Efficient loop-check for multimodal KD45n logic

Adomas Birštunas

doi:10.15388/lmr.2007.24227

Abstract

We introduce sequent calculus for multi-modal logic KD45n which uses efficient loop-check. Efficiency of the used loop-check is obtained by using marked modal operator squarei which is used as an alternative to sequent with histories ([2,3]).We use inference rules with or branches to make all rules invertible or semi-invertible. We showthe maximum height of the constructed derivation tree. Also polynomial space complexity is proved.

Highlights

Multi-modal logic KD45n is a part of the widely used BDI logic, described in [5].There is known sequent calculus for logic KD45n, but it uses inefficient loopcheck
Direct loop-check technique used in sequent calculus requires to check all the sequents in the current branch after every rule application
We introduce sequent calculus which uses only invertible or semi-invertible rules

Summary

Introduction

There is known sequent calculus for logic KD45n, but it uses inefficient (direct) loopcheck. Direct loop-check technique used in sequent calculus requires to check all the sequents in the current branch after every rule application. It means, that most of the time is used to compare sequents instead of applying inference rules. The main goal of the efficient loop-check is to make it work ‘locally’. This can be achieved by using some properties of the used logic. Sequents with histories can be used for this Such an approach is used in [2,3], where efficient loop-check for some modal logics is shown. Sequent calculus with an axiom , φ → , φ and rules (∨L), (∨R), (&L), (&R), (¬L), (¬R), (W eak), ( i) we call KD45n

Birštunas

Complexity

Conclusion