Safra's Determinization Research Articles

From LTL to deterministic automata

We present a new algorithm to construct a (generalized) deterministic Rabin automaton for an LTL formula $$\varphi $$ź. The automaton is the product of a co-Buchi automaton for $$\varphi $$ź and an array of Rabin automata, one for each $${\mathbf {G}}$$G-subformula of $$\varphi $$ź. The Rabin automaton for $${\mathbf {G}}\psi $$Gź is in charge of recognizing whether $${\mathbf {F}}{\mathbf {G}}\psi $$FGź holds. This information is passed to the co-Buchi automaton that decides on acceptance. As opposed to standard procedures based on Safra's determinization, the states of all our automata have a clear logical structure, which allows for various optimizations. Experimental results show improvement in the sizes of the resulting automata compared to existing methods.

Exploiting the Temporal Logic Hierarchy and the Non-Confluence Property for Efficient LTL Synthesis

The classic approaches to synthesize a reactive system from a linear temporal logic (LTL) specification first translate the given LTL formula to an equivalent omega-automaton and then compute a winning strategy for the corresponding omega-regular game. To this end, the obtained omega-automata have to be (pseudo)-determinized where typically a variant of Safra's determinization procedure is used. In this paper, we show that this determinization step can be significantly improved for tool implementations by replacing Safra's determinization by simpler determinization procedures. In particular, we exploit (1) the temporal logic hierarchy that corresponds to the well-known automata hierarchy consisting of safety, liveness, Buechi, and co-Buechi automata as well as their boolean closures, (2) the non-confluence property of omega-automata that result from certain translations of LTL formulas, and (3) symbolic implementations of determinization procedures for the Rabin-Scott and the Miyano-Hayashi breakpoint construction. In particular, we present convincing experimental results that demonstrate the practical applicability of our new synthesis procedure.

A tighter analysis of Piterman's Büchi determinization

Determinization and complementation are two fundamental problems in automata theory. Very recently, Piterman improved Safra's determinization and, presented a new construction which produces parity automata with a smaller size. We give a tighter analysis on that determinization construction and show that the number of states of the resulting deterministic automaton is bounded by 2 n ( n ! ) 2 instead of 2 n ! n n .

From Nondeterministic B\"uchi and Streett Automata to Deterministic Parity Automata

In this paper we revisit Safra's determinization constructions for automata on infinite words. We show how to construct deterministic automata with fewer states and, most importantly, parity acceptance conditions. Determinization is used in numerous applications, such as reasoning about tree automata, satisfiability of CTL*, and realizability and synthesis of logical specifications. The upper bounds for all these applications are reduced by using the smaller deterministic automata produced by our construction. In addition, the parity acceptance conditions allows to use more efficient algorithms (when compared to handling Rabin or Streett acceptance conditions).

Experiments with deterministic [formula omitted]-automata for formulas of linear temporal logic

This paper addresses the problem of generating deterministic ω -automata for formulas of linear temporal logic, which can be solved by applying well-known algorithms to construct a nondeterministic Büchi automaton for the given formula on which we then apply a determinization algorithm. We study here in detail Safra's determinization algorithm, present several heuristics that attempt to decrease the size of the resulting automata and report on experimental results.

Determinization and Memoryless Winning Strategies

We show that Safra's determinization of ω -automata with Streett (strong fairness) acceptance condition also gives memoryless winning strategies in infinite games, for the player whose acceptance condition is the complement of the Streett condition. Both determinization and memorylessness are essential parts of known proofs of Rabin's tree automata complementation lemma. Also, from Safra's determinization construction, along with its memoryless winning strategy extension, a single exponential complementation of Streett tree automata follows. A different single exponential construction and proof first appeared in [N. Klarlund (1992), Progress measures, immediate determinacy, and a subset construction for tree automata, in “Proceedings, 7th IEEE Symposium on Logics in Computer Science”].

Progress measures, immediate determinacy, and a subset construction for tree automata

Using the concept of progress measure, we give a new proof of Rabin's fundamental result that the languages defined by tree automata are closed under complementation. To do this we show that for certain infinite games based on tree automata, an immediate determinacy property holds for the player who is trying to win according to a Rabin acceptance condition. Immediate determinancy is stronger than the forgetful determinacy of Gurevich and Harrington, which depends on more information about the past, but applies to another class of games. Next, we show a graph theoretic duality theorem for winning conditions. Finally, we present an extended version of Safra's determinization construction. Together, these ingredients and the determinacy of Borel games yield a straightforward recipe for complementing tree automata. Our construction is almost optimal, i.e. the state space blow-up is essentially exponential- thus roughly the same as for automata on finite or infinite words. To our knowledge, no prior constructions have been better than double exponential.

Progress measures, immediate determinacy, and a subset construction for tree automata

<p>Using the concept of progress measure, we give a new proof of Rabin's fundamental result that the languages defined by tree automata are closed under complementation.</p><p>To do this we show that for certain infinite games based on tree automata, an <em>immediate determinacy</em> property holds for the player who is trying to win according to a Rabin acceptance condition. Immediate determinacy is stronger than the <em> forgetful determinacy</em> of Gurevich and Harrington, which depends on more information about the past, but applies to another class of games.</p><p>Next, we show a graph theoretic duality theorem for winning conditions. Finally, we present an extended version of Safra's determinization construction. Together, these ingredients and the determinacy of Borel games yield a straightforward recipe for complementing tree automata.</p><p>Our construction is almost optimal, i.e. the state space blow-up is essentially exponential --- thus roughly the same as for automata on finite or infinite words.</p><p>To our knowledge, no prior constructions have been better than double exponential.</p>