Ecsedi-Tóth, Péter

Varga, A.

English

Ecsedi-Tóth Péter

Varga A.

University of Szeged

An effective theorem proving algorithm B y P . ECSEDI—TÔTH a n d A . V A R G A Summary A simple procedure is presented which is a sound and complete calculus for first-order logic. The method can. easily be implemented on computers and it has considerable advantage in manual work. Introduction Nowadays computer science becomes more and more 'scientific' after its 'naive' age of the last 30 years. Much work has been devoted to create 'software industry', however, for although there is a shortage of really effective methods. Research of artificial intelligence gives rise to a hope of solving this problem. First program-correctness-proving procedures were suggested by Turing, Floyd, Manna and others in their works on 'modeling' intelligent processes of human thinking. The soul of these procedures is a pure mathematical logical tool: a theorem-proving algorithm, or in other words an automatic deduction. After the activity of Godel, Skolem, Turing, and Herbrand [5, 9; 10, 6], J. A. Robinson proposed an elegant practical realization of automatic deduction in [8]. Since 1965, when the first paper on Robinson's resolution principle appeared, many theorem-proving algorithms, each of them is an 'improvement' of the re-solution principle, have been published.. However, almost all these methods have two common disadvantages. Namely, they have.a deep combinatorial nature and one has to do a not-so-few unnecessary steps in pre-procedure. The input of a theorem-prover, working on the basis of a version of the re-solution principle, is a formula in the first-order logic. (First-order logic is one of the most important, tools for 'software industry', but theorem-provers of other types of languages e.g. zero- or second-order logic, non-classical (modal) logic, fuzzy logic and other multi-valued logics etc. are also, known.) This input formula has to be in prenex normal form, in disjunctive normal form and in Skolem-normal form. In this paper we present a procedure which can be applied to formulae not being in disjunctive normal form (skolemization and prenex normal form are needed). • 250 P. Ecsedi-Toth and A. Varga Our method has some similarity, although it is essentially different f rom that, to the resolution principle. Our procedure is a 'valuation' or rather a 'com-putation of logical value' of the given formula, and thus it has less combinatorial character. Some other resolution-like algorithms eliminating the steps needed to get a disjunctive normal form of the given input formula, are also known [7], [2, 3] but our method over the 'computation' property, mentioned above, differs in the following points, as well. It offers: i) Easier treating of quantified variables; ii) More effectiveness thanks to the less combinatorial nature of our method; iii) Easier manual and also machine execution. In the first section the notions, notations and facts needed for the development are surveyed. The second section deals with the essence of the material. In the third one the procedure is defined for first-order logic. Completeness theorems are only stated, the long but not too complicated proofs will appear elsewhere. § 1. Preliminaries By a type T we mean an ordered quintuple T = (I, J, K, t\ t") where I, J, K are pairwise disjoint sets (sets of relation symbols, function symbols and constant symbols, respectively); t', t" are functions for which Domain ( t ' ) = I , Range (t')acQ, Domain {t")=J, Range (t")cz(o. The set of r-type formulae of zero- and first-order (defined in the standard way) will be denoted by o Fz and !F l . S 1 is the class of T-type algebraic structures (the class of