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Infinitary Axiomatization of the Equational Theory of Context-Free Languages

Niels Bjørn Bugge Grathwohl, Fritz Henglein, Dexter Kozen
2013 Electronic Proceedings in Theoretical Computer Science  
We give a natural complete infinitary axiomatization of the equational theory of the context-free languages, answering a question of Leiß (1992).  ...  Acknowledgments We thank ZoltánÉsik, Hans Leiß, and the anonymous referees for helpful comments.  ...  Conclusion We have given a natural complete infinitary axiomatization of the equational theory of the context-free languages.  ... 
doi:10.4204/eptcs.126.4 fatcat:ieors3r6rzbsne7ivy3k6ynhje

Behavioural theories and the proof of behavioural properties

Michel Bidoit, Rolf Hennicker
1996 Theoretical Computer Science  
the standard theory of some corresponding class of algebras.  ...  This reduction is the basis of a method for proving behavioural theorems whenever an axiomatization of the behavioural equality is provided.  ...  Preliminary results of parts of this work were developed in [3-51.  ... 
doi:10.1016/0304-3975(96)00039-4 fatcat:d6cp7kb2cfg4bgbj7pfdkl62k4

A Finite Equational Axiomatization of the Functional Algebras for the Lambda Calculus

Antonino Salibra, Robert Goldblatt
1999 Information and Computation  
In the main result of this paper it is shown that the equations (between contexts) valid in every lambda theory have an explicit finite equational axiomatization.  ...  A lambda theory satisfies an equation between contexts, where a context is a *-term with some"holes" in it, if all the instances of the equation fall within the lambda theory.  ...  So, the explicit finite equational axiomatization for the variety of LAA's provides also an explicit axiomatization of the equations between contexts valid in every lambda theory.  ... 
doi:10.1006/inco.1998.2745 fatcat:r7nodanz25b57bnzpjk7rb54om

Implications of positive formulas in modules (RIMS) [article]

Philipp Rothmaler
2019 arXiv   pre-print
In this survey the role of implications of positive formulas -- finitary and infinitary -- is dicussed, in general and in module categories, where they seem of particular importance.  ...  A list of algebraic examples is given, some old, some rather new, and properties are derived from the particular shape of implications involved.  ...  As atomic formulas are simply linear equations in this context, positive primitive formulas are existentially quantified finite systems of (homogeneous!) linear equations.  ... 
arXiv:1904.06016v1 fatcat:xt5zyimjknbshl2r6f3tox4ece

Unification theory

Jörg H. Siekmann
1989 Journal of symbolic computation  
Unification theory provides the formal framework for investigations into the properties of this operation.  ...  This operation, called unification in work on deduction, is the "addition-and-multiplication" of AI-systems and is consequently often supported by special purpose hardware or by a fast instruction set  ...  Usually we say "theory E" and mean the equational theory T(E) axiomatized by E. Obviously the axiomatization for an equational theory is not unique.  ... 
doi:10.1016/s0747-7171(89)80012-4 fatcat:mjwsvawtirdvbc5dxdflp4n2zi

Infinitary Action Logic: Complexity, Models and Grammars

Wojciech Buszkowski, Ewa Palka
2008 Studia Logica: An International Journal for Symbolic Logic  
Here we prove a stronger form of *-elimination for the logic of *-continuous action lattices and the Π 0 1 −completeness of the equational theories of action lattices of subsets of a finite monoid and  ...  Action logic of Pratt [21] can be presented as Full Lambek Calculus FL [14, 17] enriched with Kleene star *; it is equivalent to the equational theory of residuated Kleene algebras (lattices).  ...  The equational theory of algebras of regular languages is Π 0 1 −complete. The equational theory of algebras of languages is not in Σ 0 1 ∪ Π 0 1 .  ... 
doi:10.1007/s11225-008-9116-7 fatcat:ru6ppfxfmfh4bbysajf3ksbwym

We must know – We shall know [article]

Jacob Zimbarg Sobrinho
2020 arXiv   pre-print
In this manner, after developing the initial steps of this change, I can hint that the solution goes in the direction of the coincidence of both classes, i.e., P=NP.  ...  In this article, I focus on the resiliency of the P=?NP problem. The main point to deal with is the change of the underlying logic from first to second-order logic.  ...  The concept of free and bound occurrences of variables can be carried through straightforwardly to the new context as well as the substitution of variables, free or bound, without collisions.  ... 
arXiv:2004.09235v1 fatcat:oiympk24rjgejjv5ls4pugiqvm

Page 6058 of Mathematical Reviews Vol. , Issue 2001I [page]

2001 Mathematical Reviews  
The theory of LLAs can be regarded as axiomatizing the equations that hold between the contexts of the lambda calculus, as opposed to lambda terms with free variables.  ...  The genericity lemma of finitary lambda-calculus is generalized to the infinitary lambda-calculus. The consistency of the infinitary lambda-calculus is shown by algebraic means.  ... 

