Deciding Regular Intersection Emptiness of Complete Problems for PSPACE and the Polynomial Hierarchy

Güler, Demen; Krebs, Andreas; Lange, Klaus-Jörn; Wolf, Petra

doi:10.1007/978-3-319-77313-1_12

Cited by 8 publications

(15 citation statements)

References 10 publications

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“…So we can speak about an NP-complete graph property Φ and ask how complex it is to decide the satisfiability for G F if the input is a marked graph (F, µ). 6 A first song about a remotely similar theme was released in 1991 by Salt 'n' Pepa.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

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Properties of Graphs Specified by a Regular Language

Diekert¹,

Fernau²,

Wolf³

2021

Preprint

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Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property Φ. What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question if is there exists one graph satisfying Φ? We approach this question by using formal languages for specifying families of graphs. In particular, we show that certain graph properties can be decided by studying the syntactic monoid of the specification language L if a certain torsion condition is satisfied. This condition holds trivially if L is regular.In order to show our results, we split L into a finite union of subsets Lα. Every Lα defines in a natural way a single finite graph Fα where some edges and vertices are marked. The marked graph in turn defines a set of graphs with a geometric description using the notion of graph retraction and where Fα appears as an induced subgraph. ACM Subject ClassificationTheory of computation → Formal languages and automata theory Keywords and phrases Graph properties, Regular languages, Periodic semigroups Funding Petra Wolf : DFG project FE 560/9-1 1 The notation int Reg refers to intersection non-emptiness with regular languages.

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Section: Conclusion and Open Problemsmentioning

confidence: 99%

“…For example, is it true that some graphs in ρ(L) are planar? Solving this type of decision problems was the starting point of the line of research in [6,14,15,16]. These so-called int Reg -problems 1 have been studied independently as regular realizability problems, for instance, in [2,13].…”

Section: Introductionmentioning

confidence: 99%

Properties of Graphs Specified by a Regular Language

Diekert¹,

Fernau²,

Wolf³

2021

Preprint

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“…We will follow the ideas presented in [9] of investigating what kinds of loops can occur in the automaton without violating the encoding format, namely loops inside a coefficient, loops over whole coefficients, and loops over whole inequalities. Definition 3.…”

Section: Construction Of the Condensed Automatonmentioning

confidence: 99%

“…If we consider regular sets of instances, this task can be formalised as checking whether a given regular language of P -instances (represented by a deterministic finite automaton) and the fixed language of positive P -instances have a non-empty intersection. This was the original viewpoint of the line of research introduced by Güler et al [9,23], where this problem is called the int Reg -problem of P (or int Reg (P ) for short). 2 The int Reg -problem has independently been studied under the name regular realizability problem RR(L), where the filter language L plays the role of problem P as defined above, i. e., RR(L) = int Reg (L) (see [1,15,17,19,20,21,16]).…”

Section: Introductionmentioning

confidence: 99%

“…In contrast to these research questions, the line of work initiated in [9,23] focuses on classical (hard) computational problems as filter languages and respective decision procedures heavily take advantage of the regularity of the set of input instances. Investigating the int Reg -problem for NP-complete problems shows that the decidability of their int Reg -problem is not trivial, e. g., int Reg (SAT) is decidable [9], whereas int Reg (Bounded Tiling) is not [23]. 3 This is particularly interesting because the original hardness proofs of SAT and Bounded Tiling are both given by directly encoding Turing-machine computations into a problem instance [3,6].…”

Section: Introductionmentioning

confidence: 99%

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On the Decidability of Finding a Positive ILP-Instance in a Regular Set of ILP-Instances

Wolf

2019

Descriptional Complexity of Formal Systems

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The regular intersection emptiness problem for a decision problem P (intReg(P )) is to decide whether a potentially infinite regular set of encoded P-instances contains a positive one. Since intReg(P ) is decidable for some NP-complete problems and undecidable for others, its investigation provides insights in the nature of NP-complete problems. Moreover, the decidability of the intReg-problem is usually achieved by exploiting the regularity of the set of instances; thus, it also establishes a connection to formal language and automata theory. We consider the intReg-problem for the well-known NP-complete problem Integer Linear Programming (ILP). It is shown that any DFA that describes a set of ILP-instances (in a natural encoding) can be reduced to a finite core of instances that contains a positive one if and only if the original set of instances did. This result yields the decidability of intReg(ILP).

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