[Paper Review] Deterministic Automata for the (F,G)-fragment of LTL

Jan Křetínský, Javier Esparza|arXiv (Cornell University)|Apr 23, 2012

Formal Methods in Verification9 references4 citations

TL;DR

This paper presents a direct, deterministic translation from the (F,G)-fragment of Linear Temporal Logic (LTL) into deterministic Muller automata, bypassing non-deterministic B"uchi automata and Safra-style determinization. The method uses boolean combinations of formulas from the closure of the input formula, enabling logical state merging and compact generalized Rabin acceptance conditions, resulting in significantly smaller automata—e.g., 1560 states vs. over a million for ltl2dstar—especially for fairness conditions.

ABSTRACT

When dealing with linear temporal logic properties in the setting of e.g. games or probabilistic systems, one often needs to express them as deterministic omega-automata. In order to translate LTL to deterministic omega-automata, the traditional approach first translates the formula to a non-deterministic Büchi automaton. Then a determinization procedure such as of Safra is performed yielding a deterministic ω-automaton. We present a direct translation of the (F,G)-fragment of LTL into deterministic ω-automata with no determinization procedure involved. Since our approach is tailored to LTL, we often avoid the typically unnecessarily large blowup caused by general determinization algorithms. We investigate the complexity of this translation and provide experimental results and compare them to the traditional method.

Motivation & Objective

• To eliminate the need for non-deterministic B"uchi automata and subsequent Safra-style determinization when translating LTL formulas.
• To develop a direct, deterministic translation for the (F,G)-fragment of LTL that avoids the large state-space blowup typical of general determinization.
• To construct automata with a logically structured state space that enables efficient optimization via logical equivalence.
• To provide a compact representation of acceptance conditions using generalized Rabin, and to enable efficient transformation to standard Rabin automata.
• To empirically evaluate the method against existing tools like ltl2dstar, especially for probabilistic model checking and fairness conditions.

Proposed method

• Construct a deterministic Muller automaton where states are boolean combinations of formulas from the closure of the input LTL formula.
• Use logical equivalence to merge states with identical semantic properties, reducing state space size without additional computation.
• Represent the Muller acceptance condition compactly as a generalized Rabin condition to reduce complexity.
• Provide an efficient transformation from generalized Rabin to standard Rabin automata using a known construction.
• Leverage the logical structure of the state space to enable optimizations based on propositional equivalence and tautology detection.
• Implement the construction and compare its performance with ltl2dstar on benchmark formulas, particularly fairness conditions.

Experimental results

Research questions

• RQ1 Can a direct translation from the (F,G)-fragment of LTL to deterministic automata be constructed without relying on non-deterministic B"uchi automata and Safra-style determinization?
• RQ2 Does the resulting automaton achieve a significantly smaller state space than existing tools like ltl2dstar, especially for fairness conditions?
• RQ3 Can the acceptance condition be represented compactly using generalized Rabin, and is the transformation to standard Rabin automata efficient?
• RQ4 To what extent can logical equivalences between subformulas be exploited to reduce the state space in the deterministic automaton?
• RQ5 Is the state space structure of the proposed automaton more amenable to further optimizations than those from general determinization?

Key findings

• For the fairness formula V3i=1(GFai →GFbi), the proposed method generates a deterministic Rabin automaton with only 1560 states, compared to over one million states produced by ltl2dstar.
• The state space of the constructed automaton is significantly smaller than that of ltl2dstar, particularly for infinitary properties such as fairness conditions.
• The use of generalized Rabin acceptance conditions allows for a compact representation of the acceptance condition, reducing the number of acceptance sets.
• The method enables direct state merging based on logical equivalence, reducing the state space without additional computational cost.
• The automaton's state space has a clear logical structure, making it more amenable to further optimizations than the 'messy' state space of Safra-based constructions.
• The approach is competitive with state-of-the-art tools and shows substantial advantages for classes of formulas relevant to probabilistic model checking.