Partial derivatives on graphs for Kleene allegories

Nakamura, Yoshiki

doi:10.1109/lics.2017.8005132

Cited by 6 publications

(2 citation statements)

References 18 publications

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“…For Kleene lattices (•, ∨, ∧, * ), complexity is, to the best of the author's knowledge, an open problem. However, there are decidability results on more specific classes of Kleene lattices [3,5,37,12] (lattices in all these classes are distributive, which is not generally true for Kleene lattices), which makes it plausible that the logic of Kleene lattices is also decidable. In contrast, lattice operations are not crucial for undecidability: the logic of *-continuous residuated monoids with iteration (•, \, /, * ) is also Π 0 1 -complete [33].…”

Section: Introduction 1action Lattices and Their Logicsmentioning

confidence: 99%

The Logic of Action Lattices is Undecidable

Kuznetsov

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. This logic involves Kleene star, axiomatized by an induction scheme. For a stronger system which uses an ω-rule instead (infinitary action logic) Buszkowski and Palka (2007) have proved Π 0 1 -completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by D. Kozen in 1994. In this article, we show that it is undecidable, more precisely, Σ 0 1 -complete. We also prove the same complexity results for all recursively enumerable logics between action logic and infinitary action logic; for fragments of those only one of the two lattice (additive) connectives; for action logic extended with the law of distributivity.Here the lower bound is due to Buszkowski and the upper one is due to Palka; recently A. Das and D. Pous [11] gave another proof of the Π 0 1 upper bound for ACT ω , based on non-well-founded proofs.

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Section: Introduction 1action Lattices and Their Logicsmentioning

confidence: 99%

The Logic of Action Lattices is Undecidable

Kuznetsov

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…The case of converse was dealt with by Ésik et al: the equational theories of REL and LANG differ in the presence of converse but both can be axiomatised [6,18], and they remain PSpace-complete [10]. The case of intersection (with or without converse or the various constants) is significantly more difficult, and remains partly open, see [3,9,34,16]. In this paper we focus on the addition of a constant ⊤, interpreted as the full language in LANG and as the full relation in REL.…”

Section: Introductionmentioning

confidence: 99%

On Tools for Completeness of Kleene Algebra with Hypotheses

Pous

Rot

Wagemaker

2021

Lecture Notes in Computer Science

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In the literature on Kleene algebra, a number of variants have been proposed which impose additional structure specified by a theory, such as Kleene algebra with tests (KAT) and the recent Kleene algebra with observations (KAO), or make specific assumptions about certain constants, as for instance in NetKAT. Many of these variants fit within the unifying perspective offered by Kleene algebra with hypotheses, which comes with a canonical language model constructed from a given set of hypotheses. For the case of KAT, this model corresponds to the familiar interpretation of expressions as languages of guarded strings. A relevant question therefore is whether Kleene algebra together with a given set of hypotheses is complete with respect to its canonical language model. In this paper, we revisit, combine and extend existing results on this question to obtain tools for proving completeness in a modular way. We showcase these tools by reproving completeness of KAT and KAO, and prove completeness of a new variant of KAT where the collection of tests only forms a distributive lattice.

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Expressive Power and Succinctness of the Positive Calculus of Relations

Nakamura

2020

Lecture Notes in Computer Science

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In this paper, we study the expressive power and succinctness of the positive calculus of relations . We show that (1) the calculus has the same expressive power as that of three-variable existential positive (first-order) logic in terms of binary relations, and (2) the calculus is exponentially less succinct than three-variable existential positive logic, namely, there is no polynomial-size translation from three-variable existential positive logic to the calculus, whereas there is a linear-size translation in the converse direction. Additionally, we give a more fine-grained expressive power equivalence between the (full) calculus of relations and three-variable first-order logic in terms of the quantifier alternation hierarchy. It remains open whether the calculus of relations is also exponentially less succinct than three-variable first-order logic.

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Partial derivatives on graphs for Kleene allegories

Cited by 6 publications

References 18 publications

The Logic of Action Lattices is Undecidable

The Logic of Action Lattices is Undecidable

On Tools for Completeness of Kleene Algebra with Hypotheses

Expressive Power and Succinctness of the Positive Calculus of Relations

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