Rankers over Infinite Words

Temporal logics have gained prominence in computer science as a property specification language for reactive systems. There is even an IEEE standard temporal logic supported by a consortium of Electronic Design Tool developers. Such systems maintain ongoing interaction between the environment and the system and their specification requires formulating constraints on the sequence of steps performed by the system. Unlike Classical logics which explicitly use variables to range over time points, temporal logics, which are rooted in tense logics, provide a variable-free approach which deals with time implicitly, using modalities. The work on temporal logic for specifying and proving concurrent programs began with Pnueli's initial identification of this logic for reactive systems [Pnu77]. Lamport also used temporal logic to reason about properties of distributed systems [Lam80].We work in the setting of finite and infinite words over a finite alphabet. A diverse set of modalities can be formulated to give different temporal logics. However, over time, the linear temporal logic LTL has emerged as a standard formulation. A major driver for this choice is its economy of operators while being expressive; it just uses modalities U and S. The classical result of Kamp showed that the LTL logic is expressively complete with respect to FO-definable properties of words [Kamp68]. Moreover, as shown by Sistla and Clarke, the logic has elementary PSPACE-complete satisfiability [SC85]. Yet another class of temporal logics which provides very natural form of specification are the interval temporal logics. However, their high satisfaction complexity has prevented their widespread use.It can be seen from these developments that the concerns for expressive power of the temporal logic and its algorithmic complexity have been major drivers. They directly affect the usability of model checking tools developed. Several fragments/variants of LTL have been explored to improve its usability. For example, the industry standard PSL/Sugar adds regular expressions to LTL. Various forms of counting constructs allowing quantitative constraints to be enforced have also been added to LTL and to interval temporal logics. At the same time, keeping algorithmic complexity in mind, fragments of LTL such as TL [F, P] with low satisfaction complexity have been explored [EVW02,WI09]. But there are other possibilities.One less-known such theme is that of "deterministic logics". In our own experience, while implementing a validity checker for an interval temporal logic over word models, we found marked improvements in efficiency when nondeterministic modal operators were replaced by deterministic or unambiguous ones [KP05]. This led to our interest in results on unambiguous languages, initiated by Schützenberger [Sch76]. In a subsequent paper [LPS08] we learnt that these could also be thought of as boolean combinations of deterministic and co-deterministic products over a small class, the piecewise testable languages. We expanded the scope of our work to stu...

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“…We give the construction of these formulae below and we omit the proof of above lemma which can be found in [DKL10,PS13,Shah12]. ψ…”

Section: Ranker Directionalitymentioning

confidence: 99%

Deterministic Temporal Logics and Interval Constraints

Lodaya

Pandya

2017

Electron. Proc. Theor. Comput. Sci.

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“…The evaluation of an X-ranker r extends naturally to infinite words and the L(r) over A ∞ consists of all finite or infinite words on which r is defined. For more details we refer to [3]. [f ] ω ⊆ K. The language K is a finite union of restricted unambiguous monomials A * 1 a 1 · · · A * k a k A ∞ k+1 over A ∞ , see [5,Theorem 6.6].…”

Section: Lemma 11mentioning

confidence: 99%

“…Now, K is definable over A ∞ in the first-order fragment ∆ 2 [<], see [5,Theorem 6.6]. Thus K is a Boolean combination of languages L(r) over A ∞ for X-rankers r, see [3,Theorem 3]. It follows that L = K ∩ A * is a Boolean combination of languages L(r) over A * for X-rankers r.…”

Section: Lemma 11mentioning

confidence: 99%

“…It is therefore not surprising that a large part of Theorem 12 reduces to infinite words: The proof of the implication from (1) to (2) relies on a result of Diekert and Kufleitner [5,Theorem 6.6]. The step from (2) to (3) uses a characterization of X-rankers over infinite words [3,Theorem 3]. Showing the implication from (3) to ( 4) is the most technical part.…”

Section: Is a Finite Union Of Unambiguous Monomialsmentioning

confidence: 99%

Section: Is a Finite Union Of Unambiguous Monomialsmentioning

confidence: 99%

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Regular Ideal Languages and Their Boolean Combinations

Jahn

Kufleitner

Lauser

2012

Implementation and Application of Automata

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We consider ideals and Boolean combinations of ideals. For the regular languages within these classes we give expressively complete automaton models. In addition, we consider general properties of regular ideals and their Boolean combinations. These properties include effective algebraic characterizations and lattice identities.In the main part of this paper we consider the following deterministic one-way automaton models: unions of flip automata, weak automata, and Staiger-Wagner automata. We show that each of these models is expressively complete for regular Boolean combination of right ideals. Right ideals over finite words resemble the open sets in the Cantor topology over infinite words. An omega-regular language is a Boolean combination of open sets if and only if it is recognizable by a deterministic Staiger-Wagner automaton; and our result can be seen as a finitary version of this classical theorem. In addition, we also consider the canonical automaton models for right ideals, prefix-closed languages, and factorial languages.In the last section, we consider a two-way automaton model which is known to be expressively complete for two-variable first-order logic. We show that the above concepts can be adapted to these two-way automata such that the resulting languages are the right ideals (resp. prefix-closed languages, resp. Boolean combinations of right ideals) definable in two-variable first-order logic.

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Rankers over Infinite Words

Dartois

Kufleitner

Lauser

2010

Developments in Language Theory

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Abstract. We consider the fragments FO, and ∆ 2 of first-order logic FO[<] over finite and infinite words. For all four fragments, we give characterizations in terms of rankers. In particular, we generalize the notion of a ranker to infinite words in two possible ways. Both extensions are natural in the sense that over finite words, they coincide with classical rankers and over infinite words, they both have the full expressive power of FO 2 . Moreover, the first extension of rankers admits a characterization of Σ 2 ∩ FO 2 while the other leads to a characterization of Π 2 ∩ FO 2 . Both versions of rankers yield characterizations of the fragment ∆ 2 = Σ 2 ∩ Π 2 . As a byproduct, we also obtain characterizations based on unambiguous temporal logic and unambiguous interval temporal logic.

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Rankers over Infinite Words

Cited by 6 publications

References 11 publications

Deterministic Temporal Logics and Interval Constraints

Deterministic Temporal Logics and Interval Constraints

Regular Ideal Languages and Their Boolean Combinations

Rankers over Infinite Words

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