possible models). Let M(cp) denote the following class of structures M(<p) = {SlISliES1, 91 |= <p}. (p€.iFz is a tautology (unsatisfiable) iff • M((p) = & (M(<p) = 0). The notion of algorithm is used in a naive sense. We shall suppose that a finite set of symbols Z is fixed. An effective theorem-proving algorithm 251 Let Z* = {<za, . . . , z„)|.n,< a), z ,eZ if i =i n}. By the definition of an algorithm we mean an element of Z* . ' If z£Z* then there exists a partial function z : Z * - Z * , the meaning of z. Consequently, if z1 ; z2, z 3£Z* then z1(z2) = z3 means that the result of applying the algorithm zx to the algorithm z2 is the algo-rithm z3. We will assume that the elements of 1FZ and a> are represented in Z* (for ex-ample ^ c Z * and cocZ*). Let z £ Z * and AcZ*. z enumerates the set A iff Domain (z)=to, Range (z) = A. Suppose there exist two distinguished elements in Z*, namely, z0 and zx which represent the truth-values TRUE and FALSE, respectively. If z€Z*, A^Z*. then z decides the set A iff Domain (z) = Z*, . Range (z)={za,z1) and for any z '£Z*, z(z') = z0 iff z'(-A. § 2. Deciding the set of zero-order sentences For an arbitrary zero-order sentence q>, let Sv be a sequence of prime sentences (relation symbols with no variables in their argumentum places) occurring in <p. Let : • • • . Step 1. Start with the sentence <p as initial vertex. Step 2. Choose the first not used element of S^, say p, where xp is the actually treated vertex, and substitute the truth-values TRUE and FALSE, respectively, for p in iji. Draw two edges, labelled by p and its negation, respectively, from the. vertex. -252 P. Ecsedi-Toth and A. Varga Step 3. Apply the following rules to obtain the truth-value-free version o the substituted vertex ip or a single truth-value along both of the edges: for any zero-order sentence cp, ç>Al — l A ç > - — V l ~ l ; lV (7) 0 — ; (1 ~<p)~.q> (9) (\ (0 « q>) - (j> v (10) (pA(p-*(p (11) <pA<p — 0; (p.Atp — 0 (12) <p \t (p —• (p ' (13) <pV<p — 1: <pV<p — 1 (14) ç~'<p~\ ( 15 ) ( < ? - < ? ) - 1 • ( 16 ) <P~<P (17) I — 0; 0 — 1 (18) where A, V, —, —, 0 and 1 is the symbol for conjunction, disjunction, implica-tion, equivalence, negation, truth-values FALSE and TRUE, respectively. The truth-value-free formulae or single truth-values, obtained from the actually treated vertex will be the two new vertices x and rj. Let the sequence of zero-order prime sentences of x (l) be Sx (Sn) with the same order as in Sv. Note that Sx (S, = - { p } ) . Step 4. Do steps 2 and 3 for the vertices x and r/ if they are not single truth-values! If each of the vertices newly made is a truth-value, the procedure terminates. An effective theorem-proving algorithm 253 The vertices which are truth-values will be called final-vertices. Example 1. Assume that the given zero-order sentence is q> = (A (a, b)AC) - ((¿(a) VC) — A (a, b)) where •t'(A) = 2, t'(B)=\, t'(C) = 0, t"(a) = 0, t"(b) = 0. The tree produced by the algorithm is indicated in Fig. 1, and its "computation" in Fig. 2. Fig. 1 Let o/d e c , 0gdec be algorithms such that 0 OFDEC -OFZ {zo> ZI}> OGDEC'-OFZ — {ZO> ZI}> ii) ?6ect \ _ fa a " the final-vertices of z(q>) are 1 oJ ec(<P) - | Z i otherwise, -dec/- \ _ fa if all the final-vertices of z(q>) are 0 oS VP) - \Zl otherwise. Note that 0 / d e c and 0gdec do exist, i.e., the final vertices of z(q>) do not depend on the order of Sv. Theorem. [11]. i) o / d e c decides the set of zero-order tautologies; ii) o&dec decides the set of zero-order unsatisfiable sentences. 