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A Formalization of the Theorem of Existence of First-Order Most General Unifiers

2012
*
Electronic Proceedings in Theoretical Computer Science
*

This work presents

doi:10.4204/eptcs.81.5
fatcat:uuj3bilz2nbgrjavwy2zvzmlwi
*a**formalization**of**the**theorem**of**existence**of**most**general**unifiers*in*first*-*order*signatures in*the*higher-*order*proof assistant PVS. ...*The**formalization*was applied inside*a*PVS development for term rewriting systems that provides*a*complete*formalization**of**the*Knuth-Bendix Critical Pair*theorem*, among other relevant*theorems**of**the*... Introduction*A**formalization*in*the*proof assistant PVS*of**the**theorem**of**existence**of**most**general**unifiers*(mgu's) in*first*-*order*theories is presented. ...##
###
Verification of the Completeness of Unification Algorithms à la Robinson
[chapter]

2010
*
Lecture Notes in Computer Science
*

*The*methodology is based on

*a*previously developed

*formalization*

*of*

*the*

*theorem*

*of*

*existence*

*of*

*most*

*general*

*unifiers*for

*unifiable*terms over

*first*-

*order*signatures. ... This work presents

*a*

*general*methodology for verification

*of*

*the*completeness

*of*firstorder unification algorithmsà la Robinson developed in

*the*higher-

*order*proof assistant PVS. ... Introduction In

*a*previous development, done in

*the*PVS proof assistant [ORS92] ,

*a*

*formalization*

*of*

*the*

*theorem*

*of*

*existence*

*of*

*most*

*general*

*unifiers*(mgu's) for

*unifiable*terms over

*first*-

*order*theories ...

##
###
A Mechanized Proof of a Textbook Type Unification Algorithm

2020
*
Revista de Informática Teórica e Aplicada
*

As

doi:10.22456/2175-2745.100968
fatcat:ghwyjml3bjcmdp6irgvfg54q2y
*a**first*step towards*a**formalization**of**a*type inference algorithm for such programming languages, we present*a**formalization*in Coq*of**a*type unification algorithm that follows classic algorithms presented ... We also report on*the*use*of*such*formalization*to build*a*correct type inference algorithm for*the*simply typed λ-calculus. ... Author contributions All authors have contributed equally to*the*development*of*this work. ...##
###
Verifying the unification algorithm in LCF

1985
*
Science of Computer Programming
*

Correctness

doi:10.1016/0167-6423(85)90009-7
fatcat:4d7iq4udt5bq5hspavwfr6yiju
*of*unification is expressed using predicates for such properties as idempotence and*most*-*generality*.*The*verification is presented as*a*series*of*lemmas. ... Manna and Waldinger's theory*of*substitutions and unification has been verified using*the*Cambridge LCF*theorem*prover. ... Dhingra read drafts*of*this paper;*a*referee made detailed comments. ...##
###
Page 5565 of Mathematical Reviews Vol. , Issue 90J
[page]

1990
*
Mathematical Reviews
*

Finally, Noetherian theories have

*a*well-founded subsumption*ordering*, resulting in*the**existence**of*complete sets*of**most**general**unifiers*. ... We describe*a**formalization*and proof*of**the*Church-Rosser*theorem*that was carried out with*the*Boyer-Moore*theorem*prover. ...##
###
Automation of Higher-Order Logic
[chapter]

2014
*
Handbook of the History of Logic
*

Besides

doi:10.1016/b978-0-444-51624-4.50005-8
fatcat:jfcztdvymjfujg3bzb2rq2qyzy
*the*readers*of*this chapter, we thank Zakaria Chihani, Julian Röder, Leon Weber, and Max Wisnieswki for proofreading*the*document. ...*The**first*author has been supported by*the*German Research Foundation under Heisenberg grant BE2501/9-1 and*the*second author has been supported by*the*ERC Advanced Grant ProofCert. ...*The*main result about pattern unification is that-like*first*-*order*unification-deciding*unifiability*is decidable and*most**general**unifiers**exist*for solvable problems. ...##
###
Unification: a multidisciplinary survey

1989
*
ACM Computing Surveys
*

*The*paper is intended for readers with

*a*

*general*computer science background-no specific knowledge

*of*any

*of*

*the*above topics is assumed. ... Other topics covered are resolution, higher

*order*logic,

*the*occur check, infinite terms, feature structures, equational theories, inheritance, parallel algorithms,

*generalization*, lattices, and other ... My thanks also go to

*the*reviewers for making many suggestions relating to both accuracy and clarity. ...

##
###
A Machine Checked Model of Idempotent MGU Axioms For Lists of Equational Constraints

2010
*
Electronic Proceedings in Theoretical Computer Science
*

We present

doi:10.4204/eptcs.42.3
fatcat:rm3wjoab4ffqnejyzsc2r3yvki
*formalized*proofs verifying that*the**first*-*order*unification algorithm defined over lists*of*satisfiable constraints*generates**a**most**general**unifier*(MGU), which also happens to be idempotent ... All*of*our proofs have been*formalized*in*the*Coq*theorem*prover. ...*The*entire*formalization*(all seven axioms) is done in Coq 8.1.pl3 version in around 5000 lines*of*specifications and tactics, and is available online at http://www.cs.uwyo.edu/~skothari. ...##
###
Deductive synthesis of the unification algorithm

1981
*
Science of Computer Programming
*

*The*deductive appmuch is

*a*

*formal*program construction method in which

*the*derivation

*of*

*a*program from

*a*given specification is regarded as

*a*

*theorem*-proving task. ... To construct

*a*program whose output satisfies

*the*conditions

*of*

*the*specification, we prove

*a*

*theorem*stating

*the*

*existence*

*of*such an output.