Page 4351 of Mathematical Reviews Vol. , Issue 90H [page]

1990 Mathematical Reviews  
The decidability of “F is X-weakly separable” is proved in Part 2 for sensible theories and F A-free. {For the entire collection see MR 90d:03005.} Martin W.  ...  This construction is not free, however, in that the functors employed are not adjoints. The author then gives his own free construction.  ... 

Proof systems for structured algebraic specifications: An overview [chapter]

Rolf Hennicker, Martin Wirsing
1997 Lecture Notes in Computer Science  
As underlying language we choose an ASL-like kernel language which includes reachability and observability operators. Three different kinds of proof systems are studied.  ...  The drawback of the non-compositional proof systems is that they do not reflect the modular structure of specifications.  ...  to the use of infinitary axiomatizations of the observational equality.  ... 
doi:10.1007/bfb0036169 fatcat:5zpoo6maefhsbg7gafodmeao2q

On the Complexity of the Equational Theory of Relational Action Algebras [chapter]

Wojciech Buszkowski
2006 Lecture Notes in Computer Science  
We also show that the equational theory of action algebras (lattices) of regular languages is Π 0 1 −complete.  ...  In [9] it is shown that the equational theory of *-continuous action algebras (lattices) is Π 0 1 −complete.  ...  Consequently, in the language with residuals, one obtains a finite, equational axiomatization of the algebra of regular expressions.  ... 
doi:10.1007/11828563_7 fatcat:gjnpithemffu7gfsaofr3bjdhq

Remarks on Barr's theorem: Proofs in geometric theories [article]

Michael Rathjen
2016 arXiv   pre-print
More concretely, a constructive proof of the Hauptsatz for L_∞ω is given and is put to use to arrive at a simple proof of (A) that is formalizable in constructive set theory and Martin-Loef type theory  ...  In the literature one also finds mysterious comments about the capacity of this theorem to remove the axiom of choice from derivations.  ...  Acknowledgement Part of the material is based upon research supported by the EPSRC of the UK through grant No. EP/K023128/1.  ... 
arXiv:1603.03374v1 fatcat:fvhvylfkv5edlkvae3s6vpor64

Local variable scoping and Kleene algebra with tests

Kamal Aboul-Hosn, Dexter Kozen
2008 The Journal of Logic and Algebraic Programming  
We show that the proof system is sound and complete relative to the underlying equational theory without local variables. We illustrate the use of the system with several examples.  ...  based on Kleene algebra with tests for proving the equivalence of programs in this language.  ...  Any views or conclusions expressed herein are those of the authors and do not necessarily represent the official policies or endorsements of the National Science Foundation or the United States government  ... 
doi:10.1016/j.jlap.2007.10.007 fatcat:kofjsa2xpnfk7fjcczajyz67iu

COMPLETENESS AND CATEGORICITY (IN POWER): FORMALIZATION WITHOUT FOUNDATIONALISM

JOHN T. BALDWIN
2014 Bulletin of Symbolic Logic  
The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki, Hrushovski, Kazhdan, and Shelah to flesh out the argument that the main impact of formal  ...  AbstractWe propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models)  ...  The Skolem hull can accomodate some infinitary languages. .  ... 
doi:10.1017/bsl.2014.3 fatcat:mgy3v22mz5b6fhk6h7s5je47d4

On classes of structures axiomatizable by universal d-Horn sentences and universal positive disjunctions

Guillermo Badia, João Marcos
2018 Algebra Universalis  
The classical example of this kind of characterization result is the HSP theorem, but there are myriad other examples (e.g., the characterization of elementary classes using isomorphic images, ultraproducts  ...  These include, in particular, the classes of fields, of non-trivial rings, and of directed graphs without loops where every two elements are adjacent.  ...  An early incarnation of this paper was presented at the 94th Workshop on General Algebra (AAA94) in Novi Sad, Serbia, in June 2017. We are indebted to the audience there for their comments.  ... 
doi:10.1007/s00012-018-0522-z fatcat:bwdwevelhre4rodvj3n4wx5ib4
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