bvious that t he prediction of a “Univ erse Bi-P olar D isorder Constant κ ” relat ing spacetime/timespace fro m one side to Ψ (particle)/Ψ ( an ti-particle) from the other can b e tested b y exp erimen t. In addi- tion, sev eral new h yp otheses/new explanations hav e b een (naturally) generated: 38 1. S pT B P D regards grav itat io nal w av es (spacetime/timespace Bi-Polar Dis- order) in curv ed space time a s a result of f ermion spin-like angular motion (in a Hilb ert space) of ﬂat spacetime and ﬂat timespace. 2. The mathematical foundatio ns of quan tum mec hanics (Hilbert spaces) could b e uniﬁed with that of general relativity . The view of t wo Lorentzian man- ifolds in Bi-P ola r Disorder ( B P D - confo rmally related) can b e restricted to t w o ﬂat spacetime/timespace a t quantum lev el. So, space/time dic hotomy (spacetime vs. timespace) a t b oth the sup er-galactic and the sub-a t o mic lev els. 3. A quan tum in t erpretation of the Big Bang and t he expansion of the univ erse due to the new dic hot o m y of par t icle/anti-particle vs. spacetime/timespace. 4. Exte nding the P auli exclusion principle from the sub-atomic lev el to the sup er-galactic. 5. When the ( p o or) autho r ﬁrst learnt of the 72 0 degrees for a n electron to return to its original state, it w as no surprise unlik e many others. This is a natural S pT B P D quantum view of in-or der and disor d er states. 6. When the (p o or) author learn t that “CP violation” is not complete sym- metry , he felt that w as absolutely nor ma l and consisten t with S pT B P D . S pT B P D predicts Bi-P olar Disorder in natur e rather than symmetry . But in order to main tain the or der, the re has to b e disorder (from the Big Bang to the expansion o f the Unive rse), resulting whenev er the symmetry attempts t o b ecome complete, it could recur somewhere else incomplete. The answe rs are p ositiv e for S pT B P D : It mak es a new prediction κ that can b e found b y exp eriment and it provides new h yp otheses. Not only this, but also it pro vides (natural) and consisten t explanations for unexplained phenomenon. The story ab o ut t he implications of the celebrated G ¨ o del’s Incompleteness theorem is indeed a sad one as detailed in the pa p er b y Rev erend F ather Professor Stanley L. Jaki [2]. A more bizarre story is exp ected for fuzzy lo gic pr o gr am ming where new dic hotomies ha ve b een iden tiﬁed and mathematically related to well-es tablished ones: Syn tax/Seman tics vs. Spacetime/Timespace vs. Particle/An ti- particle vs. W a v e/Pa rt icle Bi-Polar Disorder , not W av e/P article dualit y! Whether it is a prop osal fo r a ﬁnal theory , the answ er is simply “No”. In [2], it has b een conﬁrmed that ev en after considering G ¨ o del’s Incompleteness theo- rem’s implicatio ns to ph ysics, a ﬁnal theory is p ossible; but it is no t p ossible to pro v e this fact rigourously . But G ¨ o del’s Incompleteness theorem w as fo r Pe ano’s arithmetic, i.e. natural nu mbers. Phys ics mus t employ real n um b ers. So, if 39 P eano’s arithmetic has inﬁnite num b er of axioms, the author presen ts the hy - p othesis that a ﬁnal theory w ouldn’t nee d only inﬁnite n umber o f axioms, but also a mathematical langua ge whose a lphab et is inﬁnite! References: 1. Rafee Ebrahim K amouna, “F uzzy Logic Programming”, F uzzy Sets a nd Systems, 1998. 2. Stanley L. Jaki:“A Late Aw ak ening to G ¨ o del in Ph ysics”, pirate.sh u.edu/ jakistan/JakiGo del.p df, accessed 15/0 2/2008. 3. Stephe n W. Hawk ing, “G ¨ o del and the End of Ph ysics”, www.dam tp.cam.ac.uk/ strtst/dirac/ha wking, accessed 15/ 0 2/2008. 4. h t t p:/ /en.wikipedia.org /wiki/Causal structure. 5. h t t p:/ /en.wikipedia.org /wiki; k eyw ords: “Introduction to General Relativ- it y” and “General Relativit y”. 40

The Kleene-Rosser Paradox, The Liars Paradox & A Fuzzy Logic Programming Paradox Imply SAT is (NOT) NP-complete

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