An optimal construction of Hanf sentences

Bollig, Benedikt; Kuske, Dietrich

doi:10.1016/j.jal.2012.01.002

Cited by 18 publications

(31 citation statements)

References 33 publications

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“…In [2], Bollig and Kuske presented a 3-fold exponential algorithm that transforms a given FO(σ)-formula into a ν d -equivalent formula in Hanf normal form. In the slightly more general setting of ν-bounded structures, their proof yields the following (a proof will be included in the full version of this paper).…”

Section: Hanf Normal Formmentioning

confidence: 99%

“…A closer analysis of the properties of the algorithms of (1) and [2] then shows that the overall running time is still 3-fold exponential in the size of the input formula Φ. A small additional observation also allows us to handle arbitrary formulas Φ rather than just sentences.…”

Section: Introductionmentioning

confidence: 97%

“…In particular, there are very general algorithmic meta-theorems stating that first-order model checking is fixed-parameter tractable for various classes of structures, and that first-order definable optimisation problems on such classes have polynomial time approximation schemes. In the context of such algorithms, questions about the efficiency of the normal forms have recently attracted interest [2], [5], [6], [13], [14], [20].…”

Section: Introductionmentioning

confidence: 99%

“…The algorithm from [6], at first sight, seems to be non-elementary, but it actually is 4-fold exponential [3]. Finally, [2] presents a 3-fold exponential algorithm and proves a matching lower bound.…”

Section: Introductionmentioning

confidence: 99%

“…(2) For the 3-fold exponential Gaifman normal form algorithm for structures of bounded degree, we can then combine (1) with the 3-fold exponential Hanf normal form construction of [2]: First, transform an arbitrary given first-order sentence Φ into a Hanf normal form Φ H equivalent to Φ on the class of all structures of degree at most d. Then, Φ H is a Boolean combination of sentences of the form ( * ) which, according to (1), can be transformed into Gaifman normal form. A closer analysis of the properties of the algorithms of (1) and [2] then shows that the overall running time is still 3-fold exponential in the size of the input formula Φ.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An Optimal Gaifman Normal Form Construction for Structures of Bounded Degree

Heimberg

Kuske

Schweikardt

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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Abstract-This paper's main result presents a 3-fold exponential algorithm that transforms a first-order formula ϕ together with a number d into a formula in Gaifman normal form that is equivalent to ϕ on the class of structures of degree at most d. For structures of polynomial growth, we even get a 2-fold exponential algorithm.These results are complemented by matching lower bounds: We show that for structures of degree 2, a 2-fold exponential blow-up in the size of formulas cannot be avoided. And for structures of degree 3, a 3-fold exponential blow-up is unavoidable.As a result of independent interest we obtain a 1-fold exponential algorithm which transforms a given first-order sentence ϕ of a very restricted shape into a sentence in Gaifman normal form that is equivalent to ϕ on all structures.

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Section: Hanf Normal Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An Optimal Gaifman Normal Form Construction for Structures of Bounded Degree

Heimberg

Kuske

Schweikardt

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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Parameterized Synthesis for Fragments of First-Order Logic Over Data Words

Bérard¹,

Bollig²,

Lehaut³

et al. 2020

Lecture Notes in Computer Science

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We study the synthesis problem for systems with a parameterized number of processes. As in the classical case due to Church, the system selects actions depending on the program run so far, with the aim of fulfilling a given specification. The difficulty is that, at the same time, the environment executes actions that the system cannot control. In contrast to the case of fixed, finite alphabets, here we consider the case of parameterized alphabets. An alphabet reflects the number of processes, which is static but unknown. The synthesis problem then asks whether there is a finite number of processes for which the system can satisfy the specification. This variant is already undecidable for very limited logics. Therefore, we consider a first-order logic without the order on word positions. We show that even in this restricted case synthesis is undecidable if both the system and the environment have access to all processes. On the other hand, we prove that the problem is decidable if the environment only has access to a bounded number of processes. In that case, there is even a cutoff meaning that it is enough to examine a bounded number of process architectures to solve the synthesis problem. Partly supported by ANR FREDDA (ANR-17-CE40-0013).

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An Automaton over Data Words That Captures EMSO Logic

Bollig

2011

CONCUR 2011 – Concurrency Theory

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Abstract. We develop a general framework for the specification and implementation of systems whose executions are words, or partial orders, over an infinite alphabet. As a model of an implementation, we introduce class register automata, a one-way automata model over words with multiple data values. Our model combines register automata and class memory automata. It has natural interpretations. In particular, it captures communicating automata with an unbounded number of processes, whose semantics can be described as a set of (dynamic) message sequence charts. On the specification side, we provide a local existential monadic second-order logic that does not impose any restriction on the number of variables. We study the realizability problem and show that every formula from that logic can be effectively, and in elementary time, translated into an equivalent class register automaton.

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An optimal construction of Hanf sentences

Cited by 18 publications

References 33 publications

An Optimal Gaifman Normal Form Construction for Structures of Bounded Degree

An Optimal Gaifman Normal Form Construction for Structures of Bounded Degree

Parameterized Synthesis for Fragments of First-Order Logic Over Data Words

An Automaton over Data Words That Captures EMSO Logic

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