Linear Circuits, Two-Variable Logic and Weakly Blocked Monoids

Behle, Christoph; Krebs, Andreas; Mercer, Mark

doi:10.1007/978-3-540-74456-6_15

Cited by 8 publications

(3 citation statements)

References 21 publications

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“…In [7] it was shown that a certain restricted version of the block product of our constant size circuit classes would actually yield linear size circuit classes (over the same base). Here having equations for all languages captured by this circuit class, not just the regular ones, would pay off greatly.…”

Section: Resultsmentioning

confidence: 97%

“…In the last section, we introduced bases for circuits that were defined by a variety of languages. Here we will define an unary operation · P arb on varieties mapping a variety of commutative regular languages V to the variety of languages V P arb recognized by constant size circuits over the base V. This rather strange looking notation comes from the algebraic background where similar ideas have been used on the algebraic side in [7]. Using the algebraic tools from that paper one could show that constant size circuit families recognize the same languages as the finitely typed groups in the block product V P arb , where V are the (typed) monoids corresponding to the gate types and P arb are the typed monoids corresponding arbitrary predicates and hence to the non-uniform wiring of the circuit family.…”

Section: The Block Product For Varieties Of Languagesmentioning

confidence: 99%

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Using Duality in Circuit Complexity

Krebs

2016

Language and Automata Theory and Applications

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We investigate in a method for proving separation results for abstract classes of languages. A well established method to characterize varieties of regular languages are identities. We use a recently established generalization of these identities to non-regular languages by Gehrke, Grigorieff, and Pin: so called equations, which are capable of describing arbitrary Boolean algebras of languages.While the main concern of their result is the existence of these equations, we investigate in a general method that could allow to find equations for language classes in an inductive manner.Thereto we extend an important tool -the block product or substitution principle -known from logic and algebra, to non-regular language classes. Furthermore, we abstract this concept by defining it directly as an operation on (non-regular) language classes. We show that this principle can be used to obtain equations for certain circuit classes, given equations for the gate types.Concretely, we demonstrate the applicability of this method by obtaining a description via equations for all languages recognized by circuit families that contain a constant number of (inner) gates, given a description of the gate types via equations.

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Section: Resultsmentioning

confidence: 97%

Section: The Block Product For Varieties Of Languagesmentioning

confidence: 99%

Using Duality in Circuit Complexity

Krebs

2016

Language and Automata Theory and Applications

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“…A survey dedicated to the class DA and its numerous appearances can be found in [76]. See [73,77] for FO with modular quantifiers, or [7,31] for applications of FO in circuit complexity, or [9] for FO over words with data. A decomposition technique in terms of so-called block products for monoids in DA has been introduced in [72].…”

Section: Remark 1 Sometimes Da Is Given By the Identity (Uv) U V (Uv)mentioning

confidence: 99%

A Survey on Small Fragments of First-Order Logic Over Finite Words

Diekert

Gastin

Kufleitner

2008

Int. J. Found. Comput. Sci.

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We consider fragments of first-order logic over finite words. In particular, we deal with first-order logic with a restricted number of variables and with the lower levels of the alternation hierarchy. We use the algebraic approach to show decidability of expressibility within these fragments. As a byproduct, we survey several characterizations of the respective fragments. We give complete proofs for all characterizations and we provide all necessary background. Some of the proofs seem to be new and simpler than those which can be found elsewhere. We also give a proof of Simon's theorem on factorization forests restricted to aperiodic monoids because this is simpler and sufficient for our purpose. PreambleThere are many brilliant surveys on formal language theory [36,41,48,85,86]. Quite many surveys cover first-order and monadic second-order definability. But there are also nuggets below. There are deep theorems on proper fragments of firstorder definability. The most prominent fragment is FO 2 ; it is the class of languages which are defined by first-order sentences which do not use more than two names for variables. Although various characterizations are known for this class, there seems to be little knowledge in a broad community. A reason for this is that the proofs are spread over the literature and even in the survey [76] many proofs are referred to the original literature which in turn is sometimes quite difficult to read. This is our starting point. We restrict our attention to fragments strictly below first-order definability. We concentrate on algebraic and formal language theoretic characterizations for those fragments where decidability results are known, although we do not discuss complexity issues here. We give a clear preference to full proofs rather than to state all results. In our proofs we tried to be minimalistic. All technical concepts which are introduced are also used in the proofs for the main results.

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