A refined calculus for Intuitionistic Propositional Logic - DISCo

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calculi) rules to treat signed formulas of the kind T(A → B) according to themain connective of A; see also [3, 9], where calculi with analogous properties aregiven. In these cases, the re-use of formulas is avoided by replacing T(A → B)with ”simpler” formulas built up from the subformulas of A → B; moreover,suitable measures on formulas are defined, which guarantee the termination ofderivations. But, if on the one hand decision procedures for these calculi do notneed loop-checking mechanisms, on the other hand the rules to treat formulasof the kind T(A ∨ B → C) and T((A → B) → C) give rise to proofs that maybe of exponential length in the size of the formula to be proved. This problemis overcome in Hudelmaier’s sequent calculi described in [8], where proofs havelinear length and the related decision procedures require O(n log n)-space. In thispaper we refer to the calculus LG of [8], which has a natural translation in thetableau setting. The novelties of LG essentially regard the treatment of formulasof the kind T(A → B). To save space, in some rules of LG the repetitions offormulas is avoided by introducing new propositional variables. Moreover, LGprovides rules to handle the sets containing both T(A → B) and FA, providinga rule for every main connective of A. We remark that in [8], the O(n log n)-spaceresult is proved for the calculus LE, which improves LG by providing a compactnotation to represent the pairs of formulas FA, T(A → B).The calculus T Int we introduce in this paper is a refinement of Hudelmaiercalculus LG. Here, we improve Hudelmaier’s rules by giving rules to treat formulasof the kind T((A → B) → C), for all the main connectives of B, withoutintroducing rules treating pairs of signed formulas. As discussed in the paper,even if T Int has the same computational performances of Hudelmaier’s calculi,it allows us to define a “better” decision procedure due to the following facts: (i)in general T Int proofs have width which is less than that of the correspondingLG proofs; (ii) T Int rules introduce a lower degree of non determinism. Thus,both the search space and the dimension of the proofs of T Int are narrower thanLG. The new rules of T Int exploit the ideas used in [4] to get a calculus havingproofs of depth bounded by 3n, which gives rise to an O(n log n)-space decisionprocedure.Finally, we point out that our reasoning is based on semantic tools, whereas [8]uses syntactic techniques (to prove the equivalence between LG and Gentzen calculus,the author has to introduce some intermediate calculi and prove theirequivalence). As a by-product, our decision procedure allows us to build acounter-model of A whenever A is not intuitionistically valid.To conclude, we remark that since [3, 8, 9] the main works on the improvementof decision procedures for Intuitionistic Logic were faced with the definition of“efficient” implementations, see e.g. [6, 1]. Here we reconsider the problem froma purely logical point of view.The paper is structured as follows: in the next section we introduce the notationand the preliminary definitions. In Section 3 we describe T Int and we discussthe main differences with respect to Hudelmaier’s calculus LG. In Section 4 weprove T Int is sound, while in Section 5 we prove the completeness and somecomputational complexity properties.2
2 Notation and PreliminariesWe consider the propositional language L based on a denumerable set of propositionalvariables (atoms) PV and the logical connectives ¬, ∧, ∨, →. We writeA ∈ L to mean that A is a formula of L.Kripke models are the main tool to semantically characterize propositionalintuitionistic logic Int (see e.g. [2, 5] for the details). A Kripke model for L isa structure K = 〈P, ≤, ρ, ⊩〉, where 〈P, ≤, ρ〉 is a finite poset with minimumelement ρ, and ⊩ is the forcing relation, namely a binary relation on P × PVsatisfying the monotonicity condition: α ⊩ p and α ≤ β implies β ⊩ p. Theforcing relation is extended to arbitrary formulas of L as follows:1. α ⊩ A ∧ B iff α ⊩ A and α ⊩ B;2. α ⊩ A ∨ B iff α ⊩ A or α ⊩ B;3. α ⊩ A → B iff, for every β ∈ P such that α ≤ β, β ⊩ A implies β ⊩ B;4. α ⊩ ¬A iff, for every β ∈ P such that α ≤ β, β ⊩ A does not hold.We write α A to mean that α ⊩ A does not hold. It is easy to check thatthe monotonicity property holds for arbitrary formulas, i.e., for every formulaA ∈ L, α ⊩ A and α ≤ β implies β ⊩ A. A formula A is valid in a Kripke modelK = 〈P, ≤, ρ, ⊩〉 iff ρ ⊩ A (by monotonicity property, this means that α ⊩ A forevery α ∈ P ). It is well-known [2, 5] that propositional intuitionistic logic Intcoincides with the set of formulas valid in all Kripke models.3 The Tableau CalculusThe tableau calculus T Int we introduce in this section, is a refinement of theone introduced in [9] and [4]. It works on signed formulas, namely expressionsof the kind TA, FA or F c A, where A ∈ L. Signed formulas have a naturalinterpretation in Kripke semantics. Given a Kripke model K = 〈P, ≤, ρ, ⊩〉, anelement α ∈ P and a signed formula H, we say that α realizes H in K, and wewrite K, α ✄ H, iff one of the following conditions holds:– H = TA and α ⊩ A;– H = FA and α A;– H = F c A and α ⊩ ¬A.K, α̸ ✄H means that K, α ✄ H does not hold. Given a set S of signed formulas,we write K, α ✄ S to mean that K, α ✄ H for every H ∈ S; we say that S isrealizable if K, α ✄ S for some K and α. We call the certain part of S the setS c = {TA | TA ∈ S} ∪ {F c A | F c A ∈ S}We remark that, by the monotonicity property, K, α ✄ S and α ≤ β impliesK, β ✄ S c .The rules of the tableau calculus T Int are shown in Tables 1-3. In the ruleswe write S, H as a shorthand for S ∪ {H}. Every rule applies to a set of signed3
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