Reasoning about Data Repetitions with Counter Systems

Demri, Stéphane; Figueira, Diego; Praveen, M.

doi:10.2168/lmcs-12(3:1)2016

Cited by 18 publications

(33 citation statements)

References 45 publications

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“…We recall the syntax and semantics of the logic of repeating values from [10,11]. This logic extends the usual propositional linear temporal logic with the ability to reason about repetitions of data values from an infinite domain.…”

Section: Logic Of Repeating Valuesmentioning

confidence: 99%

“…This has led to the study of linear temporal logic extended with these kind of tests, called Logic of Repeating Values (LRV) [10]. The satisfiability problem for LRV is inter-reducible with the reachability problem for Vector Addition Systems with States (VASS), and when the logic is restricted to testing remote repetitions only in the future, it is inter-reducible with the coverability problem for VASS [10,11]. These connections also extend to data trees and branching VASS [3].…”

Section: Introductionmentioning

confidence: 99%

“…Contributions By combining known relations between satisfiability of (fragments of) LRV and (control state) reachability in VASS [10,11] with existing knowledge about realizability games ( [19] and numerous papers expanding on it), it is not difficult to show that realizability games for LRV are related to games on VASS. Using known results about undecidability of games on VASS, it is again not difficult to show that realizability games for LRV are undecidable.…”

Section: Introductionmentioning

confidence: 99%

“…Our first contribution in this paper is to identify that the corresponding asymmetry in LRV realizability is to give only Boolean variables to one of the players and let the logic test only for remote repetitions in the past (and disallow testing for remote repetitions in the future). Once this identification of the fragment is made, the proof of its inter-reducibility with single-sided VASS games follows more or less along expected lines by adapting techniques developed in [10,11].…”

Section: Introductionmentioning

confidence: 99%

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Playing with Repetitions in Data Words Using Energy Games

Figueira

Praveen

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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We introduce two-player games which build words over infinite alphabets, and we study the problem of checking the existence of winning strategies. These games are played by two players, who take turns in choosing valuations for variables ranging over an infinite data domain, thus generating multi-attributed data words. The winner of the game is specified by formulas in the Logic of Repeating Values, which can reason about repetitions of data values in infinite data words. We prove that it is undecidable to check if one of the players has a winning strategy, even in very restrictive settings. However, we prove that if one of the players is restricted to choose valuations ranging over the Boolean domain, the games are effectively equivalent to single-sided games on vector addition systems with states (in which one of the players can change control states but cannot change counter values), known to be decidable and effectively equivalent to energy games.Previous works have shown that the satisfiability problem for various variants of the logic of repeating values is equivalent to the reachability and coverability problems in vector addition systems. Our results raise this connection to the level of games, augmenting further the associations between logics on data words and counter systems.We denote by Z the set of integers and by N the set of non-negative integers. We let N + denote the set of integers that are strictly greater than 0. For any set S, we denote by S * (resp. S ω ) the set of all finite (resp. countably infinite) sequences of elements in S. For a sequence σ ∈ S * , we denote its length by |σ|. We denote by P(S) (resp. P + (S)) the set of all subsets (resp. non-empty subsets) of S. Logic of repeating valuesWe recall the syntax and semantics of the logic of repeating values from [10,11]. This logic extends the usual propositional linear temporal logic with the ability to reason about repetitions of data values from an infinite domain. We let this logic use both Boolean variables (i.e., propositions) and data variables ranging over an infinite data domain D. The Boolean variables can be simulated by data variables. However, we need to consider fragments of the logic, for which explicitly having Boolean variables is convenient. Let BVARS = {q, t, . . .} be a countably infinite set of Boolean variables ranging over { , ⊥}, and let DVARS = {x, y, . . .} be a countably infinite set of 'data' variables ranging over D. We denote by LRV the logic whose formulas are defined as follows: 1A valuation is the union of a mapping from BVARS to { , ⊥} and a mapping from DVARS to D. A model is a finite or infinite sequence of valuations. We use σ to denote models and σ(i) denotes the i th valuation in σ, where i ∈ N + . For any model σ and position i ∈ N + , the satisfaction relation |= is defined inductively as follows. The semantics of temporal operators next (X), previous (X −1 ), until (U), since (S) and the Boolean connectives are defined in the usual way, but for the sake of completeness we provide their 1 In a pr...

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Section: Logic Of Repeating Valuesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Playing with Repetitions in Data Words Using Energy Games

Figueira

Praveen

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…We can adapt this proof and formulate the necessary conditions for a data word to encode a solution using only the attributes x z y. With ideas from [DFP13] we can also omit using past-time operators. Moreover, this result can be generalised to arbitrary quasi-orderings that contain three attributes x z y.…”

Section: Semantics Of Ltl ↓mentioning

confidence: 99%

On Freeze LTL with Ordered Attributes

Decker

Thoma

2016

Lecture Notes in Computer Science

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This paper is concerned with Freeze LTL, a temporal logic on data words with registers. In a (multi-attributed) data word each position carries a letter from a finite alphabet and assigns a data value to a fixed, finite set of attributes. The satisfiability problem of Freeze LTL is undecidable if more than one register is available or tuples of data values can be stored and compared arbitrarily. Starting from the decidable one-register fragment we propose an extension that allows for specifying a dependency relation on attributes. This restricts in a flexible way how collections of attribute values can be stored and compared. This conceptual dimension is orthogonal to the number of registers or the available temporal operators. The extension is strict. Admitting arbitrary dependency relations, satisfiability becomes undecidable. Tree-like relations, however, induce a family of decidable fragments escalating the ordinal-indexed hierarchy of fast-growing complexity classes, a recently introduced framework for non-primitive recursive complexities. This results in completeness for the class Fε 0 . We employ nested counter systems and show that they relate to the hierarchy in terms of the nesting depth.

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A Hypersequent Calculus with Clusters for Data Logic over Ordinals

Lick

2019

Lecture Notes in Computer Science

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We study freeze tense logic over well-founded data streams. The logic features past-and future-navigating modalities along with freeze quantifiers, which store the datum of the current position and test data (in)equality later in the formula. We introduce a decidable fragment of that logic, and present a proof system that is sound for the whole logic, and complete for this fragment. Technically, this is a hypersequent system enriched with an ordering, clusters, and annotations. The proof system is tailored for proof search, and yields an optimal coNP complexity for validity and a small model property for our fragment.

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Reasoning about Data Repetitions with Counter Systems

Cited by 18 publications

References 45 publications

Playing with Repetitions in Data Words Using Energy Games

Playing with Repetitions in Data Words Using Energy Games

On Freeze LTL with Ordered Attributes

A Hypersequent Calculus with Clusters for Data Logic over Ordinals

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