Affine Parikh automata

Cadilhac, Michaël; Finkel, Alain; McKenzie, Pierre

doi:10.1051/ita/2012013

Cited by 20 publications

(15 citation statements)

References 13 publications

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“…In the next step, we exploit the decomposition into a treeshaped constraint system and an alternation-bounded constraint system to reduce satisfiability to non-emptiness of counter automata. To this end, we use a type of counter automata known as Parikh automata [25,7]. In terms of expressiveness, these are equivalent to the classical reversal-bounded counter automata [20], but their syntax makes them convenient for our purposes.…”

Section: Assertions Of Types (V) and (Vi)mentioning

confidence: 99%

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Decidability, complexity, and expressiveness of first-order logic over the subword ordering

Halfon

Schnoebelen

Zetzsche

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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We consider first-order logic over the subword ordering on finite words, where each word is available as a constant. Our first result is that the Σ 1 theory is undecidable (already over two letters).We investigate the decidability border by considering fragments where all but a certain number of variables are alternation bounded, meaning that the variable must always be quantified over languages with a bounded number of letter alternations. We prove that when at most two variables are not alternation bounded, the Σ 1 fragment is decidable, and that it becomes undecidable when three variables are not alternation bounded. Regarding higher quantifier alternation depths, we prove that the Σ 2 fragment is undecidable already for one variable without alternation bound and that when all variables are alternation bounded, the entire first-order theory is decidable.

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Section: Assertions Of Types (V) and (Vi)mentioning

confidence: 99%

“…Since we can clearly formulate the condition Eq. (7) in Presburger arithmetic, P -compatibility is decidable.…”

Section: 1mentioning

confidence: 99%

Decidability, complexity, and expressiveness of first-order logic over the subword ordering

Halfon

Schnoebelen

Zetzsche

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…A deterministic version of PAs is known to be strictly weaker than arbitrary PAs [30]. PAs are also known to be equivalent to reversal-bounded multicounter machines [28] in the sense that they describe the same class of languages [30], [11].…”

Section: A Run Is Accepting If ρ(M) ∈ F × C a Word Is Accepted If Itmentioning

confidence: 99%

Path Logics for Querying Graphs: Combining Expressiveness and Efficiency

Figueira

Libkin

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

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We study logics expressing properties of paths in graphs that are tailored to querying graph databases: a data model for new applications such as social networks, the Semantic Web, biological data, crime detection, and others. The basic construct of such logics, a regular path query, checks for paths whose labels belong to a regular language. These logics fail to capture two commonly needed features: counting properties, and the ability to compare paths. It is known that regular path-comparison relations (e.g., prefix or equality) can be added without significant complexity overhead; however, adding common relations often demanded by applications (e.g., subword, subsequence, suffix) results in either undecidability or astronomical complexity.We propose, as a way around this problem, to use automata with counting functionalities, namely Parikh automata. They express many counting properties directly, and they approximate many relations of interest. We prove that with Parikh automata defining both languages and relations used in queries, we retain the low complexity of the standard path logics for graphs. In particular, this gives us efficient approximations to queries with prohibitively high complexity. We extend the best known decidability results by showing that even more expressive classes of relations are possible in query languages (sometimes with restriction on the shape of formulae). We also show that Parikh automata admit two convenient representations by analogs of regular expressions, making them usable in real-life querying.

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“…Such an automaton was successfully used to decide logics with counting as well as logics on data words [27,17]. Its expressiveness was studied in [11].…”

Section: Related Workmentioning

confidence: 99%

“Most of” leads to undecidability: Failure of adding frequencies to LTL

Bednarczyk

Michaliszyn

2021

Lecture Notes in Computer Science

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Linear Temporal Logic (LTL) interpreted on finite traces is a robust specification framework popular in formal verification. However, despite the high interest in the logic in recent years, the topic of their quantitative extensions is not yet fully explored. The main goal of this work is to study the effect of adding weak forms of percentage constraints (e.g. that most of the positions in the past satisfy a given condition, or that $$\sigma $$ σ is the most-frequent letter occurring in the past) to fragments of LTL. Such extensions could potentially be used for the verification of influence networks or statistical reasoning. Unfortunately, as we prove in the paper, it turns out that percentage extensions of even tiny fragments of LTL have undecidable satisfiability and model-checking problems. Our undecidability proofs not only sharpen most of the undecidability results on logics with arithmetics interpreted on words known from the literature, but also are fairly simple. We also show that the undecidability can be avoided by restricting the allowed usage of the negation, and discuss how the undecidability results transfer to first-order logic on words.

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Affine Parikh automata

Cited by 20 publications

References 13 publications

Decidability, complexity, and expressiveness of first-order logic over the subword ordering

Decidability, complexity, and expressiveness of first-order logic over the subword ordering

Path Logics for Querying Graphs: Combining Expressiveness and Efficiency

“Most of” leads to undecidability: Failure of adding frequencies to LTL

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