Papers by Paliath Narendran
ArXiv, 2016
We investigate properties of convergent and forward-closed string rewriting systems in the contex... more We investigate properties of convergent and forward-closed string rewriting systems in the context of the syntactic criteria introduced in \cite{LynchMorawska} by Christopher Lynch and Barbara Morawska (we call these $LM$-Systems). Since a string rewriting system can be viewed as a term-rewriting system over a signature of purely monadic function symbols, we adapt their definition to the string rewriting case. We prove that the subterm-collapse problem for convergent and forward-closed string rewriting systems is effectively solvable. Therefore, there exists a decision procedure that verifies if such a system is an $LM$-System. We use the same construction to prove that the \emph{cap problem} from the field of cryptographic protocol analysis, which is undecidable for general $LM$-systems, is decidable when restricted to the string rewriting case.
Theoretical Computer Science, 2000
We show that simultaneous rigid E-uni cation, or SREU for short, is decidable and in fact EXPTIME... more We show that simultaneous rigid E-uni cation, or SREU for short, is decidable and in fact EXPTIME-complete in the case of one variable. This result implies that the 8 98 fragment of intuitionistic logic with equality is decidable. Together with a previous result regarding the undecidability of the 99-fragment, we obtain a complete classi cation of decidability of the prenex fragment of intuitionistic logic with equality, in terms of the quanti er pre x. It is also proved that SREU with one variable and a constant bound on the number of rigid equations is Pcomplete. Moreover, we consider a case of SREU where one allows several variables, but each rigid equation either contains one variable, or has a ground left-hand side and an equality between two variables as a righthand side. We show that SREU is decidable also in this restricted case.
Theoretical Computer Science, 1985
Almract. The 'Recursive Path Ordering' (RIO) scheme of Dershowitz is a powerful way of extending ... more Almract. The 'Recursive Path Ordering' (RIO) scheme of Dershowitz is a powerful way of extending a partial order on a set of function symbols to a well-founded partial order on their set of terms. We prove that, given a pair of terms, the problem of deciding whether they can be made RPO-comparable, by choosing a partial order on their function symbols, is NP-complete.
Journal of Symbolic Computation, 1989
The question of whether a monoid presented by a finite Thue system is cancellative is shown to be... more The question of whether a monoid presented by a finite Thue system is cancellative is shown to be undecidable (its negation is semidecidable), even when the Thue system is Church-Rosser. A decision procedure is described for the cjise of monadic Church-Rosser Thue systems and general commutative Thue systems. 'The results in this paper were first presented at the Workshop for Combinatorial Algorithms in Algebraic Structures, held in October 1985 at the Europaische Akademie, Otzenhausen, West Germany, under the auspices of Universitat Kaiserslautern. Work by the first author was supported in part by the National Science Foundation under grant DCR 84-08461. Work by the second author was supported in part by the National Science Foundation under grant DCR 84-01898.
Bull. EATCS, 2014
The scientific community receives with sadness the news of the death of Robert McNaughton, a pion... more The scientific community receives with sadness the news of the death of Robert McNaughton, a pioneer of theoretical computer science who has shaped our field by his ingenious contributions reported in a large number of lucid and highly influential papers. Bob McNaughton grew up in Brooklyn in New York City. His undergraduate degree is from Columbia University and his doctorate from Harvard, where his advisor was Willard Van Orman Quine. Several of his fellow Quine students became distinguished logicians:
We study the recursive path ordering (RPO) in the context of string-rewriting systems. We are int... more We study the recursive path ordering (RPO) in the context of string-rewriting systems. We are interested in finding a symbol ordering in RPO such that for every rule in the rewriting system, the left hand side is higher than the right hand side. We show that this SYMBOL-ORDER problem is NP-complete by a reduction from the 2-3-SAT problem. We also work on the one-rule case and show a polynomial time algorithm for such systems.
