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Infinitary Axiomatization of the Equational Theory of Context-Free Languages

2013
*
Electronic Proceedings in Theoretical Computer Science
*

We give a natural complete

doi:10.4204/eptcs.126.4
fatcat:ieors3r6rzbsne7ivy3k6ynhje
*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*. ...##
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Behavioural theories and the proof of behavioural properties

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. ...

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

1999
*
Information and Computation
*

In

doi:10.1006/inco.1998.2745
fatcat:r7nodanz25b57bnzpjk7rb54om
*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*. ...##
###
Implications of positive formulas in modules (RIMS)
[article]

2019
*
arXiv
*
pre-print

In this survey

arXiv:1904.06016v1
fatcat:xt5zyimjknbshl2r6f3tox4ece
*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*. ...##
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Unification theory

1989
*
Journal of symbolic computation
*

Unification

doi:10.1016/s0747-7171(89)80012-4
fatcat:mjwsvawtirdvbc5dxdflp4n2zi
*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. ...##
###
Infinitary Action Logic: Complexity, Models and Grammars

2008
*
Studia Logica: An International Journal for Symbolic Logic
*

Here we prove a stronger form

doi:10.1007/s11225-008-9116-7
fatcat:ru6ppfxfmfh4bbysajf3ksbwym
*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 . ...##
###
We must know – We shall know
[article]

2020
*
arXiv
*
pre-print

In this manner, after developing

arXiv:2004.09235v1
fatcat:oiympk24rjgejjv5ls4pugiqvm
*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. ...##
###
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. ...

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

1997
*
Lecture Notes in Computer Science
*

As underlying

doi:10.1007/bfb0036169
fatcat:5zpoo6maefhsbg7gafodmeao2q
*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. ...##
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On the Complexity of the Equational Theory of Relational Action Algebras
[chapter]

2006
*
Lecture Notes in Computer Science
*

We also show that

doi:10.1007/11828563_7
fatcat:gjnpithemffu7gfsaofr3bjdhq
*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. ...##
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Remarks on Barr's theorem: Proofs in geometric theories
[article]

2016
*
arXiv
*
pre-print

More concretely, a constructive proof

arXiv:1603.03374v1
fatcat:fvhvylfkv5edlkvae3s6vpor64
*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. ...##
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Local variable scoping and Kleene algebra with tests

2008
*
The Journal of Logic and Algebraic Programming
*

We show that

doi:10.1016/j.jlap.2007.10.007
fatcat:kofjsa2xpnfk7fjcczajyz67iu
*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 ...##
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COMPLETENESS AND CATEGORICITY (IN POWER): FORMALIZATION WITHOUT FOUNDATIONALISM

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*. . ...

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

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. ...

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