Academia.eduAcademia.edu

Outline

Title
Abstract
Introduction
Complexity Analysis
Conclusion
References
All Topics
Mathematics

Nominal Anti-Unification

Profile image of Temur KutsiaTemur Kutsia

2015

https://doi.org/10.4230/LIPICS.RTA.2015.57
visibility

…

description

17 pages

link

1 file

Abstract

We study nominal anti-unification, which is concerned with computing least general generalizations for given terms-in-context. In general, the problem does not have a least general solution, but if the set of atoms permitted in generalizations is finite, then there exists a least general generalization which is unique modulo variable renaming and alpha-equivalence. We present an algorithm that computes it. The algorithm relies on a subalgorithm that constructively decides equivariance between two terms-in-context. We prove soundness and completeness properties of both algorithms and analyze their complexity. Nominal anti-unification can be applied to problems where generalization of first-order terms is needed (inductive learning, clone detection, etc.), but bindings are involved.

Related papers

A variant of higher-order anti-unification
Jordi Levy

2015

We present a rule-based Huet’s style anti-unification algorithm for simply-typed lambda-terms in η-long β-normal form, which computes a least general higher-order pattern generalization. For a pair of arbitrary terms of the same type, such a generalization always exists and is unique modulo α-equivalence and variable renaming. The algorithm computes it in cubic time within linear space. It has been implemented and the code is freely available.

View PDFchevron_right
Efficiency of a good but not linear nominal unification algorithm
Daniel Friedman

EasyChair Preprints, 2018

We present a nominal unification algorithm that runs in O(n × log(n) × G(n)) time, where G is the functional inverse of Ackermann's function. Nominal unification generates a set of variable assignments if there exists one, that makes terms involving binding operations α-equivalent. We preserve names while using special representations of de Bruijn numbers to enable efficient name management. We use Martelli-Montanari style multi-equation reduction to generate these name management problems from arbitrary unification terms.

View PDFchevron_right
Anti-Unification and Natural Language Processing
Nino Amiridze, Temur Kutsia

EasyChair Preprints, 2018

Anti-unification is a well-known method to compute generalizations in logic. Given two objects, the goal of anti-unification is to reflect commonalities between these objects in the computed generalizations, and highlight differences between them. Anti-unification appears to be useful for various tasks in natural language processing. Semantic classification of sentences based on their syntactic parse trees, grounded language learning, semantic text similarity, insight grammar learning, metaphor modeling: This is an incomplete list of topics where generalization computation has been used in one form or another. The major anti-unification technique in these applications is the original method for first-order terms over fixed arity alphabets, introduced by Plotkin and Reynolds in 1970s, and some of its adaptations. The goal of this paper is to give a brief overview about existing linguistic applications of anti-unification, discuss a couple of powerful and flexible generalization computation algorithms developed recently, and argue about their potential use in natural language processing tasks.

View PDFchevron_right
Nominal Unification and Matching of Higher Order Expressions with Recursive Let
Jordi Levy

ArXiv, 2021

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. We also explore specializations like nominal letrec-matching for expressions, for DAGs, and for garbagefree expressions and determine their complexity. 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.

View PDFchevron_right
Unital Anti-Unification: Type and Algorithms
Temur Kutsia

2020

Unital equational theories are defined by axioms that assert the existence of the unit element for some function symbols. We study anti-unification (AU) in unital theories and address the problems of establishing generalization type and designing anti-unification algorithms. First, we prove that when the term signature contains at least two unital functions, anti-unification is of the nullary type by showing that there exists an AU problem, which does not have a minimal complete set of generalizations. Next, we consider two special cases: the linear variant and the fragment with only one unital symbol, and design AU algorithms for them. The algorithms are terminating, sound, complete, and return tree grammars from which the set of generalizations can be constructed. Anti-unification for both special cases is finitary. Further, the algorithm for the one-unital fragment is extended to the unrestricted case. It terminates and returns a tree grammar which produces an infinite set of gen...

View PDFchevron_right
Term-Graph Anti-Unification
Temur Kutsia

2018

We study anti-unification for possibly cyclic, unranked term-graphs and develop an algorithm, which computes a minimal complete set of generalizations for them. For bisimilar graphs the algorithm computes the join in the lattice generated by a functional bisimulation. These results generalize anti-unification for ranked and unranked terms to the corresponding term-graphs, and solve also anti-unification problems for rational terms and dags. Our results open a way to widen anti-unification based code clone detection techniques from a tree representation to a graph representation of the code.

