Two-variable Logic with a Between Relation

Krebs, Andreas; Lodaya, Kamal; Pandya, Paritosh K.; Straubing, Howard

doi:10.1145/2933575.2935308

Cited by 11 publications

(25 citation statements)

References 27 publications

(46 reference statements)

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“…A different but related between predicate (namely a(x, y), for a ∈ A, is true if there is an a-labelled position between positions x and y) was introduced and studied in [5,6,7]. Such a predicate is not definable in FO 2 (<), the two variable fragment of first-order logic (which corresponds to the well known semigroup variety DA [12]).…”

Section: Related Workmentioning

confidence: 99%

Logics for Reversible Regular Languages and Semigroups with Involution

Gastin

Manuel

Govind

2019

Developments in Language Theory

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We present MSO and FO logics with predicates 'between' and 'neighbour' that characterise various fragments of the class of regular languages that are closed under the reverse operation. The standard connections that exist between MSO and FO logics and varieties of finite semigroups extend to this setting with semigroups extended with an involution. The case is different for FO with neighbour relation where we show that one needs additional equations to characterise the class. ACM Subject Classification Theory of computation → Formal languages and automata theory;Theory of computation → Logic

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Section: Related Workmentioning

confidence: 99%

Logics for Reversible Regular Languages and Semigroups with Involution

Gastin

Manuel

Govind

2019

Developments in Language Theory

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“…Those on the left are the classes of the original dot depth hierarchy of Cohen and Brzozowski [CB71]. The logics which we have introduced in [KLPS16,KLPS18] and in this paper are at top centre. They have a nonempty intersection with every level of both the hierarchies.…”

Section: Alternation Depthmentioning

confidence: 99%

“…The situation appears to be different if we bound the size of the alphabet. In an earlier paper [KLPS16], we showed that if |A| = 2, then the alternation depth of languages in A * definable in FO 2 [<, bet] is bounded above by 3. We conjecture that for any fixed alphabet A, the alternation depth is bounded above by a linear function of |A|, but this question remains open.…”

Section: Alternation Depthmentioning

confidence: 99%

Two-variable logics with some betweenness relations: Expressiveness, satisfiability and membership

Krebs,

Lodaya,

Pandya

et al. 2019

Preprint

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We study two extensions of FO 2 [<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, 'the letter a appears between positions x and y' and 'the factor u appears between positions x and y'. These are, in a sense, the simplest properties that are not expressible using only two variables.We present several logics, both first-order and temporal, that have the same expressive power, and find matching lower and upper bounds for the complexity of satisfiability for each of these formulations. We give effective conditions, in terms of the syntactic monoid of a regular language, for a property to be expressible in these logics. This algebraic analysis allows us to prove, among other things, that our new logics have strictly less expressive power than full first-order logic FO[<]. Our proofs required the development of novel techniques concerning factorizations of words.

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“…One can also define expressively equivalent two-variable fragments of first-order logic corresponding to the classes of interval constraints, as in our earlier work [KLPS16], but we do not pursue this here.…”

Section: Sublogicsmentioning

confidence: 99%

“…In fact threshold counting can be reduced to invariant counting or to modulo counting. In our earlier paper [KLPS16], we used the first reduction as the basis for a decision procedure. Here we use the second theorem as the basis for our decision procedure.…”

Section: Expressivenessmentioning

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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Two-variable Logic with a Between Relation

Cited by 11 publications

References 27 publications

Logics for Reversible Regular Languages and Semigroups with Involution

Logics for Reversible Regular Languages and Semigroups with Involution

Two-variable logics with some betweenness relations: Expressiveness, satisfiability and membership

Deterministic Temporal Logics and Interval Constraints

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