Unification and Matching on Compressed Terms

Term unification plays an important role in many areas of computer science, especially in those related to logic. The universal mechanism of grammar-based compression for terms, in particular the so-called Singleton Tree Grammars (STG), have recently drawn considerable attention. Using STGs, terms of exponential size and height can be represented in linear space. Furthermore, the term representation by directed acyclic graphs (dags) can be efficiently simulated. The present paper is the result of an investigation on term unification and matching when the terms given as input are represented using different compression mechanisms for terms such as dags and Singleton Tree Grammars. We describe a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. For the same problem, NP-completeness is obtained when the terms are represented using the more general formalism of Singleton Tree Grammars. For first-order unification and matching polynomial time algorithms are presented, each of them improving previous results for those problems.Comment: This paper is posted at the Computing Research Repository (CoRR) as part of the process of submission to the journal ACM Transactions on Computational Logic (TOCL)

Unification on Compressed Terms

First-order term unification is an essential concept in areas like functional and logic programming, automated deduction, deductive databases, artificial intelligence, information retrieval, compiler design, etc. We build upon recent developments in grammar-based compression mechanisms for terms and investigate algorithms for first-order unification and matching on compressed terms. We prove that the first-order unification of compressed terms is decidable in polynomial time, and also that a compressed representation of the most general unifier can be computed in polynomial time. Furthermore, we present a polynomial time algorithm for first-order matching on compressed terms. Both algorithms represent an improvement in time complexity over previous results [GGSS09, GGSS08]. We use several known results on the tree grammars used for compression, called singleton tree grammars (STG)s, like polynomial time computability of several subalgorithmms: certain grammar extensions, deciding equality of represented terms, and generating their preorder traversal. An innovation is a specialized depth of an STG that shows that unifiers can be represented in polynomial spac

Linear Compressed Pattern Matching for Polynomial Rewriting (Extended Abstract)

This paper is an extended abstract of an analysis of term rewriting where the terms in the rewrite rules as well as the term to be rewritten are compressed by a singleton tree grammar (STG). This form of compression is more general than node sharing or representing terms as dags since also partial trees (contexts) can be shared in the compression. In the first part efficient but complex algorithms for detecting applicability of a rewrite rule under STG-compression are constructed and analyzed. The second part applies these results to term rewriting sequences. The main result for submatching is that finding a redex of a left-linear rule can be performed in polynomial time under STG-compression. The main implications for rewriting and (single-position or parallel) rewriting steps are: (i) under STG-compression, n rewriting steps can be performed in nondeterministic polynomial time. (ii) under STG-compression and for left-linear rewrite rules a sequence of n rewriting steps can be performed in polynomial time, and (iii) for compressed rewrite rules where the left hand sides are either DAG-compressed or ground and STG-compressed, and an STG-compressed target term, n rewriting steps can be performed in polynomial time.Comment: In Proceedings TERMGRAPH 2013, arXiv:1302.599

On Unification Modulo One-Sided Distributivity: Algorithms, Variants and Asymmetry

An algorithm for unification modulo one-sided distributivity is an early result by Tid\'en and Arnborg. More recently this theory has been of interest in cryptographic protocol analysis due to the fact that many cryptographic operators satisfy this property. Unfortunately the algorithm presented in the paper, although correct, has recently been shown not to be polynomial time bounded as claimed. In addition, for some instances, there exist most general unifiers that are exponentially large with respect to the input size. In this paper we first present a new polynomial time algorithm that solves the decision problem for a non-trivial subcase, based on a typed theory, of unification modulo one-sided distributivity. Next we present a new polynomial algorithm that solves the decision problem for unification modulo one-sided distributivity. A construction, employing string compression, is used to achieve the polynomial bound. Lastly, we examine the one-sided distributivity problem in the new asymmetric unification paradigm. We give the first asymmetric unification algorithm for one-sided distributivity