Electronic Proceedings in Theoretical Computer Science
Recently, interest has been emerging in the application of symbolic techniques to the specificati... more Recently, interest has been emerging in the application of symbolic techniques to the specification and analysis of cryptosystems. These techniques, when accompanied by suitable proofs of soundness/completeness, can be used both to identify insecure cryptosystems and prove sound ones secure. But although a number of such symbolic algorithms have been developed and implemented, they remain scattered throughout the literature. In this paper, we present a tool, CryptoSolve, which provides a common basis for specification and implementation of these algorithms, CryptoSolve includes libraries that provide the term algebras used to express symbolic cryptographic systems, as well as implementations of useful algorithms, such as unification and variant generation. In its current initial iteration, it features several algorithms for the generation and analysis of cryptographic modes of operation, which allow one to use block ciphers to encrypt messages more than one block long. The goal of our work is to continue expanding the tool in order to consider additional cryptosystems and security questions, as well as extend the symbolic libraries to increase their applicability.
[1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science
A new algorithm for computing a complete set of uni ers for two terms involving associativecommut... more A new algorithm for computing a complete set of uni ers for two terms involving associativecommutative function symbols is presented. The algorithm is based on a non-deterministic algorithm given by the authors in 1986 to show the NP-completeness of associative-commutative uni ability. The algorithm is easy to understand, its termination can be easily established. More importantly, its complexity can be easily analyzed and is shown to be doubly exponential in the size of the input terms. The analysis also shows that there is a double-exponential upper bound on the size of a complete set of uni ers of two input terms. Since there is a family of simple associativecommutative uni cation problems which have complete sets of uni ers whose size is doubly exponential, the algorithm is optimal in its order of complexity in this sense. This is the rst associative-commutative unication algorithm whose complexity has been completely analyzed. The approach can also be used to show a single exponential complexity for computing a complete set of uni ers for terms involving associativecommutative function symbols which also have the identity. Furthermore, for uni cation in the presence of associative-commutative-idempotent operators we get a doubly exponential bound.
Unification modulo convergent term rewrite systems is an important research area with many applic... more Unification modulo convergent term rewrite systems is an important research area with many applications. In their seminal paper Lynch and Morawska gave three conditions on rewrite systems that guarantee that unifiability can be checked in polynomial time (P). We show that these conditions are tight, in the sense that relaxing any one of them will "upset the applecart," giving rise to unification problems that are not in P (unless P = NP), and in doing so address an open problem posed by Lynch and Morawska. We also investigate a related decision problem: we show the undecidability of subterm-collapse for the restricted term rewriting systems that we are considering.
Lecture Notes in Computer Science, 2015
We investigate a hierarchical combination approach to the unification problem in non-disjoint uni... more We investigate a hierarchical combination approach to the unification problem in non-disjoint unions of equational theories. In this approach, the idea is to extend a base theory with some additional axioms given by rewrite rules in such way that the unification algorithm known for the base theory can be reused without loss of completeness. Additional techniques are required to solve a combined problem by reducing it to a problem in the base theory. In this paper we show that the hierarchical combination approach applies successfully to some classes of syntactic theories, such as shallow theories since the required unification algorithms needed for the combination algorithm can always be obtained. We also consider the matching problem in syntactic extensions of a base theory. Due to the more restricted nature of the matching problem, we obtain several improvements over the unification problem. 1. E 1 is subterm collapse free and Σ 2 symbols in R 1 appear as inner constructors. Thus, if s is Σ 1-rooted then t is Σ 1-rooted.