*The*proof is restricted to be sufficrent! ... Combining

*the*above proposition and definition, we see that

*the*

*unifiers*

*of*two I-expressions are precisely

*the*instances

*of*

*a*

*most*-

*general*

*unifier*. CoroIIary (

*Most*-

*general*

*unifier*). ...

##
###
Resolution Principle and Fuzzy Logic
[chapter]

2012
*
Fuzzy Logic - Algorithms, Techniques and Implementations
*

If we want to

doi:10.5772/36161
fatcat:kkfzxhjyg5a6jliryfsigi5wxi
*generalize**the*notion*of*resolution and lift it into*first*-*order*case we have to define*first**the*notion*of*selection function for*general*clauses. ... ., G ′ n ] Fσ[G/⊥] ∨ F ′ σ[G/⊤] (3) where σ = MGU(*A*) is*the**most**general**unifier*(MGU)*of**the*set*of**the*atoms*A*= {G 1 ,...,G k , G ′ 1 ,...,G ′ n } ,G= G 1 σ. ...##
###
Page 2950 of Mathematical Reviews Vol. , Issue 81H
[page]

1981
*
Mathematical Reviews
*

*The*unification problem, i.e.,

*the*problem

*of*

*existence*

*of*

*a*common

*unifier*, is decidable for

*first*-

*order*logic and is algorithmically undecidable for third-

*order*and higher-

*order*logic. ... For \2-theories we prove

*a*cut

*theorem*and

*a*con- sistency

*theorem*. We construct an example

*of*

*a*\2-theory con- taining all translations

*of*derivations

*of*

*formal*arithmetic.” ...

##
###
Substantiation of the method of integrated group unification of machine and appliance designs

2019
*
Eastern-European Journal of Enterprise Technologies
*

Los', Doctor

doi:10.15587/1729-4061.2019.183844
fatcat:b4oeh7cld5gpbhiyhlrzamrytu
*of*Technical Sciences, for his invaluable scientific contribution to foundations*of**the*theory*of*structure design*of*process machines and appliances. ... by*a*movement*of*thought from single to*general*(*the*logical method*of*study) [21] ; -in*order*to ensure rigor*of**the**theorem**of*unification*of**the*design structures, an axiomatic method was used which ...*The**theorem**of**existence**of*designs facilitates departure from this inertia [6] . There are some reasons to use*the**theorem**of**existence**of*designs in designing. ...##
###
Natural deduction as higher-order resolution

1986
*
The Journal of Logic Programming
*

Depth-

doi:10.1016/0743-1066(86)90015-4
fatcat:d2jkwnbuyzeo7ko2ymi6qvxvpm
*first*subgoaling along inference rules constitutes*a*higher-*order*PROLOG.*a*BACKGROUND At least seven interactive*theorem*provers use*the*LCF framework. ...*A*new*theorem*prover, Isabelle, is intended to*unify*these diverging paths.*A*recurring theme will be*the*relationship between syntax and semantics. ... There are practical algorithms for computing*the**most**general**unifier**of*two expressions, or reporting that no*unifier**exists*. ...##
###
A Complete Mechanization of Second-Order Type Theory

1973
*
Journal of the ACM
*

*The*approaches followed for

*the*higher

*order*

*theorem*-proving differ mainly in

*the*degree

*of*

*generality*.

*The*

*most*

*general*approach is represented by [3, 12] . ...

*The*

*first*is that

*the*

*unifying*substitution procedure--an essential feature

*of*

*the*classic

*first*-

*order*resolution--must be

*generalized*(it is noted that for

*the*higher

*order*unification

*the*proper notion ...

*A*Forkes, whose discovery

*of*

*a*certain higher

*order*unification algorithm inspired me to develop this method, and to Dr.

*A*. Ehrenfeucht and Dr.

*A*. Gabrielian for their comments and criticisms. ...

##
###
Contraction algebras and unification of (infinite) terms

1992
*
Journal of computer and system sciences (Print)
*

We develop

doi:10.1016/0022-0000(92)90002-z
fatcat:jkeba5ult5d6rfqcjpyryoctdm
*a**general*theory*of*systems*of*equations at*the*level*of*categories in close connection with*the*Banach Principle*of*Contraction. Two applications*of*this theory are given. ...*The**first*is*the*case*of*regular systems*of*equations with arbitrary terms.*The*second is*the*case*of*systems*of*equations attached to*a*context-free grammar (sometimes called ALGOL-like systems). ...*The**existence**of**the**most**general**unifier**of*any two*unifiable*infinite terms on*a*finite set*of*variables (*a*result which also appears in [S] ) is obtained as*a*corollary*of**the*correctness*theorem*for ...
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