254 P. Ecsedi-Toth and A. Varga cp, S„ = {A(a, b), C, 5 ( a ) } A{a,b)f\C — ((5(a) VC) — A (a, b)) A (a, b) <Pu Stl = {C, B(a)} A (a, b) = {C, B(a)} 1A C - ( ( i ( a ) V C ) « 1 1(1) C - ( ( 5 ( a ) V C ) « l 1 ( 1 7 ) C — ( ( 5 ( a ) V C ) - 1 t ( 9 ) C — ( f i ( a ) V C ) «All, ^ n = { 5 ( a ) } ^12, S^is = { 5 ( a ) } O V C - ( ( 5 ( a ) V C ) -> 0 ) t (2) 0 - ( ( 5 ( a ) V C ) ~ - o ) { ( 1 8 ) 1 - ( ( « ( f l ) V C ) -- o ) i (7) ( ( 5 ( a ) V C ) ^ - o ) { (10) 5 ( a ) V C ^ 2 1 . ^ 2 1 = { ^ ( a ) } "/'22> ^ 2 2 = { 5 ( a ) } 1 - ( 5 ( a ) V I ) 0 - ( 5 ( a ) V 0 ) 5 ( a ) V I 5 ( a ) V 0 i (7) 1 ( 8 ) t ( 3 ) J (4) 5 ( a ) VI 1 1 5 ( a ) t ( 3 ) t (18) 1 0 5 ( a ) 5(a) 1 0 t (18) {(18) 0 1 fig'. 2 § 3. Tree-constructing for first-order sentences We define the well-known Skolem-extension of type T in the following way: If then where r(°> = r . T(m> = ( I , /<»>, K,t',t"("•)) is defined T ( m + 1 ) = j r W U i / * ; K i + An effective theorem-proving algorithm 255 and is a new function symbol for every <p(xl5 . . . , x„) 6 jF1'"1' and furthermore, if f$J*. XS = U {TC") I M < (O}. Theorem (Skolem). There exist algorithms ex and un such that i) u n ^ F ' - i F ' 5 ; ii) For any <pGiFr, )) = S l S , M(<p) = 0 iff M(un(<p)) = 0. This theorem is a simple consequence of the Skolem Normal Form Theorem [1, 4, 9]. Let a sentence cp be given in prenex form. Consider the formulae ex(<p) and un (cp). We define the sets P? (P™) and S " (S£n) as follows: - f " is the sequence of zero-order prime formulae of ex(<p) (uh(cp)) which are not sentences i.e., contain at least one free variable. The prime formulae % and t] differing only in free variables are distinguished in P " (P£n). 5 " (S£n) is the sequence of zero-order prime sentences of ex(<p) (un(<p)) if there exist such sentences. In the opposite case, substitute arbitrary constant symbols for the free variables of the elements of P" (P%a) and let the substituted formulae, which are now sentences, be in the sequence S | x (S£n). Example 2. Let the formula for any arbitary constant symbol a, P " = 0, = (E(a, b), E(a, c), I(b, c)>. 256 P. Ecsedi-Toth and A. Varga Denote the matrices of ex(<p) and un(<p) by ex(<p)* and un(<p)*, respectively. We recall the concept of unification [8]. Let be prime-formulae. <p and t/rare unifiable iff there exists a substitution such that the substituted cp and are identical literal by literal. Let (p be a first-order sentence in prenex form. The following generalization of z defines two binary trees zex(q>) and z""((p): Step 1. Start with the formula ex(<p)* (un(<p)*) to obtain the initial vertex of z"(<p) (and £""(</>)), respectively. Step 2. Choose the first element of S" say p, where ^ is the actually-treated vertex and substitute p first by 1 then by 0. Draw two edges from labelled by p and its negation, respectively. Seek for the elements of P" (P$n) which can be unified with p, and execute the unifying substitution for all such elements. Step 3. Apply the tautologies listed in § 2 as rules (1)—(18) to obtain the truth-value-free version of the substituted vertex or to obtain a single truth-value along both edges. These truth-value-free formulae or the single truth-values will be the two new vertices y_ and r¡. Then construct the sequences S" (S™), S¡¡* (S™), P" (P»% Pn" (P™), Step 4. Repeat the steps for every vertices newly obtained if they are not single truth-values. If all of the vertices are truth-values the procedure terminates. Note that zex(<p) and zun(<p) exist for any first-order sentence <p if it is in prenex form. The trees do not depend on the order of S " and S£a. If <p is a tautology or it is unsatisfiable then zex(cp) and z""((p) are finite trees. Example 3. Let us construct the trees zex(<p) and z""((p) for the formula of Example 2. They are indicated by Fig. 3 and Fig. 4. Let i / c o m , 1gcom be algorithms such that i i) i f " " " : — Z*, 1 G