Languages in general algebras

Shepard, C. D.

doi:10.1145/800169.805430

Cited by 4 publications

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“…This is to be expected from the general theory of BFV resolutions, see[Sch08].24 We recall that the arity of an operation is the number of arguments upon which it depends, see[She69] for the origin of the term.…”

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confidence: 90%

A Lie-Rinehart algebra in general relativity

Blohmann¹,

Schiavina²,

Weinstein³

2022

Preprint

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We construct a Lie-Rinehart algebra over an infinitesimal extension of the space of initial value fields for Einstein's equations. The bracket relations in this algebra are precisely those of the constraints for the initial value problem. The Lie-Rinehart algebra comes from a slight generalization of a Lie algebroid in which the algebra consists of sections of a sheaf rather than a vector bundle. (An actual Lie algebroid had been previously constructed by Blohmann, Fernandes, and Weinstein over a much larger extension.) The construction uses the BV-BFV (Batalin-Fradkin-Vilkovisky) approach to boundary value problems, starting with the Einstein equations themselves, to construct an L ∞ -algebroid over a graded manifold which extends the initial data. The Lie-Rinehart algebra is then constructed by a change of variables. One of the consequences of the BV-BFV approach is a proof that the coisotropic property of the constraint set follows from the invariance of the Einstein equations under space-time diffeomorphisms.

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confidence: 90%

A Lie-Rinehart algebra in general relativity

Blohmann¹,

Schiavina²,

Weinstein³

2022

Preprint

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“…Eilenberg and Wright [69] present these results in a category theoretic form. For various classes of subsets in general algebras we refer also to Wagner [249], Lescanne [150], Marchand [175], Shepard [220], and Steinby [227]. Dubinsky [67] discusses equational and recognizable subsets of nondeterministic algebras.…”

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confidence: 99%

Tree Automata

Gécseg,

Steinby

2015

Preprint

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Proof. Let C = {a ∈ A | aϕ = aψ}. Then H ⊆ C by the assumption. If m ≥ 0, σ ∈ Σ m and a 1 , . . . , a m ∈ C, then σ A (a 1 , . . . , a m ) ∈ C:Hence C is closed and we get C = A. This implies ϕ = ψ. ✷We define now two concepts closely related to homomorphisms, namely congruences and quotient algebras.Definition 1.2.7 A congruence (relation) of A is an equivalence relation on A which is invariant with respect to all operations σ A (σ ∈ Σ). A relation ̺ ⊆ A × A is said to be invariant with respect to an m-ary operation f :The set of all congruences of an algebra A is denoted by C(A).Every algebra A has at least the trivial congruences δ A and ι A . For ̺ ∈ C(A), the ̺-class a̺ of an element a ∈ A is also called a congruence class (modulo ̺). The partition A/̺ of A defined by the congruence classes is compatible in the sense that for all m ≥ 0, σ ∈ Σ m and a 1 ̺, . . . , a m ̺ ∈ A/̺ there is a class a̺ such that σ A (a 1 ̺, . . . , a m ̺) ⊆ a̺.

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“…This is the general nature of the translations we will consider. Such syntactically based translations of formal languages, frequently called syntax directed translations, have been studied recently by Thatcher [21], A h o and Ullman [l, 2], Lewis and Stearns [13], and in an algebraic setting by Shepard [17]. F o r a rather complete bibliography, see [21].…”

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confidence: 99%

Semantic preserving translations

Benson

1974

Math. Systems Theory

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Let X1, X2 be derivation systems (free x-categories) generated by context free grammars. Let Xo be a translation category with x-functors fi: Xo --~ X, i = l, 2." Let T be an ~*-theory, a generalization of algebraic theories. Let It: X~ -~ T be algebraic interpretations of the derivations systems, giving the semantics of derivation systems. The translation category Xo is shown to preserve the common semantics through the translation if there is a natural transformation from the functor ]'2 °/2 to the functor fl o 11. This is used to show that certain elementary conditions on well-behaved generalized 2 sequential machine maps (g2sm maps) result in semantics preservation by the g2sm maps.1. Introduction. Consider two formal languages, L1 and L 2. If one does not consider the grammars for these languages, then a translation from L 1 to L2 is a relation from L1 to L 2. However, this notion gives no insight as to why two sentences, ~o 1 e LI and to 2 E L2, are or are not related by the translation. If we accept the idea that each syntactic structure of a sentence determines exactly one meaning, then it is correct to consider a translation as mapping each syntactic structure of sentences in L~ to one or more syntactic structures of various sentences in L 2. If one further admits the possibility that not all syntactic structures of the first language system are translatable, we see that a translation can be viewed as a relation from the syntactic structures of L 1 to the syntactic structures of L 2. This is the general nature of the translations we will consider. Such syntactically based translations of formal languages, frequently called syntax directed translations, have been studied recently by Thatcher [21], A h o and Ullman [l, 2], Lewis and Stearns [13], and in an algebraic setting by Shepard [17]. F o r a rather complete bibliography, see [21].Translations must be effective to be of interest in current formal studies of natural and p r o g r a m m i n g languages. A natural method of obtaining the

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Languages in general algebras

Abstract: This paper presents the results of recent investigations into the question of when the context-free (CF) languages of an algebra are all recognizable.

Cited by 4 publications

References 3 publications

A Lie-Rinehart algebra in general relativity

A Lie-Rinehart algebra in general relativity

Tree Automata

Semantic preserving translations

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