Variants of unification considering compression and context variables

Term unification is a basic operation in several areas of computer science, specially in those related to logic. Generally speaking, it consists on solving equations over expressions called terms. Depending on the kind of variables allowed to occur in the terms and under which conditions two terms are considered to be equal, several frameworks of unification such as first-order unification, higher-order unification, syntactic unification, and unification modulo theories can be distinguished. Moreover, other variants of term unification arise when we consider nontrivial representations for terms. In this thesis we study variants of the classic first-order syntactic term unification problem resulting from the introduction of context variables, i.e. variables standing for contexts, and/or assuming that the input is given in some kind of compressed representation. More specifically, we focus on two of such compressed representations: the well-known Directed Acyclic Graphs (DAGs) and Singleton Tree Grammars (STGs). Similarly as DAGs allow compression by exploiting the reusement of repeated instances of a subterm in a term, STGs are a grammar-based compression mechanism based on the reusement of repeated (multi)contexts. An interesting property of the STG formalism is that many operations on terms can be efficiently performed directly in their compressed representation thus taking advantage of the compression also when computing with the represented terms. Although finding a minimal STG representation of a term is computationally difficult, this limitation has been successfully overcome is practice, and several STG-compressors for terms are available. The STG representation has been applied in XML processing and analysis of Term Rewrite Systems. Moreover, STGs are a very useful concept for the analysis of unification problems since, roughly speaking, allow to represent solutions in a succinct but still efficiently verifiable form.La unificació de termes és una operación básica en diverses àrees de la informática, especialment en aquelles relacionades amb la lógica. En termes generals, consisteix en resoldre equacions sobre expressions anomenades termes. Depenent del tipus de variables que apareguin en els termes i de sota quines condicions dos termes són considerats iguals, podem definir diverses variants del problema d'unificació de termes. A més a més, altres variants del problema d'unificació de termes sorgeixen quan considerem representacions no trivials per als termes. S'estudia variants del problema clàssic d'unificació sintáctica de primer ordre resultants de la introducció de variables de context i/o de l'assumpció de que l'entrada és donada en format comprimit. Ens centrem en l'estudi de dues representacions comprimides: els grafs dirigits acíclics i les gramàtiques d'arbre singletó. Similarment a com els grafs dirigits acíclics permeten compressió mitjançant el reús d'instàncies repetides d'un mateix subterme, les gramàtiques d'arbres singletó són un sistema de compressió basat en el reús de (multi)contexts. Una propietat interessant d'aquest formalisme és que moltes de les operacions comuns sobre termes es poden realizar directament sobre la versió comprimida d'aquests de forma eficient, sense cap mena de descompressió prèvia. Tot i que trobar una representació minimal d'un terme amb una gramática singletó és una tasca computacionalment difícil, aquesta limitació ha estat resolta de forma satisfactòria en la pràctica, hi ha disponibles diversos compressors per a termes. La representació de termes amb gramàtiques singletó ha estat útil per al processament de documents XML i l'anàlisi de sistemes de reescriptura de termes. Les gramàtiques singletó són concepte molt útil per a l'anàlisi de problemas d'unificació, ja que permeten representar solucions de forma comprimida sense renunciar a que puguin ser verificades de forma eficient. A la primera part d'aquesta tesi s'estudien els problemas d'unificació i matching de primer ordre sota l'assumpció de que els termes de l'entrada són representats fent servir gramàtiques singletó. Presentem algorismes de temps polinòmic per a aquests problemas, així com propostes d'implementacions i resultats experimentals. Aquests resultats s'utilitzen més endevant en aquesta tesi per a l'anàlisi de problemes d'unificació i matching amb variables de contexte i entrada comprimida. A la resta d'aquesta tesi ens centrem en variants d'unificació amb variables de contexte, que són un cas particular de variables d'ordre superior. Més concretament, a la segona part d'aquesta tesi, ens centrem en un cas particular d'unificació de contextes on nomès es permet una sola variable de context en l'entrada. Aquest problema s'anomena "one context unification". Mostrem que aquest problema es pot resoldre en temps polinòmic indeterminista, no només quan els termes de l'entrada són representats de forma explícita, sino també quan es representen fent servir grafs dirigits acíclics i gramàtiques singletó. També presentem un resultat parcial recent en la tasca de trobar un algorisme de temps polinòmic per a one context unification. Al final de la tesi, estudiem el problema de matching per a entrades amb variables de contexte, que és un cas particular d'unificació on només un dels dos termes de cada equació té variables. En contrast amb el problema general, matching de contextes ha estat classificat com a problema NP-complet. Mostrem que matching de contextes és NP-complet fins i tot quan es fan servir gramàtiques com a formalisme de representació de termes. Això implica que afegir compressió no ens porta a un augment dràstic de la complexitat del problema. També demostrem que la restricció de matching de contextes on el nombre de variables de contexte està afitat per una constant fixa del problema es pot resoldre en temps polinòmic, fins i tot quan es fan servir grafs dirigits acíclics com formalisme de representació de terme

Unification and matching on compressed terms

Term unification plays an important role in many areas of computer science, especially in those related to logic. The universal mechanism of grammar-based compression for terms, in particular the so-called singleton tree grammars (STGAs), have recently drawn considerable attention. Using STGs, terms of exponential size and height can be represented in linear space. Furthermore, the term representation by directed acyclic graphs (dags) can be efficiently simulated. The present article is the result of an investigation on term unification and matching when the terms given as input are represented using different compression mechanisms for terms such as dags and singleton tree grammars. We describe a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. For the same problem, NP-completeness is obtained when the terms are represented using the more general formalism of singleton tree grammars. For first-order unification and matching polynomial time algorithms are presented, each of them improving previous results for those problems.Peer Reviewe

Unification on Compressed Terms

First-order term unification is an essential concept in areas like functional and logic programming, automated deduction, deductive databases, artificial intelligence, information retrieval, compiler design, etc. We build upon recent developments in grammar-based compression mechanisms for terms and investigate algorithms for first-order unification and matching on compressed terms. We prove that the first-order unification of compressed terms is decidable in polynomial time, and also that a compressed representation of the most general unifier can be computed in polynomial time. Furthermore, we present a polynomial time algorithm for first-order matching on compressed terms. Both algorithms represent an improvement in time complexity over previous results [GGSS09, GGSS08]. We use several known results on the tree grammars used for compression, called singleton tree grammars (STG)s, like polynomial time computability of several subalgorithmms: certain grammar extensions, deciding equality of represented terms, and generating their preorder traversal. An innovation is a specialized depth of an STG that shows that unifiers can be represented in polynomial spac