Proceedings of the 16th International Symposium on Principles and Practice of Declarative Programming, 2014
Recent advances in the automated analysis of cryptographic protocols have aroused new interest in... more Recent advances in the automated analysis of cryptographic protocols have aroused new interest in the practical application of unification modulo theories, especially theories that describe the algebraic properties of cryptosystems. However, this application requires unification algorithms that can be easily implemented and easily extended to combinations of different theories of interest. In practice this has meant that most tools use a version of a technique known as variant unification. This requires, among other things, that the theory be decomposable into a set of axioms B and a set of rewrite rules R such that R has the finite variant property with respect to B. Most theories that arise in cryptographic protocols have decompositions suitable for variant unification, but there is one major exception: the theory that describes encryption that is homomorphic over an Abelian group. In this paper we address this problem by studying various approximations of homomorphic encryption over an Abelian group. We construct a hierarchy of increasingly richer theories, taking advantage of new results that allow us to automatically verify that their decompositions have the finite variant property. This new verification procedure also allows us to construct a rough metric of the complexity of a theory with respect to variant unification, or variant complexity. We specify different versions of protocols using the different theories, and analyze them in the Maude-NPA cryptographic protocol analysis tool to assess their behavior. This gives us greater * Jose Meseguer and Fan Yang have been partially
Proceedings. Thirteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.98CB36226)
A term rewrite system (TRS) terminates i its rules are contained in a reduction ordering >. In or... more A term rewrite system (TRS) terminates i its rules are contained in a reduction ordering >. In order to deal with any set of equations, including inherently non-terminating ones (like commutativity), TRS have been generalised to ordered TRS (E; >), where equations of E are applied in whatever direction agrees with >. The con uence of terminating TRS is well-known to be decidable, but for ordered TRS the decidability of con uence has been open. Here we show that the con uence of ordered TRS is decidable if ordering constraints for > can be solved in an adequate way, which holds in particular for the class of LPO orderings. For sets E of constrained equations, con uence is shown to be undecidable. Finally, ground reducibility is proved undecidable for ordered TRS.
Lecture Notes in Computer Science, 2003
Theoretical Computer Science, 1998
For finite convergent term-rewriting systems it is shown that the equational unification problem ... more For finite convergent term-rewriting systems it is shown that the equational unification problem is recursively independent of the equational matching problem, the word matching problem, and the 2nd-order equational matching problem. Apart from the latter these results are derived by considering term-rewriting systems on signatures that contain unary function symbols only (i.e., string-rewriting systems). Also for this special case 2nd-order equational matching is shown to be reducible to lst-order equational matching. In addition, we present some new decidability results for simultaneous equational matching and unification. Finally, we compare the word unijication problem to the 2nd-order equational unzjication problem.
ACM SIGSOFT Software Engineering Notes, 1985
Lecture Notes in Computer Science, 1999
A new decision procedure for the existential fragment of ordering constraints expressed using the... more A new decision procedure for the existential fragment of ordering constraints expressed using the recursive path ordering is presented. This procedure is nondeterministic and checks whether a set of constraints is solvable over the given signature, i.e., the signature over which the terms in the constraints are defined. It is shown that this nondeterministic procedure runs in polynomial time, thus establishing the membership of this problem in the complexity class NP for the first time.
Theoretical Computer Science, 1993
It is investigated as to how far the various decidability results for finite, monadic, and conflu... more It is investigated as to how far the various decidability results for finite, monadic, and confluent string-rewriting systems can be carried over to the class of finite monadic string-rewriting systems that are only weakly confluent. Here a monadic string-rewriting system R on some alphabet z is called weakly confluent if it is confluent on all the congruence classes [a]s, with ao,Su {e}. After establishing that the property of weak confluence is tractable for finite monadic string-rewriting systems, we prove that many decision problems that are tractable for finite, monadic, and confluent systems are, in fact, undecidable for finite monadic systems that are only weakly confluent. An example is the word problem. On the other hand, for finite, monadic, and weakly confluent systems that present groups, the validation problem for linear sentences is decidable. Many decision problems, among them the word problem and the generalized word problem, can be expressed through linear sentences and, hence, they all are decidable in this setting. The paper closes with
Theoretical Computer Science, 1989
Based on a careful analysis of reduction sequences in monadic Thue systems we show that some unif... more Based on a careful analysis of reduction sequences in monadic Thue systems we show that some uniform decision problems, among them the uniform conjugacy problem, are decidable in polynomial time for finite monadic Church-Rosser Thue systems. On the other hand, an example of a decision probiem is exhibited that is undecidab!e even for this class of Thue systems.
Theoretical Computer Science, 1985
We present a single-axiom Thue system with a decidable word problem for which there does not exis... more We present a single-axiom Thue system with a decidable word problem for which there does not exist any finite equivalent canonical system. However, an equivalent finite canonical system for this Thue system can be obtained if new symbols are introduced in the presentation. This result settles an open question by Jantzen (1982) who asked whether every Thue system with a decidable word problem has an equivalent finite canonical System. We also discuss relationships between Thue systems and term rewriting systems.
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Papers by Paliath Narendran