Associative-commutative Function Research Articles

On the parameterized complexity of associative and commutative unification

This article studies the parameterized complexity of the unification problem with associative, commutative, or associative-commutative functions with respect to the parameter “number of variables”. It is shown that if every variable occurs only once then both of the associative and associative-commutative unification problems can be solved in polynomial time, but that in the general case, both problems are W[1]-hard even when one of the two input terms is variable-free. For commutative unification, an algorithm whose time complexity depends exponentially on the number of variables is presented; moreover, if a certain conjecture is true then the special case where one input term is variable-free belongs to FPT. Some related results are also derived for a natural generalization of the classic string and tree edit distance problems that allows variables.

Deciding knowledge in security protocols under equational theories

The analysis of security protocols requires precise formulations of the knowledge of protocol participants and attackers. In formal approaches, this knowledge is often treated in terms of message deducibility and indistinguishability relations. In this paper we study the decidability of these two relations. The messages in question may employ functions (encryption, decryption, etc.) axiomatized in an equational theory. One of our main positive results says that deducibility and indistinguishability are both decidable in polynomial time for a large class of equational theories. This class of equational theories is defined syntactically and includes, for example, theories for encryption, decryption, and digital signatures. We also establish general decidability theorems for an even larger class of theories. These theorems require only loose, abstract conditions, and apply to many other useful theories, for example with blind digital signatures, homomorphic encryption, XOR, and other associative–commutative functions.

A term equality problem equivalent to graph isomorphism

We demonstrate that deciding if two terms containing otherwise uninterpreted associative, commutative, and associative-commutative function symbols and commutative variable-binding operators are equal is polynomially equivalent to determining if two graphs are isomorphic. The reductions we use provide insight into this result and suggest polynomial time special cases.

Associative-commutative unification

Unification in equational theories, that is, solving equations in varieties, is of special relevance to automated deduction. Major results in term rewriting systems, as in ( Peterson & Stickel, 1981; Hsiang, 1982), depend on unification in presence of associative-commutative functions. ( Stickel, 1975; Stickel, 1981) gave an associative-commutative unification algorithm, but its termination in the general case was still questioned. The first part of this paper is an introduction to unification theory, the second part concerns the solving of homogeneous linear diophantine equations, and the third contains a proof of termination and correctness of the associative-commutative unification algorithm for the general case.

Abstracts of Technical Reports Computer Science Branch, Corporate Research and Development, General Electric Company, Schenectady, NY 12345 (GE)

The unification problem for terms containing associative and commutative functions is of great importance in theorem provers based on the rewriting approach and resolution methods. The complexity of checking whether two such terms are unifiable was known to be NP-hard. It is proved that the problem is NP-complete by describing a nondeterministic polynomial time algorithm for it. It is also shown that if in addition, an associative-commutative function is assumed to be idempotent and/or have an identity, the problem still remains NP-complete.

A Unification Algorithm for Associative-Commutative Functions

article Free Access Share on A Unification Algorithm for Associative-Commutative Functions Author: Mark E. Stickel RCA, David Sarnoff Research Center, Princeton, NJ and Carnegie-Mellon University, Pittsburgh, Pennsylvania RCA, David Sarnoff Research Center, Princeton, NJ and Carnegie-Mellon University, Pittsburgh, PennsylvaniaView Profile Authors Info & Claims Journal of the ACMVolume 28Issue 3July 1981 pp 423–434https://doi.org/10.1145/322261.322262Published:01 July 1981Publication History 201citation950DownloadsMetricsTotal Citations201Total Downloads950Last 12 Months79Last 6 weeks4 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF