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6О НЕКОТОРЫХ ПРОПОЗИЦИОНАЛЬНЫХ СИСТЕМАХ ВЫВОДОВ ДЛЯ РАЗЛИЧНЫХ ЛОГИК
Чубарян А.А.
Факультет информатики и прикладной математики,Ереванский государственный университет,
Ереван, доктор физ.мат.наук,профессор
Читоян А.С.
Факультет информатики и прикладной математики,Ереванский государственный университет,
Ереван,аспирант
ON SOME PROPOSITIONAL PROOF SYSTEMS FOR
VARIOUS LOGICS
Chubaryan A.A.
Department of Informatics and Applied Mathematics, Yerevan State University, Yerevan, Doctor of Science,
Professor Tshitoyan A.S.
Department of Informatics and Applied Mathematics, Yerevan State University, Yerevan, PhD student
АННОТАЦИЯ
Для Классической, Интуиционистской и Минимальной (Иогансона) логик введены семейства некоторых пропозициональных систем выводов с полным правилом подстановки и с ограниченным правилом подстановки, а также исследованы относительные эффективности этих систем для каждой из рассматриваемых логик. Доказано, что для каждой из указанных логик введенные системы с полным правилом подстановки полиномиально эквивалентны по длине с соответствуюшими системами Фреге, но для каждого £ >1 системы с ^-ограниченным правилом подстановки, где количество связок подставляемой формулы не более ■£, выводы в форме дерева могут иметь по длине вывода экспоненциальное ускорение относительно систем с аналогичным ограничением 1-1.
ABSTRACT
The families of some prepositional proof systems with full substitution rule and with restricted substitution rules are introduced for Classical, Intuitionistic and Minimal (Johansson's) logics, and the efficiencies of introduced systems are compared for every mentioned logic. We show that for each of mentioned logics the introduced system with full substitution rule is polyomially equivalent to Frege systems by size, but for every £ >1 systems with l-restricted substitution rule, where the number of connectives for substituted formula is bounded by l, proofs in tree form can have an exponential speed-up over the one bounded by l-1.
Ключевые слова: пропозициональные сыстемы выводов, полиномиальная сводимость, экспоненциальное ускорение, определяюшие дизъюнктивные нормальные формы.
Keywords: prepositional proof systems, proof complexity, polynomial simulation, exponential speed-up, determinative disjunctive normal form.
1. Introduction. Cut-free sequent system and resolution system for Classical, Intuitionistic and Minimal (Johansson's) prepositional logics (CPL, IPL,MPL) are the most frequently used proof systems for automated theorem proving, but they are «weak» systems. The main attractive feature of the resolution method is its single inference rule and for the cut-free systems - its feasible subformula property. But there exist some formulas, which require exponential proof complexities in these systems. It is well-known that cut rule and substitution rule can provoke the essential proof speed-up and some kinds of restricted cut rule and restricted substitution rule possess such properties also [4,5,12,13 and some others]. Particularly in [1] for CPL it is shown that Sequent system (in tree form) with cut rule, where the number of connectives of cut formulas is bounded by (l+ 1), has an exponential speed-up over
the one, bounded by l. Analogous results for multi-suc-cedent systems of IPL and MPL are obtained in [6]. The efficiencies of resolution systems of CPL, IPL,MPL with full substitution rule and of the family of resolution systems with l-restricted substitution rule, where the number of connectives for substituted formula is bounded by l, are investigated in [6]. The speed-up of proof sizes, analogous to above mentioned, are obtained for these systems also. The families of elimination systems with full substitution rule and with restricted substitution rules were used as the bridges between the sequent and resolution systems. As the elimination systems are very useful for proving of exponential lower bounds of proof sizes for some tautologies, it is interesting to investigate just these systems. In this paper for every mentioned logic we compare the proof complexity in propositional elimination systems with
lull substitution rule (SEC, SEI and SEM ) and with £-restricted substitution rule (SfEC, SfEI and SfEM).
