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

Andréia B Avelar, André L Galdino, Flávio LC de Moura, Mauricio Ayala-Rincón
2012 Electronic Proceedings in Theoretical Computer Science  
This work presents 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.  ... 
doi:10.4204/eptcs.81.5 fatcat:uuj3bilz2nbgrjavwy2zvzmlwi

Verification of the Completeness of Unification Algorithms à la Robinson [chapter]

Andréia B. Avelar, Flávio L. C. de Moura, André Luiz Galdino, Mauricio Ayala-Rincón
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  ... 
doi:10.1007/978-3-642-13824-9_10 fatcat:ne54gywxnre4jpqld5mosc5fkm

A Mechanized Proof of a Textbook Type Unification Algorithm

André Rauber Du Bois, Rodrigo Ribeiro, Maycon Amaro
2020 Revista de Informática Teórica e Aplicada  
As 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.  ... 
doi:10.22456/2175-2745.100968 fatcat:ghwyjml3bjcmdp6irgvfg54q2y

Verifying the unification algorithm in LCF

Lawrence C. Paulson
1985 Science of Computer Programming  
Correctness 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.  ... 
doi:10.1016/0167-6423(85)90009-7 fatcat:4d7iq4udt5bq5hspavwfr6yiju

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]

Christoph Benzmüller, Dale Miller
2014 Handbook of the History of Logic  
Besides 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.  ... 
doi:10.1016/b978-0-444-51624-4.50005-8 fatcat:jfcztdvymjfujg3bzb2rq2qyzy

Unification: a multidisciplinary survey

Kevin Knight
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.  ... 
doi:10.1145/62029.62030 fatcat:u4ieq3eekzg25fgpxsaebyktkm

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

Sunil Kothari, James Caldwell
2010 Electronic Proceedings in Theoretical Computer Science  
We present 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  ... 
doi:10.4204/eptcs.42.3 fatcat:rm3wjoab4ffqnejyzsc2r3yvki

Deductive synthesis of the unification algorithm

Zohar Manna, Richard Waldinger
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).  ... 
doi:10.1016/0167-6423(81)90004-6 fatcat:h5t3cvxdrjf5zjdxzn2vw7ltpm

Resolution Principle and Fuzzy Logic [chapter]

Hashim Habiballa
2012 Fuzzy Logic - Algorithms, Techniques and Implementations  
If we want to 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.  ... 
doi:10.5772/36161 fatcat:kkfzxhjyg5a6jliryfsigi5wxi

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

Gennadii Golub, Nataliya Tsyvenkova, Viacheslav Chuba, Anna Holubenko, Marina Tereshchuk
2019 Eastern-European Journal of Enterprise Technologies  
Los', Doctor 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.  ... 
doi:10.15587/1729-4061.2019.183844 fatcat:b4oeh7cld5gpbhiyhlrzamrytu

Natural deduction as higher-order resolution

Lawrence C. Paulson
1986 The Journal of Logic Programming  
Depth-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.  ... 
doi:10.1016/0743-1066(86)90015-4 fatcat:d2jkwnbuyzeo7ko2ymi6qvxvpm

A Complete Mechanization of Second-Order Type Theory

Tomasz Pietrzykowski
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.  ... 
doi:10.1145/321752.321764 fatcat:zvdikkc5xnd3tahlznengmuxjy

Contraction algebras and unification of (infinite) terms

Razvan Diaconescu
1992 Journal of computer and system sciences (Print)  
We develop 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  ... 
doi:10.1016/0022-0000(92)90002-z fatcat:jkeba5ult5d6rfqcjpyryoctdm
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