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Higher-Order Unification: A structural relation between Huet's method and the one based on explicit substitutions

Flávio L.C. de Moura, Mauricio Ayala-Rincón, Fairouz Kamareddine
2008 Journal of Applied Logic  
We compare two different styles of Higher-Order Unification (HOU): the classical HOU algorithm of Huet for the simply typed λ-calculus and HOU based on the λσ-calculus of explicit substitutions.  ...  We prove that there exists an important structural correspondence between Huet's HOU and the λσ-HOU method: for each (sub-)problem in the unification tree there exists a counterpart in the derivation tree  ...  Acknowledgements We would like to thank the careful reading, constructive corrections and detailed suggestions of one of the referees.  ... 
doi:10.1016/j.jal.2006.10.001 fatcat:3ym4njp7wnfkbd4dm4virhbh5m

Higher-order unification with dependent function types [chapter]

Conal M. Elliott
1989 Lecture Notes in Computer Science  
to include dependent function types, the resulting calculus ( ) forms the basis of a very elegant and expressive Logical Framework, encompassing the syntax, rules, and proofs for a wide class of logics  ...  , and the greatly enriched expressive power of . It can be used to considerable advantage in many of the current applications of Huet's algorithm, and has important new applications as well.  ...  9 Acknowledgment I am very grateful to Frank Pfenning for originally suggesting the problem in , and for several very helpful discussions yielding many useful ideas, in particular the idea of approximate  ... 
doi:10.1007/3-540-51081-8_104 fatcat:vn7qtt37m5a3xadhfrnp7pocry

Higher Order Unification via Explicit Substitutions

Gilles Dowek, Thérèse Hardin, Claude Kirchner
2000 Information and Computation  
At last we show how to relate uni cation in -calculus and in a calculus with explicit substitutions.  ...  This di culty comes from the interaction between the substitutions initiated by -reduction and the ones initiated by the uni cation process.  ...  This work has been supported partly by the French Inter-PRC operation \M ecanisation du raisonnement", the Esprit basic research actions Confer and Types and the Esprit working group CCL.  ... 
doi:10.1006/inco.1999.2837 fatcat:smdrvm24ybfozgesugh2y67qlm

An Implementation of the Language Lambda Prolog Organized around Higher-Order Pattern Unification [article]

Xiaochu Qi
2009 arXiv   pre-print
Lambda Prolog achieves its functionality by extending a Prolog-like language by using typed lambda terms as data structures that it then manipulates via higher-order unification and some new program-level  ...  A key idea is the orientation of the computation model of Lambda Prolog around a restricted version of higher-order unification with nice algorithmic properties and appearing to encompass most interesting  ...  This form of unification, known as higher-order unification, differs from the one used in a language like Prolog in that equality between terms is based not just on identity but also on the conversion  ... 
arXiv:0911.5203v1 fatcat:6d2kzgj6b5d5th6dhiijcv4mqe

Higher-order unification revisited: Complete sets of transformations

Wayne Snyder, Jean Gallier
1989 Journal of symbolic computation  
In this paper, we reexamine the problem of general higher-order unification and develop an approach based on the method of transformations on systems of terms which has its roots in Herbrand's thesis,  ...  Introduction Higher-order unification is a method for unlfylng terms in the Simple Theory of Types [6], that is, given two typed lambda-terms el and e~, finding a substitution cr for the ~ree variables  ...  In this paper, we reexamine the problem of general higher-order unification and develop an approach based on the method of transformations on systems o]: terms which has its roots in Herbrand's thesis,  ... 
doi:10.1016/s0747-7171(89)80023-9 fatcat:v3icruent5eurja2gwclo3pti4

Practical Higher-Order Pattern Unification with On-the-Fly Raising [chapter]

Gopalan Nadathur, Natalie Linnell
2005 Lecture Notes in Computer Science  
In particular, a unification algorithm is described that proceeds by recursively descending through the structures of terms, performing raising and other transformations on-the-fly and only as needed.  ...  Higher-order pattern unification problems arise often in computations carried out within systems such as Twelf, λProlog and Isabelle.  ...  Acknowledgements This work began while the first author was on a sabbatical visit to the Protheo group at LORIA and INRIA, Nancy and the Comete and Parsifal groups at Ecole Polytechnique and INRIA, Saclay  ... 
doi:10.1007/11562931_28 fatcat:6tr7ribftjb4ppsvtfxsf4zvbq

Automation of Higher-Order Logic [chapter]

Christoph Benzmüller, Dale Miller
2014 Handbook of the History of Logic  
We thank Chad Brown for sharing notes that he has written related to the material in this chapter.  ...  Besides the readers of this chapter, we thank Zakaria Chihani, Julian Röder, Leon Weber, and Max Wisnieswki for proofreading the document.  ...  Higher-order substitutions and normal forms One of the challenges posed by higher-order substitution is that the many normal forms on which theorem provers often rely are not stable under such substitution  ... 
doi:10.1016/b978-0-444-51624-4.50005-8 fatcat:jfcztdvymjfujg3bzb2rq2qyzy