We show that for each of mentioned logics the SE-type system for tree form proofs is polynomially equivalent to Frege systems by size, but for every £ > 0 Sf+iE-type system has exponential speed-up over the SfE-type for tree form proofs. We give also some new algorithm for construction of determinative conjuncts and determinative disjunctive normal form for some propositional tautology on the base of it proof in cutfree sequent system for every mentioned logic.
The summary of part for this paper is presented in
[3].
2 .Preliminaries. We will use the current concepts of the unit Boolean cube (En), a propositional formula, a classical tautology, a proof system for classical prop-ositional logic and proof complexity. The particular choice of a language for presented propositional formulas is immaterial in this consideration. However, because of some technical reasons we assume that the language contains the propositional variables pt (i > 1) and (or) pi. (i > 1; j > 1), logical connectives —, &, v, 3 and parentheses (,). Following the usual terminology we call the variables and negated variables literals for CPL. The conjunct K (term) can be represented simply as a set of literals (no conjunct contains a variable and its negation simultaneously).
2.1. Determinative disjunctive normal forms and elimination proof systems, based on them.
In [2] the following notions were introduced.
Each of the following trivial identities for a prop-ositional formula y is called replacement-rule:
0 & y = 0, y & 0 = 0, 1 & y = y, y & 1 = y,
0 V y = y, y V 0 = y, 1 V y = 1, y
V 1 = 1,
0 3 y = 1, y 3 0 = xp, 1 3 y = y, y
3 1 = 1,
0 = 1, 1 = 0, $ = y.
Application of a replacement-rule to some word consists in the replacing of some its subwords, having the form of the left-hand side of one of the above identities, by the corresponding right-hand side.
Let 9 be a propositional formula, P = {p1, p2,..., pn] be the set of all variables of 9 and P' = iPi1,Pi2, ■■■ ,Pim} (1 <m <n) be some subset of P.
Definition 1. Given a = {a1,a2,. ,am} c Em, the conjunct K^ = {pilai,pi2^2,.,pimam} is called 9 — 1 -determinative (9 — 0 -determinative) if assigning Oj (1 < j < m) to each p£ and successively using replacement-rules, we obtain the value of 9 (1 or 0) independently of the values of the remaining variables.
9 — 1 -determinative conjunct and 9 — 0-deter-minative conjunct are called also 9 -determinative or determinative for 9.
Definition 2. 1-determinative for 9 conjunct = {Pi^.Pi/2, ■■■,Pim°m} is called minimal determinative if no subset of Ka is determinative for 9.
Definition 3. DNF D = {K1,K2, ...,Kf} is called determinative DNF (dDNF) for 9 if
9 = D and every conjunct K^ (1 < i < j) is 1-determinative for 9.
Note that if minimal 9 — 1 -determinative con-
junct has m literals, then dDNF for ф has at least 2m conjuncts. This property is very important for obtaining some good lower bounds for proof steps in the systems, investigated in this paper.
In [2] some classical propositional proof system EC is introduced by the first co-author. The axioms of EC aren't fixed, but for every formula (p each conjunct from some dDNF of (p can be used as an axiom.
The classical elimination rule (C£-rule) infers К' U К" from terms K' U {p} and K' U {p}, where K' and K''are terms and p is a variable.
The proof in EC is a finite sequence of terms such that every term in the sequence is one of the axioms of EC or is inferred from earlier terms in the sequence by C£ -rule.
DNF D = {K1,K2,..., Kt} is called full (tautology) if using C£ -rule the empty conjunct (0) can be proved from the axioms {K1, K2, ...,K{}.
As the intuitionistic (minimal) validity is determined by derivability in some intuitionistic (minimal) propositional proof calculus, the above definition of dDNF for classical tautologies is not applicable here. Some algorithm for construction of dDNF of a classical tautology (p on the base of resolution refutation for ф is described in [2]. The analogies of the (p — determinative DNF for IPL and MPL (<p — I-determinative DNF and (p — M-determinative DNF accordingly) were constructed in [7] on the base of proofs in intuitionistic and minimal resolution systems RI and RM. Note that only variables with one or double negations are the literals in I-determinative conjuncts (p з! and (p d!) d! type formulas are literals in M-determinative con-juncts). Taking into consideration that for every classical tautology ^ the formula —— ^ ((( ^ d!) d!))