View PDFchevron_right
A Unifying Model of Variables and Names
Kidane Yemane

2005

We investigate a category theoretic model where both "variables" and "names", usually viewed as separate notions, are particular cases of the more general notion of distinction. The key aspect of this model is to consider functors over the category of irreflexive, symmetric finite relations. The models previously proposed for the notions of "variables" and "names" embed faithfully in the new one, and initial algebra/final coalgebra constructions can be transferred from the formers to the latter. Moreover, the new model allows for defining distinction-aware simultaneous substitutions as clone-like structures. Finally, we apply this model to develop the first semantic interpretation of the FOλ ∇ logic.

View PDFchevron_right
Higher-Order Pattern Anti-Unification in Linear Time
Temur Kutsia

Journal of Automated Reasoning, 2016

We present a rule-based Huet's style anti-unification algorithm for simply typed lambda-terms, which computes a least general higher-order pattern generalization. For a pair of arbitrary terms of the same type, such a generalization always exists and is unique modulo α-equivalence and variable renaming. With a minor modification, the algorithm works for untyped lambda-terms as well. The time complexity of both algorithms is linear.

View PDFchevron_right
A general theory of semi-unification
Assaf Kfoury

1993

An extended study of the theory of semi-unification is presented. Various restrictions on the terms allowed for substitution give rise to different cases of semi-unification. Semi-unification on finite and regular terms has already been considered in the literature. This thesis introduces a general case of semi-unification where substitutions are allowed on non-regular terms, and the equivalence of this general case to a well-known undecidable data base dependency problem is proved, thus establishing the undecidability of general semi-unification. Type reconstruction for the simply typed lambda calculus with polymorphic recursion is closely related to semi-unification. It has already been proved that semi-unification on finite terms is equivalent to this type reconstruction problem when only finite types are permitted. The equivalence of semi-unification on regular terms to type reconstruction when recursive types are permitted is proved, which proves the undecidability of this type...

View PDFchevron_right
Correcting a widespread error in unification algorithms
Peter Norvig

Software: Practice and Experience, 1991

The unification of two patterns both containing variables is an ubiquitous operation in Logic Programming and in many Artificial Intelligence applications. Thus, many texts present unification algorithms. Unfortunately, at least seven of these presentations are incorrect. The common error occurs when logic variables are represented as binding lists; implementations that destructively update variable cells do not manifest the error. This note gives the examples that uncover the error and presents a correction. KEY WORDS Unification Logic Programming Prolog Lisp The Problem Correct unification algorithms have been known and employed at least since Robinson's algorithm ¢ ¡£ was published in 1965. The problem with the Robinson algorithm is that it applies substitutions, replacing variables with their bindings, at each intermediate step of the algorithm. While correct, the algorithm creates many copies of intermediate structures, generating unnecessary garbage. There are two common solutions to this problem. The first, used by most PROLOG implementors, is to represent logic variables as cells that can be destructively updated. When a variable is unified with a value, the variable's cell is modified to point at the value. A trail of these modifications must be kept so that they can be undone later, if the computation must backtrack.

View PDFchevron_right

References (31)

