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Register a new userLTL with the Freeze Quantifier and Register Automata
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Added by
Ranko Lazic
Publication date
2006
fields
Informatics Engineering
and research's language is
English
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A data word is a sequence of pairs of a letter from a finite alphabet and an element from an infinite set, where the latter can only be compared for equality. To reason about data words, linear temporal logic is extended by the freeze quantifier, which stores the element at the current word position into a register, for equality comparisons deeper in the formula. By translations from the logic to alternating automata with registers and then to faulty counter automata whose counters may erroneously increase at any time, and from faulty and error-free counter automata to the logic, we obtain a complete complexity table for logical fragments defined by varying the set of temporal operators and the number of registers. In particular, the logic with future-time operators and 1 register is decidable but not primitive recursive over finite data words. Adding past-time operators or 1 more register, or switching to infinite data words, cause undecidability.
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Constraint LTL, a generalisation of LTL over Presburger constraints, is often used as a formal language to specify the behavior of operational models with constraints. The freeze quantifier can be part of the language, as in some real-time logics, but this variable-binding mechanism is quite general and ubiquitous in many logical languages (first-order temporal logics, hybrid logics, logics for sequence diagrams, navigation logics, logics with lambda-abstraction etc.). We show that Constraint LTL over the simple domain (N,=) augmented with the freeze quantifier is undecidable which is a surprising result in view of the poor language for constraints (only equality tests). Man
Controller synthesis for general linear temporal logic (LTL) objectives is a challenging task. The standard approach involves translating the LTL objective into a deterministic parity automaton (DPA) by means of the Safra-Piterman construction. One of the challenges is the size of the DPA, which often grows very fast in practice, and can reach double exponential size in the length of the LTL formula. In this paper we describe a single exponential translation from limit-deterministic Buchi automata (LDBA) to DPA, and show that it can be concatenated with a recent efficient translation from LTL to LDBA to yield a double exponential, enquote{Safraless} LTL-to-DPA construction. We also report on an implementation, a comparison with the SPOT library, and performance on several sets of formulas, including instances from the 2016 SyntComp competition.
We prove near-optimal trade-offs for quantifier depth versus number of variables in first-order logic by exhibiting pairs of $n$-element structures that can be distinguished by a $k$-variable first-order sentence but where every such sentence requires quantifier depth at least $n^{Omega(k/log k)}$. Our trade-offs also apply to first-order counting logic, and by the known connection to the $k$-dimensional Weisfeiler--Leman algorithm imply near-optimal lower bounds on the number of refinement iterations. A key component in our proof is the hardness condensation technique recently introduced by [Razborov 16] in the context of proof complexity. We apply this method to reduce the domain size of relational structures while maintaining the minimal quantifier depth to distinguish them in finite variable logics.
We solve some decision problems for timed automata which were recently raised by S. Tripakis in [ Folk Theorems on the Determinization and Minimization of Timed Automata, in the Proceedings of the International Workshop FORMATS2003, LNCS, Volume 2791, p. 182-188, 2004 ] and by E. Asarin in [ Challenges in Timed Languages, From Applied Theory to Basic Theory, Bulletin of the EATCS, Volume 83, p. 106-120, 2004 ]. In particular, we show that one cannot decide whether a given timed automaton is determinizable or whether the complement of a timed regular language is timed regular. We show that the problem of the minimization of the number of clocks of a timed automaton is undecidable. It is also undecidable whether the shuffle of two timed regular languages is timed regular. We show that in the case of timed Buchi automata accepting infinite timed words some of these problems are Pi^1_1-hard, hence highly undecidable (located beyond the arithmetical hierarchy).
We present a unified translation of LTL formulas into deterministic Rabin automata, limit-deterministic Buchi automata, and nondeterministic Buchi automata. The translations yield automata of asymptotically optimal size (double or single exponential, respectively). All three translations are derived from one single Master Theorem of purely logical nature. The Master Theorem decomposes the language of a formula into a positive boolean combination of languages that can be translated into {omega}-automata by elementary means. In particular, Safras, ranking, and breakpoint constructions used in other translations are not needed.
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