Место для уравнения. is intuitionistic (minimal) tautology, some future results for CPL can be proved for IPL (MPL) also.
By analogy the corresponding proof system EI (EM) can be constructed for IPL (MPL). As axiom it is considered every I-determinative (M-determinative) conjunct from some I-determinative (M-determinative) DNF for any formula ф.
For EI (EM) we take the following inference rule
К'Up К" Up
L — rule
M, — rule
K' U K" K' U (p 31) 31 K" U (p 31)
K' U K"
where K' and K'' are terms and p is a variable.
We must introduce for every mentioned logics the notion of generalized literal as result of replacement of propositional variable in a literal of this logic by any disjunction of propositional variables and notion of generalized conjunct as a set of literals and (or) generalized literals.
Substitution rule for the set of conjuncts C, at least one conjunct of which contains the variable p: —^â, where S(C)fi denotes the set of results for substitution of a formula A, which is any disjunction of
propositional variables, instead of variable p everywhere in the conjuncts of the set C. Note that S(C)p is generalized conjunct
The generalized elimination rule for a formula
. C1 U {4} C2 U {4} . . . .. . _ .
A:-, where A is a literal or a formula,
C1 U c2
substituted on any step .
The system SEC (SEI, SEM) is the system EC (EI, EM) with substitution rule and corresponding generalized resolution rule.
For given £ the system SfEC (SfEI, SfEM) is the system SEC (SEI, SEM) with the following restriction: the number of connectives of substituted formulas is bounded by
2.2. Other proof systems. We use also the well known Frege systems JFC for CPL [11] and JFI (JM) for IPL (MPL), given in [7]
Gentzen style sequent system LK is defined as in [10] .The notions of cedents, sequents,antecedents and succedents are defined as usual. Axiom of this system is p ^ p for propositional variable p. Rules of inference are
j r^A,A and BJ^A 2 A,r^A,B ^ A,B,r^A ^ r^A,A or T^A,B
ADBJ^A
^A,ADB AvBJ^A r^A,AvB AJ^A
. A&BJ^A r^A,A&B .-A,r^A 8.^A,—A
^^^ (structural rule)-— (cut rule),
where A and B are formulas, r, r', A and A' are the sequences of formulas and besides r( A) is subset of
r'(A').
By LK- we denote the system LK without cut rule and the system LK with cut rule, where the number of connectives in cut formulas is bounded by l , we denote by LKl.
The analogous multi-succedent systems LI (LI- ,LI l ) and LM (LM - ,LM l ) for IPL and MPL are described in [9].
Note, that the systems EC, EI, EM are cut systems, cut-formulas of which are a variable - formula with 0 connective.
2.3.Polynomial simulation, exponential speedup. In [8] the following notions are given.
Definition 4. Let ® be a proof system for proposi-tional calculus, which is sound and complete, and let P be a proof in ® (®-proof). The size of P, defined as the number of all symbols, used in P, is denoted by size(P).
Definition 5. Let and be proof systems for propositional calculus.
1) p-simulates ®2 if there exists a polynomial p such that for any formula 9 and any proof P2 of 9 in ®2 there exists a ®1-proof P1 of 9 (translated into language) so that size(P0 < p(size(P2)),
2) and ®2 are p-equivalent iff and ®2 p-simulate each other,
3) has exponential speed-up over the system ®2, if there exists a sequence of such formulas 9n (or their representation), that for any ®2-proof (P2)n of formula 9n and some ®1-proof (POn of 9n size((P2)n) >
20(size((Pi)n )).
We use also the well-known notion of proof in tree form (see, for example, [1]) .
3. Main results. In order to prove the main results of our paper, we must give some auxiliary notions, propositions and an algorithm.
For the sets of conjuncts C = {K1, K2,...,Kn} and C'={K'1, K'2,K'm} we denote by C x C' the set of unions for following conjunct pairs {KiU Kj / 1<i<n, 1<j<m}. For given formula 9 by (9)1 ((9)0) we denote the set of all 9-1-determinative (9-0-determinative) conjuncts.