  1. María Alpuente, Santiago Escobar, José Meseguer, and Pedro Ojeda. A modular equational generalization algorithm. In Michael Hanus, editor, LOPSTR, volume 5438 of LNCS, pages 24-39. Springer, 2008.
  2. R TA 2 0 1 5
  3. 2 Alexander Baumgartner and Temur Kutsia. A library of anti-unification algorithms. In Eduardo Fermé and João Leite, editors, Logics in Artificial Intelligence -14th European Conference, JELIA 2014, Funchal, Madeira, Portugal, September 24-26, 2014. Proceedings, volume 8761 of Lecture Notes in Computer Science, pages 543-557. Springer, 2014.
  4. Alexander Baumgartner, Temur Kutsia, Jordi Levy, and Mateu Villaret. A variant of higher-order anti-unification. In Femke van Raamsdonk, editor, RTA, volume 21 of LIPIcs, pages 113-127. Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik, 2013.
  5. Alexander Baumgartner, Temur Kutsia, Jordi Levy, and Mateu Villaret. Nominal Anti- Unification. Technical Report 15-03, RISC, JKU Linz, April 2015.
  6. Christophe Calvès. Complexity and Implementation of Nominal Algorithms. PhD thesis, King's College London, 2010.
  7. Christophe Calvès and Maribel Fernández. A polynomial nominal unification algorithm. Theor. Comput. Sci., 403(2-3):285-306, 2008.
  8. Christophe Calvès and Maribel Fernández. Matching and alpha-equivalence check for nom- inal terms. J. Comput. Syst. Sci., 76(5):283-301, 2010.
  9. James Cheney. Relating nominal and higher-order pattern unification. In Laurent Vigneron, editor, UNIF'05, LORIA A05-R-022, pages 105-119, 2005.
  10. James Cheney. Equivariant unification. JAR, 45(3):267-300, 2010.
  11. Gilles Dowek, Murdoch J. Gabbay, and Dominic P. Mulligan. Permissive nominal terms and their unification. In 24th Italian Conference on Computational Logic, CILC'09, 2009.
  12. Gilles Dowek, Murdoch James Gabbay, and Dominic P. Mulligan. Permissive nominal terms and their unification: an infinite, co-infinite approach to nominal techniques. Logic Journal of the IGPL, 18(6):769-822, 2010.
  13. Cao Feng and Stephen Muggleton. Towards inductive generalization in higher order logic. In Derek H. Sleeman and Peter Edwards, editors, ML, pages 154-162. Morgan Kaufmann, 1992.
  14. Murdoch Gabbay and Andrew M. Pitts. A new approach to abstract syntax with variable binding. Formal Asp. Comput., 13(3-5):341-363, 2002.
  15. Murdoch J. Gabbay. A Theory of Inductive Definitions with alpha-Equivalence. PhD thesis, University of Cambridge, UK, 2000.
  16. Murdoch J. Gabbay and Andrew M. Pitts. A new approach to abstract syntax involving binders. In LICS, pages 214-224. IEEE Computer Society, 1999.
  17. Ulf Krumnack, Angela Schwering, Helmar Gust, and Kai-Uwe Kühnberger. Restricted higher-order anti-unification for analogy making. In Mehmet A. Orgun and John Thornton, editors, Australian Conference on Artificial Intelligence, volume 4830 of LNCS, pages 273- 282. Springer, 2007.
  18. Temur Kutsia, Jordi Levy, and Mateu Villaret. Anti-unification for unranked terms and hedges. In Manfred Schmidt-Schauß, editor, RTA, volume 10 of LIPIcs, pages 219-234. Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik, 2011.
  19. Jordi Levy and Mateu Villaret. Nominal unification from a higher-order perspective. In Andrei Voronkov, editor, RTA, volume 5117 of LNCS, pages 246-260. Springer, 2008.
  20. Jordi Levy and Mateu Villaret. An efficient nominal unification algorithm. In Christopher Lynch, editor, Proceedings of the 21st International Conference on Rewriting Techniques and Applications, RTA 2010, July 11-13, 2010, Edinburgh, Scottland, UK, volume 6 of LIPIcs, pages 209-226. Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik, 2010.
  21. Jordi Levy and Mateu Villaret. Nominal unification from a higher-order perspective. ACM Trans. Comput. Log., 13(2):10, 2012.
  22. Jianguo Lu, John Mylopoulos, Masateru Harao, and Masami Hagiya. Higher order gener- alization and its application in program verification. Ann. Math. Artif. Intell., 28(1-4):107- 126, 2000.
  23. Frank Pfenning. Unification and anti-unification in the calculus of constructions. In LICS, pages 74-85. IEEE Computer Society, 1991.
  24. Gordon D. Plotkin. A note on inductive generalization. Machine Intel., 5(1):153-163, 1970.
  25. Luc De Raedt. Logical and Relational Learning. Springer, 2008.
  26. John C. Reynolds. Transformational systems and the algebraic structure of atomic formulas. Machine Intel., 5(1):135-151, 1970.
  27. Ute Schmid. Inductive Synthesis of Functional Programs, Universal Planning, Folding of Finite Programs, and Schema Abstraction by Analogical Reasoning, volume 2654 of LNCS. Springer, 2003.
  28. Christian Urban. Nominal techniques in Isabelle/HOL. J. Autom. Reasoning, 40(4):327- 356, 2008.
  29. Christian Urban and James Cheney. Avoiding equivariance in alpha-prolog. In Paweł Urzyczyn, editor, Typed Lambda Calculi and Applications, volume 3461 of LNCS, pages 401-416. Springer Berlin Heidelberg, 2005.
  30. Christian Urban, Andrew M. Pitts, and Murdoch J. Gabbay. Nominal unification. In Matthias Baaz and Johann A. Makowsky, editors, CSL, volume 2803 of LNCS, pages 513- 527. Springer, 2003.
  31. Christian Urban, Andrew M. Pitts, and Murdoch J. Gabbay. Nominal unification. Theor. Comput. Sci., 323(1-3):473-497, 2004.

