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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  
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.  ...  Termination and soundness proofs of any unification algorithm are proved by reusing the formalization of this theorem and completeness should be proved according to the specific way in that non unifiable  ...  But in the current paper, the focus is to have a natural mechanical proof of the completeness of any unification algorithm in the Robinson style, reusing the general methodology for the verification of  ... 
doi:10.1007/978-3-642-13824-9_10 fatcat:ne54gywxnre4jpqld5mosc5fkm

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  
In addition, the formalization methodology has been proved of practical use in order to verify the correctness of unification algorithms in the style of the original Robinson's unification algorithm.  ...  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  ...  Verification of unification algorithms This methodology of proof of the existence of mgu's can be applied in order to formalize the completeness of unification algorithmsà la Robinson, as presented in  ... 
doi:10.4204/eptcs.81.5 fatcat:uuj3bilz2nbgrjavwy2zvzmlwi

Nominal Unification of Higher Order Expressions with Recursive Let [article]

Manfred Schmidt-Schauß and Temur Kutsia and Jordi Levy and Mateu Villaret
2016 arXiv   pre-print
A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time.  ...  We also explore specializations like nominal letrec-matching for plain expressions and for DAGs and determine the complexity of corresponding unification problems.  ...  Conclusion and Future Research We constructed a nominal letrec unification algorithm, several nominal letrec matching algorithms for variants, which all run in nondeterministic polynomial time.  ... 
arXiv:1608.03771v1 fatcat:igd5ptbomfdihfxmc3q6gbtila

Nominal Unification and Matching of Higher Order Expressions with Recursive Let [article]

Manfred Schmidt-Schauß and Temur Kutsia and Jordi Levy and Mateu Villaret and Yunus Kutz
2021 arXiv   pre-print
A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in nondeterministic polynomial time.  ...  Finally, we also provide a nominal unification algorithm for higher-order expressions with recursive let and atom-variables, where we show that it also runs in nondeterministic polynomial time.  ...  Since we want to avoid the exponential size explosion of the Robinson-style unification, keeping the good properties of Martelli Montanari-style algorithms [26] , we stick to a set of equations as data  ... 
arXiv:2102.08146v1 fatcat:ol5vdmgfizh6xktr5k3ttyg7ti

The automation of proof: a historical and sociological exploration

D. Mackenzie
1995 IEEE Annals of the History of Computing  
has been rooted primarily in the verification of computer programs and hardware 'designs.-· · IEEE  ...  to be irrelevant; and interactive theorem proving, where the proof is directly guided by a human being.  ...  "Unification"-the key idea drawn by Robinson from ,Ppiwitz3.  ... 
doi:10.1109/85.397057 fatcat:rcaohi4jafaoxivpjmdqmhsqtq

The Logic Programming Paradigm and Prolog [article]

Krzysztof R. Apt
2001 arXiv   pre-print
This is a tutorial on logic programming and Prolog appropriate for a course on programming languages for students familiar with imperative programming.  ...  In this paper he introduced the resolution principle, the notion of unification and a unification algorithm.  ...  In fact, Colmerauer and Roussel write in [3] : "There is no question that Prolog is essentially a theorem prover 'à la Robinson.'  ... 
arXiv:cs/0107013v2 fatcat:qyu54fzhbnadrc2ovrczqlrtti

Prolog at IBM: An advanced and evolving application development technology

M. Benichou, H. Beringer, J.-M. Gauthier, C. Beierle
1992 IBM Systems Journal  
the status of a variable at a specific point in the algorithm).  ...  Having observed that Prolog unification is nothing but an algorithm to solve equalities between symbolic terms, Jaffar and Lassez proved that it was pos- sible to replace this unification by a more pow  ... 
doi:10.1147/sj.314.0755 fatcat:jidyrgtlzvfihjz3pj6mum7bse

Theory-Specific Automated Reasoning [chapter]

Andrea Formisano, Eugenio G. Omodeo
2010 Lecture Notes in Computer Science  
Proof services related to the theories most often take the form of satisfiability decision or semi-decision procedures (in a sense, polyadic inference rules), while some of the services offered by the  ...  axiomatically or characterized by indication of either a privileged model or a family of intended models.  ...  Schwartz for his encouragement in pursuing research on proof-verification.  ... 
doi:10.1007/978-3-642-14309-0_3 fatcat:4yubeyj6nzevtn7orc24x5cpyi

