Generating Schemata of Resolution Proofs

Two distinct algorithms are presented to extract (schemata of) resolution proofs from closed tableaux for propositional schemata [4]. The first one handles the most efficient version of the tableau calculus but generates very complex derivations (denoted by rather elaborate rewrite systems). The second one has the advantage that much simpler systems can be obtained, however the considered proof procedure is less efficient. In [2, 4] a tableau calculus (called Stab) is presented for reasoning on schemata of propositional problems. This proof procedure is able to test the validity of logical formulæ built on a set of indexed propositional symbols, using generalized connectives such as ∨ n i=1 or ∧ n i=1, where i, n are part of the language (n denotes a parameter, i.e. an existentially quantified variable). A schema is unsatisfiable iff it is unsatisfiable for every value of n. Stab combines the usual expansion rules of propositional logic with some delayed instantiation schemes that perform a case-analysis on the value of the parameter n. Termination is ensured for a specific class of schemata, called regular, thanks to a loop detection rule which is able to prune infinite tableaux into finite ones, by encoding a form a mathematical induction (by “descente infinie”). A related algorithm, called Dpll and based on an extension of the Davis-Putnam-Logemann-Loveland procedure, is presented in [3]. In the present work, we show that resolution proofs can be automatically extracted from the closed tableaux constructed by Stab or Dpll on unsatisfiable schemata. More precisely, we present an algorithm that, given a closed tableau T for a schema φn, returns a schema of a refutation of φn in the resolution calculus [9]. In the usual propositional case, it is well-known that algorithms exist to extract resolution proofs from closed tableaux constructed either by the usual structural rules [11, 13] or by the DPLL algorithm [7, 6]. The resolution proofs are used in various applications, for instance for certification [14], for abstraction-refinement [10] or for explanations generation [8]. The present paper extends these techniques to propositional schemata. Beside the previously mentioned applications, this turned out to be particularly important in the context of the ASAP project [1] in which schemata calculi are applied to the formalisation and analysis of mathematical proofs via cut-elimination. Indeed, the algorithm used for cut-elimination, called CERES [5], explicitly relies on the existence of a resolution proof of the so-called characteristic clause set extracted from the initial proof. The cut-free proof is reconstructed from this refutation, by replacing the clauses occurring in this set by some “projections” of the original proof. While Stab and Dpll are able to detect the unsatisfiability of characteristic clause sets, as such this is completely useless since actually it is known that those sets are always unsatisfiable (see Proposition 3.2 in [5]). It is thus essential to be able to generate explicitly a representation of the resolution proof. This is precisely the aim of the present paper. Since the initial formula depends on a parameter n, its proof will also depend on n (except in very particular and trivial cases), i.e. it must be a schema of resolution proof (which will be encoded by recursive definitions). The rest of the paper is structured as follows. In Section 1 we introduce the basic notions and notations used throughout our work, in particular the logic of propositional schemata (syntax and semantics). In Section 2 we define a tableau-based proof procedure for this logic. This calculus simulates both Stab and Dpll (for the specific class of schemata considered in the present paper). In Section 3 we provide an algorithm to extract resolution proofs from closed tableaux. Similarly