A polynomial time complete disjunction property in intuitionistic propositional logic

We extend the polynomial time algorithms due to Buss and Mints[2] and Ferrari, Fiorentini and Fiorino[4] to yield a polynomial time complete disjunction property in intuitionistic propositional logic. The disjunction property, DP of the intuitionistic propositional logic Ip says that if a disjunction α0 ∨ α1 is derivable intuitionistically, then so is αi for an i. This property follows from cut-elimination in sequent calculi, normalization theorem in natural deduction, Kleene’s or Aczel’s slash Γ|C or completeness for Kripke models. Buss and Mints[2] gave a polynomial time algorithm, which extracts an i from a given derivation of α0 ∨ α1 in natural deduction such that αi is intuitionistically valid. Such a feasible algorithm based on sequent calculi is given in Buss and Pudlák[3], and Ferrari, Fiorentini and Fiorino[4] provides an algorithm for derivable sequents Γ ⇒ α0 ∨ α1 with sets Γ of Harrop formulas. The idea in these algorithms, which comes from [2], is to prove that one of formulas α0 and α1 is in a small set of sequents (immediately derivable sequents) relative to a given intuitionistic derivation of the disjunction α0 ∨ α1(Boundedness), for which there is a polynomial time algorithm testing the membership of sequents in the set, and any sequent in the set is readily seen to be intuitionistically valid. In [4] the authors introduce extraction calculi to generate the set. In [2] the proof of the Boundedness is done through a partial normalization in natural deduction, and the proof in [3] through cut-elmination. On the other side, one in [4] is based on an evaluation relation, a variant of Aczel’s slash[1], cf. [6]. In this note we consider the complexity of the DP with Harrop antecedents. We describe two proofs of Boundedness. One is obtained by a slight modification from [2], and the other is essentially the same as one in [4], but let us stress the fact that the evaluation relation is a feasible restriction of Aczel’s slash.