Departmental Papers (CIS)

Document Type

Conference Paper

Date of this Version

June 2001

Comments

Copyright © 2001 IEEE. Reprinted from Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science, 2001, pages 291-300.
Publisher URL: http://ieeexplore.ieee.org/xpl/tocresult.jsp?isNumber=20180&page=1

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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.

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Date Posted: 29 October 2004

This document has been peer reviewed.