Departmental Papers (CIS)

Document Type

Conference Paper

Subject Area

CPS Theory

Date of this Version

June 2001

Publication Title

Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science, 2001

First Page

291

Last Page

300

DOI

10.1109/LICS.2001.932505

Abstract

Deciding infinite two-player games on finite graphs with the winning condition specified by a linear temporal logic (LTL) formula, is known to be 2EXPTIME-complete. In this paper, we identify LTL fragments of lower complexity. Solving LTL games typically involves a doubly-exponential translation from LTL formulas to deterministic omega-automata. First, we show that the longest distance (length of the longest simple path) of the generator is also an important parameter, by giving an O(dlog n)-space procedure to solve a Büchi game on a graph with n vertices and longest distance d. Then, for the LTL fragment with only eventualities and conjunctions, we provide a translation to deterministic generators of exponential size and linear longest distance, show both of these bounds to be optimal, and prove the corresponding games to be PSPACE-complete. Introducing next modalities in this fragment, we provide a translation to deterministic generators still of exponential size but also with exponential longest distance, show both of these bounds to be optimal, and prove the corresponding games to be EXPTIME-complete. For the fragment resulting by further adding disjunctions, we provide a translation to deterministic generators of doubly-exponential size and exponential longest distance, show both of these bounds to be optimal, and prove the corresponding games to be EXPSPACE. Finally, we show tightness of the double-exponential bound on the size as well as the longest distance for deterministic generators for LTL even in the absence of next and until modalities.

Permission Statement

© 2001 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

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Date Posted: 29 October 2004

This document has been peer reviewed.