Steps Toward a Computational Metaphysics

Branden Fitelson, Edward N. Zalta
2007 Journal of Philosophical Logic  
After reviewing the second-order, axiomatic theory of abstract objects, we show (1) how to represent a fragment of that theory in prover9's firstorder syntax, and (2) how prover9 then finds proofs of interesting  ...  In this paper, we describe what we have discovered when the theory of abstract objects is implemented in prover9 (a first-order automated reasoning system which is the successor to otter).  ...  The unification theorem (Robinson 1963) guarantees the existence of a unique most general unifier for any two terms of first-order logic, and an algorithm for computing it (called the unification algorithm  ... 
doi:10.1007/s10992-006-9038-7 fatcat:55te44fiorfopojmmlyu57yvle

Design and implementation of a generic, logic and functional programming language [chapter]

Didier Bert, Rachid Echahed
1986 Lecture Notes in Computer Science  
As modules in LPG can be genetic, an instantiation mechanism is needed ; such a mechanism is described here as welt as the interpreter and an E-unification algorithm, thus making LPG a powerful programming  ...  This paper presents the broad outlines of LPG, a language designed for generic specification and programming.  ...  V x E V, ~(x) <-E £(x) where t <-E t' iff 3 ~, 0(t) =E t," In the case where a set of equations E possesses a canonical term rewriting system, an E-unification algorithm has been given [Hullot 80 ].  ... 
doi:10.1007/3-540-16442-1_9 fatcat:uj5eq2shyzd6vncv5u3m7uxjgm

Reference list of indexed articles

1995 Theoretical Computer Science  
Teillaud, On the randomized construction of the Delaunay tree 112 (2) (1993) 339-354 168. J.A. Aslam and A. Dhagat, On-line algorithms for ?  ...  Schmid, The average CRI-length of a controlled ALOHA collision resolution algorithm 108 (2) 11993) 291-310 Games) 108 (2) (1993) 31 I-329 I IX. A.  ... 
doi:10.1016/0304-3975(95)80003-4 fatcat:5nnk76aalje5potfoeg7hksbce

Lectures on Jacques Herbrand as a Logician [article]

Claus-Peter Wirth, Joerg Siekmann, Christoph Benzmueller, Serge Autexier
2014 arXiv   pre-print
We sketch Herbrand's two proofs of the consistency of arithmetic and his notion of a recursive function, and last but not least, present the correct original text of his unification algorithm with a new  ...  Besides the well-known correction of Herbrand's False Lemma by Goedel and Dreben, we also present the hardly known unpublished correction of Heijenoort and its consequences on Herbrand's Modus Ponens Elimination  ...  Paulson -the second readers of our previous handbook article ] on the same subject -for their numerous, extensive, deep, and most valuable and helpful suggestions for improvement of previous versions  ... 
arXiv:0902.4682v5 fatcat:pgxxo7vjcbeyvhjej4svsv6lqe

On The Resolution Semiring

Marc Bagnol
2014 Zenodo  
interpretation of proof theory withinthis semiring via the categorical axiomatization of the geometry of interactionapproach.  ...  It provides us with an algebraicand abstract setting, while being presented in a syntactic and concrete way, inwhich a theoretical study of computation can be carried on.We will review first the interactive  ...  La radicalité de son approche à la logique et à la science en général restera une inspiration, je l'espère, pour ma future vie de chercheur.  ... 
doi:10.5281/zenodo.806771 fatcat:klgfbxmvfrf3xhad2pb4tq56ee

Objective Analysis for Numerical Weather Prediction

A. Hollingsworth
1986 Journal of the Meteorological Society of Japan  
When used to verify short-range forecasts, they provide a complete description of the structure of the forecast errors.  ...  The basic job of any analysis algorithm is to filter random observation error from the data and then interpolate the filtered data to a regular grid.  ...  At ECMWF we keep a complete record of the processing of every single datum.  ... 
doi:10.2151/jmsj1965.64a.0_11 fatcat:bzas4tzwhbev3ksricf555h3fu

History of Interactive Theorem Proving [chapter]

John Harrison, Josef Urban, Freek Wiedijk
2014 Handbook of the History of Logic  
The helpful review of the official reader Larry Paulson as well as useful comments from Mike Gordon, Tom Hales and J Moore, have significantly improved the eventual form of the chapter.  ...  ACKNOWLEDGEMENTS The authors are grateful to Jörg Siekmann for inviting them to prepare this chapter and for his patience in the face of our lengthy delays.  ...  Together with unification, this framework could be used to give very simple direct implementations of some classic first-order proof search algorithms like tableauxà la leanT A P [Beckert and Posegga,  ... 
doi:10.1016/b978-0-444-51624-4.50004-6 fatcat:sngnxnl3ivectj5taj3dgsaa7m
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