C O M : 1 F T S — Z * , where elements of i F 1 ' are supposed to be in prenex normal form. ü ) /com/ \ _ [zo if a ' l the final vertices of zex(cp) are 1, i J \<P) — [undefined otherwise, -com/- \ _ \zo if a l l the final vertices of ^ ( c p ) are 0. \<P) — jundefined otherwise. Remark, i /0 0 1", igcom exist by the previous note. Theorem. Let (p^1Fx ' be a sentence in prenex form. Then i) S? = (E(a, a), I(a, a)> Я г - 5 The tree z""(ç>) of f M Ul 258 P. Ecsedi-Toth and A. Varga Fig, 4 The tree z" (p)of (x)AC(x)). Assume we want to know whether \jj is a consequence of and ç>2. There are two possibilities': to show that thé formula <p' = <p1A<joa- î// is a tautology; or to show that the formula <p" = (pxA(p2/\~\^i is unsatisfiable. The previous theorem ensures us that if the formula cp' is a tautology then ifcom((p')=z0; and if the formula <p" is unsatisfiable then 1gcom(cp")=z0. éx (cp') = (3x)(3y){([A (x) - 0 ( x ) A C ( x ) ) ] A A [A (a) A D (a)]) - [D (y) A C0>)]), Ûn (</>") = (Vx)(V>')([^W - (-S(x)AC(x))] A AU(a)A JD(a)]A[l£)(y)VlC(j ; ) ]) , where a is a Skolem-function (constant). The tree z"(<p') is indicated in Fig. 5. All the final-vertices of the tree zex(<p') are 1. Thus 1 / c o m (<p ' )=z 0 , i.e., <p' is a tautol-ogy. The tree z"D(<p") is very similar to z"(<p') but all the final-vertices are 0. Thus 1gQOm((p")=z0, i.e. <p" is unsatisfiable. An effective theorem-proving algorithm 259 P? = (A(x), B(x), C{x), D(y), C(y)) sp = (A(a),D(a))/ Fig. 5 / The tree z "(</>') RESEARCH GROUP ON MATHEMATICAL LOGIC BOLYAI INSTITUTE O F THE AND THEORY OF AUTOMATA O F THE ATTILA JÓZSEF UNIVERSITY HUNGARIAN ACADEMY O F SCIENCES H-6720 SZEGED, H U N G A R Y H-6720 SZEGED, H U N G A R Y ARADI VÉRTANÚK TERE 1. SOMOGYI U. 7. ( Acta Cybernetica III/3 260 P. Ecsedi-Töth and A. Varga: An effective theorem-proving algorithm References [1] ANDREKA, H., T. GEROELY, I. NEMETI, Easily comprehensible mathematical logic and its model theory, KFKI memo, mimeographed, Budapest, 1975. [2] CHANG, C . L., Theorem proving by generation of pseudo-semantic trees, Div. of Comput. Res. and Technology, National Institute of Health, Bethesda, Maryland, 1971. [3] CHANG, C . L . , Theorem proving with variable-constrained resolution, Information Sei., v. 4 , 1 9 7 2 , p p . 2 1 7 — 2 3 1 . [4] CHANG, C. C , H J. KEISLER, Model Theory, North-Holland Publ. Co., Amsterdam, 1973. [5] GÖDEL, K., Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatsh. Math. Phys., v. 37, 1930, pp. 349—360. [6] HERBRAND, J . , Recherches sur la theorie de la demonstration, Travaux de la Societe des Sciences et des Lettres de Varsovie, v. 33, 1930, p. 128. [7] PRAWITZ, D., Advances and problems in mechanical proof procedures, Machine Intelligence, v. 4, (edited by B. Meitzer and D. Michie), American Elsevier, New York, 1969, pp. 59—71. [8] ROBINSON, J. A., A machine oriented logic based on the resolution principle, J. Assoc. Comput. Mach., 1965, pp. 23—41. [9] SKOLEM, T., Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweis-barkeit mathematischer Sätze nebst einem Theorem über dichte Mengen, Skrifter utgitt av Videnskapsselskapet i Kristiania, I, Mat. Naturv. Kl., v. 4, 1920, p. 36. [10] TURING, A. M., Computing machinery and intelligence, Mind, v. 59, 1950, pp. 433—460. [11] VARGA, A., P. ECSEDI-TÖTH, F. MÖRICZ, An effective method for minimizing Boolean polyno-mials, Research Report, Research Group on Mathematical Logic and Theory of Automata of the Hungarian Academy of Sciences, 1977, p. 40. (Received May 17, 1977 )

An effective theorem proving algorithm

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An effective theorem proving algorithm

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