to the formulæ themselves, the constructed derivations are represented by rewrite systems. In Section 4 we introduce a second algorithm which generates simpler derivations but that requires that one of the closure rules defined in Section 2 (the so-called Loop Detection rule) be replaced by a less powerful rule, called the Global Loop Detection rule. Section 5 briefly concludes our work. 1 Propositional schemata The definitions used in the present paper differ from the previous ones, but the considered logic is equivalent to the class of regular schemata considered in [2] (it is thus strictly less expressive than general schemata, for which the satisfiability problem is undecidable). We consider three disjoint sets of symbols: a set of arithmetic variables V , a set of propositional variables Ω and a set of defined symbols Υ. Let ≺ be a total well-founded ordering on the symbols in Υ. An index expression is either a natural number or of the form n + k, where n is an arithmetic variable and k is a natural number. Let I be a set of index expressions. The set F(I) of formulæ built on I is inductively defined as follows: if p ∈ Ω ∪ Υ and α ∈ I then pα ∈ F(I); ⊤,⊥ ∈ F(I); and if φ, ψ ∈ F(I) then ¬φ, φ ∨ ψ, φ ∧ ψ, φ ⇒ ψ and φ ⇔ ψ are in F(I). Definition 1 We assume that each element υ ∈ Υ is mapped to two rewrite rules ρυ and ρ 0 υ that are respectively of the form υi+1 → φ (inductive case) and υ0 → ψ (base case), where φ ∈ F({i+ 1, i, 0}), ψ ∈ F({0}) and: 1. For every atom τα occurring in φ such that τ ∈ Υ we have either τ ≺ υ and α ∈ {i+ 1, i, 0} or τ = υ and α ∈ {0, i}. 2. For every atom τα occurring in ψ such that τ ∈ Υ we have τ ≺ υ and α = 0. ✸ We denote by R the rewrite system: {ρυ, ρ 0 υ | υ ∈ Υ}. The rules ρ 1 υ and ρ 0 υ are provided by the user, they encode the semantics of the defined symbols. Proposition 2 R is convergent. Proof. By Conditions 1 and 2 in Definition 1, the rules in R either strictly decrease the values of the defined symbols occurring in the formula w.r.t. ≺ or do not increase the value of these symbols but strictly decreases the value of their indices. Thus termination is obvious. Confluence is then immediate since the system is orthogonal. For every formula φ, we denote by φ↓R the unique normal form of φ. A schema (of parameter n) is an element of F({0, n, n+ 1}). We denote by φ{n ← k} the formula obtained from φ by replacing every occurrence of n by k. Obviously for any schema φ, φ{n ← k} ∈ F({0, k, k + 1}). A propositional formula is a formula φ ∈ F(N) containing no defined symbols. Notice that if φ ∈ F(N) then φ↓R is a propositional formula. Proposition 3 If φ ∈ F(N) then φ↓R is a propositional formula. Proof. By definition ofR, φ↓R∈ F(N). Furthermore, if φ↓R contains a defined symbol υ then either ρυ or ρ 0 υ applies, which is impossible. An interpretation is a function mapping every arithmetic variable n to a natural number and every atom of the form pk (where k ∈ N) to a truth value true or false. An interpretation I validates a propositional formula φ iff one of the following conditions holds: φ is of the form pk and I(pk) = true; φ is of the form ¬ψ and I does not validate ψ; or φ is of the form ψ1 ∨ ψ2 (resp. ψ1 ∧ ψ2) and I validates ψ1 or ψ2 (resp. ψ1 and ψ2). I validates a schema φ (written I |= φ) iff I validates φ{n ← I(n)}↓R. We write φ |= ψ if every interpretation I validating φ also validates ψ and φ ≡ ψ if φ |= ψ and ψ |= φ. Example 4 The schema p0∧ ∧ n i=1(pi−1 ⇒ pi)∧¬pn is encoded by p0∧υn∧¬pn, where υ is defined by the rules: υi+1 → (¬pi ∨ pi+1) ∧ υi and υ0 → ⊤. The schema ∨ n i=1 pi ∧ ∧ n i=1 ¬pi is encoded by τn ∧ τ ′ n , where τ and τ ′ are defined by the rules: τi+1 → pi+1 ∨ τi, τ0 → ⊥, τ ′ i+1 → ¬pi+1 ∧ τ ′ i and τ ′ 0 → ⊤. Both schemata are obviously unsatisfiable. The schema (pn ⇔ (pn−1 ⇔ (. . . (p1 ⇔ p0) . . .))) is defined by υ ′ n , where: υ i+1 → (pi+1 ⇔ υ ′ i ) and υ 0 → p0. ♣