Structure Theorem and Strict Alternation Hierarchy for FO^2 on Words

Weis, Philipp; Immerman, Neil

doi:10.2168/lmcs-5(3:4)2009

Cited by 32 publications

(39 citation statements)

References 21 publications

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“…The reader should consult [17] and [21] for the definition of FO 2 [<] and the alternation hierarchy within it. For our purposes here, they are to be viewed simply as the language classes corresponding to certain varieties of finite monoids, as discussed below.…”

Section: Preliminariesmentioning

confidence: 99%

“…In the original model-theoretic study of the alternation hierarchy in FO 2 [<], Weis and Immerman [21] and also Kufleitner and Weil [9] showed that while the hierarchy is strict, it collapses for each fixed-size alphabet. An algebraic proof of strictness was given in [17], using the identities that form the subject of the present paper.…”

Section: Collapse Of the Hierarchymentioning

confidence: 99%

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An Effective Characterization of the Alternation Hierarchy in Two-Variable Logic

Krebs

Straubing

2017

ACM Trans. Comput. Logic

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We characterize the languages in the individual levels of the quantifier alternation hierarchy of first-order logic with two variables by identities. This implies decidability of the individual levels. More generally we show that two-sided semidirect products with J as the right-hand factor preserve decidability. , the second author provided an algebraic characterization of the levels of the hierarchy, showing that they correspond to the levels of weakly iterated two-sided semidirect products of the pseudovariety J of finite J -trivial monoids. This still left open the problem of decidability of the hierarchy: effectively determining from a description of a regular language the lowest level of the hierarchy to which the language belongs. This problem was apparently solved in Almeida-Weil [2], from which explicit identities for the iterated product varieties can be extracted. However, an error in that paper called the correctness of these results into question. Here we show that the given identities do indeed characterize these pseudovarieties. In particular, since it is possible to verify effectively whether a given finite monoid satisfies one of these identities, we obtain an effective procedure for exactly determining the alternation depth of a regular language definable in two-variable logic. ACM Subject Classification

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Section: Preliminariesmentioning

confidence: 99%

Section: Collapse Of the Hierarchymentioning

confidence: 99%

An Effective Characterization of the Alternation Hierarchy in Two-Variable Logic

Krebs

Straubing

2017

ACM Trans. Comput. Logic

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“…Investigating the automata for UL, Schwentick, Thérien and Vollmer [STV02] defined Partially Ordered 2-Way Deterministic Automata (po2dfa) and showed that these exactly recognize the language class UL. Weis and Immerman have characterized UL as a boolean combination of "rankers" [WI09]. A survey paper [DGK08] describes this language class and its characterizations.…”

Section: Deterministic Logics and Unambiguous Star-free Langaugesmentioning

confidence: 99%

Deterministic Temporal Logics and Interval Constraints

Lodaya

Pandya

2017

Electron. Proc. Theor. Comput. Sci.

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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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“…5 The idea is that 5 The presented relations could also be defined by (condensed) rankers (as it is done in [14] and [15]). Rankers were introduced by Weis and Immerman [30] (thus, the WI exponent in ≡ WI m,n ) who reused these relations hold on two words u and v if both words allow for the same sequence of factorizations at the first or last occurrence of a letter. For this, we use the following recursive definition.…”

Section: Relations For the Trotter-weil Hierarchymentioning

confidence: 99%

The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

Kufleitner

Wächter

2016

Computer Science – Theory and Applications

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Abstract. For two given ω-terms α and β, the word problem for ω-terms over a variety V asks whether α = β in all monoids in V. We show that the word problem for ω-terms over each level of the Trotter-Weil Hierarchy is decidable. More precisely, for every fixed variety in the Trotter-Weil Hierarchy, our approach yields an algorithm in nondeterministic logarithmic space (NL). In addition, we provide deterministic polynomial time algorithms which are more efficient than straightforward translations of the NL-algorithms. As an application of our results, we show that separability by the so-called corners of the Trotter-Weil Hierarchy is witnessed by ω-terms (this property is also known as ω-reducibility). In particular, the separation problem for the corners of the Trotter-Weil Hierarchy is decidable.

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Structure Theorem and Strict Alternation Hierarchy for FO^2 on Words

Cited by 32 publications

References 21 publications

An Effective Characterization of the Alternation Hierarchy in Two-Variable Logic

An Effective Characterization of the Alternation Hierarchy in Two-Variable Logic

Deterministic Temporal Logics and Interval Constraints

The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

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