It is obvious that for construction of determinative conjuncts we can use following 0-determinative identities or 1-determinative identities:
(-9)0 = (9)1, (-9)1 = (9)0,
(9&y)0 = (9)0 U (y)0, (9&y)1 = (9)1 x (y)1,
(9vy)0 = (9)0 x (y)0, (9vy)1 = (9)1 U (y)1,
(93V)0 = (9)1 x (V)0, (93V)1 = (9)0 U (V)1.
We use also the well-known notions of positive and negative occurrences of subformulas (or variables) in the formula or in the sequent [10].
For construction of full 9-determinative disjunctive normal form on the base of LK--proof of tautology 9 we suggest the following Algorithm:
Let W be the tree form proof of 9 in cut-free system with the minimal size. From each occurrence of a variable in axiom we can point step by step the sequence of subformulas of 9 (from variable until the formula 9 itself), each of which (maybe with some other subformula) is used for derivation of the following one in this sequence by some of logical rule. For each step we construct the set of determinative conjuncts for corresponding subformula, using 0-determinative identities for subformulas with negative occurrence and 1-determinative identities for subformulas with positive occurrence. By using this Algorithm for mentioned non-classical logics we must take each literal with additional negation for subformulas with double negation occurrence.
Note, that 1) the size of proof in E-type system from above constructed dDNF can not be more, than the size of basic cut-free proof and 2) if we have a proof in LK system, whose cut-formulas are only some disjunctions of propositional variables, then for SE-type systems the mentioned algorithm gives generalized conjuncts, which contain not only variables as literals but also the cut (substituted) formulas as generalized literals.
Let r be a sequence of formulas. By -- r we denote the sequence of double negation of each formula from r. In minimal logic a formula —^ (—— can be represented as ^ d!
((( ?d!) d!)) . Formula ^o (o-representation of
formula is obtained from ^ by substitution instead of every subformula —y of formula ^ the formula y d !. Let r be a sequence of formulas, then by To we denote the sequence of o -representations of each formula from r.
Proposition 3.1
If sequent r ^ A is proved in LK, then sequent ——r ^ ——A ( ro ^ Ao) is proved in LI (LM).
Proof for LI is given in [10], proof for LM can be given by analogy.
Proposition 3.2(N.Arai)
For any £ >1 and arbitrary large m, there exists a valid sequent r ^ A of the size m such that:
(a) r ^ A has a proof in LK £ of size O(m2).
(b) Any LK(^ - 1) proof of r ^ A has >2Vm/ f se-quents.
Proof is given in [1]. Note, that for every £ >1 the cut-formulas are disjunctions of no more that £ +1 propositional variables, and moreover each variable contains only in one disjunction. The last property is very important for proving the following statements.
Proposition 3.3
V £ > 0 the systems SfEC and LK £ (SfEI and LI SfEM and LM are p-equivalent in tree form.
Proof. It was proved in [2], that LK ^-simulates EC (in tree form). By analogy it is not difficult to prove that for every l > 0 LKl ^-simulates SlE (in tree form). Really, we must use described in [2] algorithm not only to determinative conjuncts, but also to generalized con-juncts, obtained from determinative conjuncts by substitution rules, and then we must modeling generalized elimination rule with cut rule (both with "restricted" formulas). Here is very important that no one variable has occurrence in two or more cut formulas (substituted formulas). Analogous results for corresponding multi-succedent intuitionistic and multi-succedent minimal systems is proved on the base of corresponding algorithms, given in [7]. Reverse simulations follow from above Algorithm and notes, given after them.
Proposition 3.4
The systems LK (LM, LI) and FC (FI, FM) are p-equivalent.
Proof for the systems LK and FC is given in [11].Analogous results for corresponding intuitionistic and minimal systems are proved in [7].
Main Theorem
1) V £ > 0 the system S-mEC (S*nEI, S-mEM) has exponential speed-up over the SfEC (SfEI, SfEM) in tree form.
2) The tystems SEC (SEI, SEM) and FC (FI, FM) are p-equivalent.
Proof follows from the statements of Propositions 3.1 - 3.4.
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