Related papers

Anti-unification and Generalization: A Survey
Temur Kutsia

arXiv (Cornell University), 2023

Anti-unification (AU) is a fundamental operation for generalization computation used for inductive inference. It is the dual operation to unification, an operation at the foundation of automated theorem proving. Interest in AU from the AI and related communities is growing, but without a systematic study of the concept nor surveys of existing work, investigations often resort to developing application-specific methods that existing approaches may cover. We provide the first survey of AU research and its applications and a general framework for categorizing existing and future developments.

View PDFchevron_right
Nominal unification
AW Pitts IV, Christian Urban

Theoretical Computer Science, 2004

We present a generalisation of first-order unification to the practically important case of equations between terms involving binding operations. A substitution of terms for variables solves such an equation if it makes the equated terms α-equivalent, i.e. equal up to renaming bound names. For the applications we have in mind, we must consider the simple, textual form of substitution in which names occurring in terms may be captured within the scope of binders upon substitution. We are able to take a "nominal" approach to binding in which bound entities are explicitly named (rather than using nameless, de Bruijn-style representations) and yet get a version of this form of substitution that respects α-equivalence and possesses good algorithmic properties. We achieve this by adapting two existing ideas. The first one is terms involving explicit substitutions of names for names, except that here we only use explicit permutations (bijective substitutions). The second one is that the unification algorithm should solve not only equational problems, but also problems about the freshness of names for terms. There is a simple generalisation of classical first-order unification problems to this setting which retains the latter's pleasant properties: unification problems involving α-equivalence and freshness are decidable; and solvable problems possess most general solutions.

View PDFchevron_right
Nominal Unification from a Higher-Order Perspective
Jordi Levy

ACM Transactions on Computational Logic, 2012

Nominal Logic is a version of first-order logic with equality, name-binding, renaming via nameswapping and freshness of names. Contrarily to higher-order logic, bindable names, called atoms, and instantiable variables are considered as distinct entities. Moreover, atoms are capturable by instantiations, breaking a fundamental principle of lambda-calculus. Despite these differences, nominal unification can be seen from a higher-order perspective. From this view, we show that nominal unification can be reduced to a particular fragment of higher-order unification problems: Higher-Order Pattern Unification. This reduction proves that nominal unification can be decided in quadratic deterministic time, using the linear algorithm for Higher-Order Pattern Unification. We also prove that the translation preserves most generality of unifiers.

View PDFchevron_right
(Nominal) Unification by Recursive Descent with Triangular Substitutions
ramana kumar

Interactive Theorem Proving, 2010

We mechanise termination and correctness for two unification algorithms, written in a recursive descent style. One computes unifiers for first order terms, the other for nominal terms (terms including α-equivalent binding structure). Both algorithms work with triangular substitutions in accumulator-passing style: taking a substitution as input, and returning an extension of that substitution on success. This style of algorithm has performance benefits and has not been mechanised previously. The algorithms use nested recursion so the termination proofs are non-trivial; the termination relation is also slightly different from usual.

View PDFchevron_right
An efficient nominal unification algorithm
Jordi Levy

2010

Nominal Unification is an extension of first-order unification where terms can contain binders and unification is performed modulo α-equivalence. Here we prove that the existence of nominal unifiers can be decided in quadratic time. First, we linearly-reduce nominal unification problems to a sequence of freshness and equalities between atoms, modulo a permutation, using ideas as Paterson and Wegman for first-order unification. Second, we prove that solvability of these reduced problems may be checked in quadratic time. Finally, we point out how using ideas of Brown and Tarjan for unbalanced merging, we could solve these reduced problems more efficiently.

View PDFchevron_right

Related topics

  • Mathematics
  • Computer Science
    • 580 California St., Suite 400
    San Francisco, CA, 94104
    © 2025 Academia. All rights reserved