Turing machines with atoms, constraint satisfaction problems, and descriptive complexity

Klin, Bartek; Lasota, Sławomir; Ochremiak, Joanna; Toruńczyk, Szymon

doi:10.1145/2603088.2603135

Cited by 7 publications

(9 citation statements)

References 16 publications

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“…Finally, in Section V we show how locally finite templates streamline the (previously unpleasantly technical) proof of the main result in [10], a characterization of those linearly patched structures over which the least fixpoint logic LFP captures polynomial time computations.…”

Section: + 21 =mentioning

confidence: 97%

“…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%

“…In a previous paper [10], we studied the expressive power of various logics over a certain class of structures, which we now recall in a slightly simplified version. As an application of the methods developed in this paper, we briefly say how the main result of [10] can be re-developed in the setting of locally finite CSPs, showing the key ideas of the proof much more clearly and clearing them from a clutter of technicalities.…”

Section: Order-invariant Logicsmentioning

confidence: 99%

“…As an application of the methods developed in this paper, we briefly say how the main result of [10] can be re-developed in the setting of locally finite CSPs, showing the key ideas of the proof much more clearly and clearing them from a clutter of technicalities.…”

Section: Order-invariant Logicsmentioning

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%

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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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Section: + 21 =mentioning

confidence: 97%

Section: Related Workmentioning

confidence: 99%

Section: Order-invariant Logicsmentioning

confidence: 99%

Section: Order-invariant Logicsmentioning

confidence: 99%

Section: Order-invariant Logicsmentioning

confidence: 99%

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Locally Finite Constraint Satisfaction Problems

Klin

Kopczyński

Ochremiak

et al. 2015

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

Self Cite

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From urelements to Computation

Ciancia

2016

IFIP Advances in Information and Communication Technology

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Around 1922-1938, a new permutation model of set theory was defined. The permutation model served as a counterexample in the first proof of independence of the Axiom of Choice from the other axioms of Zermelo-Fraenkel set theory. Almost a century later, a model introduced as part of a proof in abstract mathematics fostered a plethora of research results, ranging from the area of syntax and semantics of programming languages to minimization algorithms and automated verification of systems. Among these results, we find Lawvere-style algebraic syntax with binders, final-coalgebra semantics with resource allocation, and minimization algorithms for mobile systems. These results are also obtained in various different ways, by describing, in terms of category theory, a number of models equivalent to the permutation model. We aim at providing both a brief history of some of these developments, and a mild introduction to the recent research line of "nominal computation theory", where the essential notion of name is declined in several different ways. Research partially funded by the European Commission FP7 ASCENS (nr. 257414), and EU QUANTICOL (nr. 600708).

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Limitations of Algebraic Approaches to Graph Isomorphism Testing

Berkholz

Grohe

2015

Lecture Notes in Computer Science

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We investigate the power of graph isomorphism algorithms based on algebraic reasoning techniques like Gröbner basis computation. The idea of these algorithms is to encode two graphs into a system of equations that are satisfiable if and only if if the graphs are isomorphic, and then to (try to) decide satisfiability of the system using, for example, the Gröbner basis algorithm. In some cases this can be done in polynomial time, in particular, if the equations admit a bounded degree refutation in an algebraic proof systems such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on the polynomial calculus degree over all fields of characteristic = 2 and also linear lower bounds for the degree of Positivstellensatz calculus derivations.We compare this approach to recently studied linear and semidefinite programming approaches to isomorphism testing, which are known to be related to the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system that lies between degree-k Nullstellensatz and degree-k polynomial calculus.

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Turing machines with atoms, constraint satisfaction problems, and descriptive complexity

Cited by 7 publications

References 16 publications

Locally Finite Constraint Satisfaction Problems

Locally Finite Constraint Satisfaction Problems

From urelements to Computation

Limitations of Algebraic Approaches to Graph Isomorphism Testing

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