Converting non-classical matrix proofs into sequent-style systems

Abstract

Abstract. We present a uniform algorithm for transforming matrix proofs in classical, constructive, and modal logics into sequent style proofs. Making use of a similarity between matrix methods and Fitting's prefixed tableaus we first develop a procedure for extracting a prefixed sequent proof from a given matrix proof. By considering the additional restrictions on the order of rule applications we then extend this procedure into an algorithm which generates a conventional sequent proof. Our algorithm is based on unified representations of matrix characterizations for various logics as well as of prefixed and usual sequent calculi. The peculiarities of a logic are encoded by certain parameters which are summarized in tables to be consulted by the algorithm.

A Multi-level Approach to Program Synthesis

“…After reducing the β-position β 1 the reduction ordering is split into two suborderings ∝ 1 , ∝ 2 and the conversion continues separately on each of the suborderings. For this we have developed an operation split (∝ , β 1 ) [37] which first splits the reduction ordering ∝ . Secondly non-normal form reductions are applied to each of the ∝ i in order to delete redundancies from ∝ i which are no longer relevant for the corresponding branch of the sequent proof.…”

“…Adding two wait-labels dynamically to α 6 and α 5 completes ∝ and avoids this deadlock during traversal. For a more detailed presentation of this approach as well as for an algorithmic realization we refer to [37].…”

“…In [37,38] we have shown that complete redundancy deletion after splitting at β-positions cannot be performed efficiently when only the spanning mating is given from the matrix proof. Efficiency means that the selection of proof-relevant subrelations from the ∝ i should avoid any additional search.…”

“…Our approach for reconstructing LJ mc -proofs from MJ -proofs has been uniformly extended to various non-classical logics [37,21] for which matrix characterizations exist. A uniform representation of different logics and proofs within logical calculi as well as abstract descriptions for integrating special properties of these logics in a uniform way, e.g.…”

Problem-oriented applications of automated theorem proving

“…Other approaches try to present them even in natural language [DGH + 94]. Similar approaches have been developed for non-classical logics as well [SK95,SK96].…”