A refined calculus for Intuitionistic Propositional Logic - DISCo

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S, T(A ∧ B) S, F(A ∧ B) S, F c(A ∧ B)S, TA, TB T∧ S, FA | S, FB F∧ S, F cA | S, F Fc∧cBS, T(A ∨ B) S, F(A ∨ B) S, F c(A ∨ B)S, TA | S, TB T∨ S, FA, FB F∨ S, F cA, F cB Fc∨Tables 2 and 3S, F(A → B) S, F c(A → B)S c, TA, FB F→ S c, TA, F cBFc→S, T(¬A) S, F(¬A) S, F c(¬A)S, F cA T¬ S c, TA F¬ S c, TAS c = {TA | TA ∈ S} ∪ {F cA | F cA ∈ S}Fc¬Table 1. The T Int calculusformulas, but only acts on the signed formula H explicitly indicated in thepremise; we call H the major premise of the rule, whereas we call all the othersigned formulas minor premises of the rule. The sets in the consequence areobtained by decomposing in some way the major premise of the rule and eithercopying all the minor premises (see e.g. the rule T∧ of Table 1) or only copyingthe certain part of the minor premises (see e.g. the rule F → of Table 1). Wecall splitting rules the rules of T Int having two sets in the consequence.Some rules require additional conditions in order to be applied. The ruleT → certain of Table 2 can be applied only if S = S c (that is, the set Sof minor premises does not contain F-signed formulas). The rule MP (modusponens) of Table 2, having T(A → B) as major premise, requires the presenceof TA among the minor premises. We point out that in [8] this rule is given inthe restricted form where A is a propositional variable.Finally, we notice that some rules of Tables 2 and 3 require the introductionof a new atom q, namely a propositional variable q not occurring in the premisesof the rule. This expedient goes back to [8] and avoids duplications of non atomicformulas in the consequence of a rule (namely, repetitions of subformulas of themajor premise); for instance, without the introduction of q, the consequenceof T → ∨ should be S, T(A → C), T(B → C), where C occurs twice, and thisprevents the definition of a linear complexity measure on sets of signed formulas.A set S of signed formulas is contradictory if either {TA, FA} ⊆ S or{TA, F c A} ⊆ S, for some formula A. Clearly, contradictory sets cannot berealized. A proof table (or proof tree) for S is a finite tree τ with S as root andsuch that all the children of a node S ′ of τ are the sets in the consequence of arule applied to S ′ . If all the leaves of τ are contradictory sets, we say that τ is a4
S, TA, T(A → B)S, TA, TBMPS, T(A → B)T→certain if S = ScS, F cA | S, TBS, T((A ∧ B) → C)S, T(A → (B → C)) T→∧S, T(¬A → B)S c, TA | S, TB T→¬S, T((A ∨ B) → C)T→∨ with q a new atomS, T(A → q), T(B → q), T(q → C)Table 2. Rules for T →closed proof table for S. A closed proof table is a proof of the calculus: a formulaA is provable in T Int iff there exists a closed proof table for {FA}.To conclude this section we discuss the main novelties of our calculus; inparticular we consider the differences among T Int and the tableau calculi of [9,4] and the sequent calculi introduced in [8]. For sequent calculi we present therules adopting the standard translation into tableau rules.First of all we notice that the rules of Tables 1 and 2 essentially coincidewith those described in [9], where the sign F c is introduced to characterizeintuitionistic negation. As for the rules of Table 3, they replace the ruleS, T((A → B) → C)T→→S c , TA, FB, T(B → C) | S, TCof [9], that goes back to [3] and [11] (given in a sequent calculus style), and therule F io →→ of [4] shown at the end of this section.The rule T →→ has been introduced by Vorob’ev to avoid loop-checking inthe decision procedure. On the other hand, the double occurrence of formulaB in the leftmost conclusion of T →→ gives rise to deductions that may beof exponential depth in the length of the formula to be proved (see [8, 6] for adetailed discussion). In [8] the problem is solved by introducing, beside the ruleT →→, some rules to treat the leftmost conclusion of T →→, according to themain connective of B. Moreover, the calculus LG of [8] provides rules to handlethe pairs of formulas FB, T(B → C), according to the main connective of B.5
Page 1 and 2: A refined calculus forIntuitionisti
Page 3: 2 Notation and PreliminariesWe cons
Page 8 and 9: equired. A deeper discussion about
Page 10 and 11: The previous lemma is the main step
Page 12 and 13: Kripke model K = 〈P, ≤, ρ, ⊩
Page 14: 8. J. Hudelmaier. An O(n log n)-spa

A refined calculus for Intuitionistic Propositional Logic - DISCo

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