Turing Machines with Atoms

Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modeled as coalgebras. The fact that bisimulations up to context can be safely used in any language specified by GSOS rules can also be seen as an instance of our framework, using the well-known observation by Turi and Plotkin that such languages form bialgebras. In the second part of the paper, we provide a new categorical treatment of weak bisimilarity on labeled transition systems and we prove the soundness of up-to context for weak bisimulations of systems specified by cool rule formats, as defined by Bloom to ensure congruence of weak bisimilarity. The weak transition systems obtained from such cool rules give rise to lax bialgebras, rather than to bialgebras. Hence, to reach our goal, we extend the categorical framework developed in the first part to an ordered setting. B Daniela Petrişan

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“…In contrast to (strict) models (see (8)), in a lax model the converse does not hold: not all the transitions are derivable from the GSOS rules.…”

Section: S Xmentioning

confidence: 95%

A general account of coinduction up-to

et al. 2016

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“…Net models similar to Petri nets with data have been continuously proposed since the 80s, including, among the others, high-level Petri nets [13], colored Petri nets [17], unordered and ordered data nets [21], ν-Petri nets [25], and constraint multiset rewriting [5,8,9]. Petri nets with data can be also considered as a reinterpretation of the classical definition of Petri nets in sets with atoms [3,4], where one allows for orbit-finite sets of places and transitions instead of just finite ones. The decidability and complexity of standard problems for Petri nets over various data domains has attracted a lot of attention recently, see for instance [14,21,22,24,25].…”

Section: Relatedmentioning

confidence: 99%

WQO Dichotomy for 3-Graphs

Lasota

Piórkowski

2018

Lecture Notes in Computer Science

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Abstract. We investigate data-enriched models, like Petri nets with data, where executability of a transition is conditioned by a relation between data values involved. Decidability status of various decision problems in such models may depend on the structure of data domain. According to the WQO Dichotomy Conjecture, if a data domain is homogeneous then it either exhibits a well quasi-order (in which case decidability follows by standard arguments), or essentially all the decision problems are undecidable for Petri nets over that data domain.We confirm the conjecture for data domains being 3-graphs (graphs with 2-colored edges). On the technical level, this results is a significant step beyond known classification results for homogeneous structures.

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“…We choose to define our sets by first order formulas, but all results we show here could be reformulated in terms of group actions, orbit-finite sets and finite supports, studied in [21]. In fact, we used that terminology in most previous work on computation theory over sets with atoms [10], [23], [24], of which the present paper is a natural continuation.…”

Section: Related Workmentioning

confidence: 99%

“…Various extensions of first order logic can be also evaluated over linearly p-patched structures, in particular the Least Fix Point logic (LFP), a well studied extension of first order logic by a fixpoint operation [8], [35]. It turns out that over linearly p-patched structures, LFP is equivalent to LFP+C (a further extension by a counting mechanism) and to polynomial time Turing Machines with Atoms -an analogue of Turing machines in the realms of sets with atoms [10], [24].…”

Section: Order-invariant Logicsmentioning

confidence: 99%

Locally Finite Constraint Satisfaction Problems

Klin

Kopczyński

Ochremiak

et al. 2015

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

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Abstract-First-order definable structures with atoms are infinite, but exhibit enough symmetry to be effectively manipulated. We study Constraint Satisfaction Problems (CSPs) where both the instance and the template are definable structures with atoms.As an initial step, we consider locally finite templates, which contain potentially infinitely many finite relations. We argue that such templates occur naturally in Descriptive Complexity Theory.We study CSPs over such templates for both finite and infinite, definable instances. In the latter case even decidability is not obvious, and to prove it we apply results from topological dynamics. For finite instances, we show that some central results from the classical algebraic theory of CSPs still hold: the complexity is determined by polymorphisms of the template, and the existence of certain polymorphisms, such as majority or Maltsev polymorphisms, guarantees the correctness of classical algorithms for solving finite CSP instances.

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Turing Machines with Atoms

Cited by 32 publications

References 12 publications

A general account of coinduction up-to

A general account of coinduction up-to

WQO Dichotomy for 3-Graphs

Locally Finite Constraint Satisfaction Problems

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