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.  ...  Scheme variables may have as decisive an impact on LCF as they did on provers based on Herbrand's theorem. The Davis-Putnam method generated ground clauses.  ... 
doi:10.1016/0743-1066(86)90015-4 fatcat:d2jkwnbuyzeo7ko2ymi6qvxvpm

Comparing Approaches To Resolution Based Higher-Order Theorem Proving

Christoph Benzmüller
2002 Synthese  
In particular we focus on Andrews' higher-order resolution (Andrews 1971), Huet's constrained resolution (Huet 1972), higher-order E-resolution, and extensional higher-order resolution (Benzmüller and  ...  We investigate several approaches to resolution based automated theorem proving in classical higher-order logic (based on Church's simply typed λ-calculus) and discuss their requirements with respect to  ...  I am also grateful to Michael Kohlhase for many fruitful discussions on extensional higher-order theorem proving.  ... 
doi:10.1023/a:1020840027781 fatcat:sghuiqlg6vd3dhgr2f6clo4ssq

Higher-Order Multi-Valued Resolution

Michael Kohlhase, Ortwin Scheja
1999 Journal of Applied Non-Classical Logics  
This paper introduces a multi-valued variant of higher-order resolution and proves it correct and complete with respect to a variant of Henkin's general model semantics.  ...  This resolution method is parametric in the number of truth values as well as in the particular choice of the set of connectives (given by arbitrary truth tables) and even substitutional quantifiers.  ...  Since higher-order reasoning with presuppositions is ubiquitous in natural language [Kra95, Mus89] , a higher-order version of [KK97] based on the results of this paper would provide a computational  ... 
doi:10.1080/11663081.1999.10510980 fatcat:62pazeuxp5bzjjbaqz7uoadsji

Unification: a multidisciplinary survey

Kevin Knight
1989 ACM Computing Surveys  
Other topics covered are resolution, higher order logic, the occur check, infinite terms, feature structures, equational theories, inheritance, parallel algorithms, generalization, lattices, and other  ...  The unification problem and several variants are presented. Various algorithms and data structures are discussed.  ...  My thanks also go to the reviewers for making many suggestions relating to both accuracy and clarity.  ... 
doi:10.1145/62029.62030 fatcat:u4ieq3eekzg25fgpxsaebyktkm

Extensional Higher-Order Paramodulation in Leo-III [article]

Alexander Steen, Christoph Benzmüller
2021 arXiv   pre-print
Reasoning with primitive equality is enabled by adapting paramodulation-based proof search to higher-order logic.  ...  Leo-III supports reasoning in polymorphic first-order and higher-order logic, in all normal quantified modal logics, as well as in different deontic logics.  ...  Further related systems include higher-order extensions of SMT solvers [11] , and there is ongoing work to lift first-order ATP systems based on superposition to fragments of HOL, including E [85, 102  ... 
arXiv:1907.11501v2 fatcat:hv3uysew6fdrtixvtv5d7ydgb4

A treatment of higher-order features in logic programming [article]

Gopalan Nadathur
2004 arXiv   pre-print
Another aspect relates to the treatment of a unification operation that has a branching character and that sometimes calls for the delaying of the solution of unification problems.  ...  The logic programming paradigm provides the basis for a new intensional view of higher-order notions.  ...  Acknowledgements Earlier versions of the ideas in this paper have benefitted from discussions with Bharat Jayaraman and Debra Wilson.  ... 
arXiv:cs/0404020v1 fatcat:aykcccwc2fe5vh6c36u62qeme4

Modular Higher-Order Equational Preunification

1996 Journal of symbolic computation  
In this paper we present a procedure for such preunification, which is based on three transformations and parameterized by a first-order equational unification procedure that admits free function symbols  ...  Preunification of simply typed λ-terms with respect to the equivalence relation induced by α-, β-and η-conversion and an arbitrary first-order equational theory is useful in higher-order proof and logic  ...  Acknowledgements We sincerely thank the anonymous referees for their extremely detailed and constructive comments on several previous versions of this paper.  ... 
doi:10.1006/jsco.1996.0059 fatcat:4zielh7f2ja57ld3dymoyft6fy

Extensional Paramodulation for Higher-Order Logic and Its Effective Implementation Leo-III

Alexander Steen
2019 Künstliche Intelligenz  
The completeness proof hereby unifies and simplifies existing abstract consistency techniques for a formulation of higher-order logic that is based on primitive equality as sole logical connective.  ...  An evaluation on a heterogeneous set of benchmark problems confirms that Leo-III is one of the most effective and versatile higher-order automated reasoning systems to date. vii  ...  The here presented unification rules are a variant of Huet's unification procedure [Hue75] which is known to be sound and complete with respect to higher-order unification [Sny91, SG89] .  ... 
doi:10.1007/s13218-019-00628-8 fatcat:b6yfenxgxndb3myskqushsm6uy
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