On word problems in equational theories

The Knuth-Bendix procedure for word problems in universal algebra is known to be very effective when it is applicable. However, the procedure will fail if it generates equations which cannot be oriented into rules (i.e. the system is not noetherian), or if it generates infinitely many rules (i.e. the system is not confluent). In 1980 Huet showed that even if the system is not confluent, the Knuth-Bendix procedure still yiels a demi-decision procedure for word problems, provided that every equation can be oriented. In this paper we show that even if there are non-orientable equations, the Knuth-Bendix procedure can still be modified into a reasonably efficient semi-decision procedure for word problems in equational theories. Thus, we have lifted the noetherian requirement in the Knuth-Bendix procedure. Several confluence results are also given in the paper together with some experiments. Our method can also be extended to more general theories. Comparison with related works is also given. The proof of completeness, which is an interesting subject by itself, employs a new proof technique which utilizes a notion of transfinite semantic trees which is designed for proving refutational completeness of theorem proving methods in general

On word problems in equational theories

The Knuth-Bendix procedure for word problems in universal algebra is known to be very effective when it is applicable. However, the procedure will fail if it generates equations which cannot be oriented into rules (i.e. the system is not noetherian), or if it generates infinitely many rules (i.e. the system is not confluent). In 1980 Huet showed that even if the system is not confluent, the Knuth-Bendix procedure still yiels a demi-decision procedure for word problems, provided that every equation can be oriented. In this paper we show that even if there are non-orientable equations, the Knuth-Bendix procedure can still be modified into a reasonably efficient semi-decision procedure for word problems in equational theories. Thus, we have lifted the noetherian requirement in the Knuth-Bendix procedure. Several confluence results are also given in the paper together with some experiments. Our method can also be extended to more general theories. Comparison with related works is also given. The proof of completeness, which is an interesting subject by itself, employs a new proof technique which utilizes a notion of transfinite semantic trees which is designed for proving refutational completeness of theorem proving methods in general

A Symbolic Intruder Model for Hash-Collision Attacks

In the recent years, several practical methods have been published to compute collisions on some commonly used hash functions. In this paper we present a method to take into account, at the symbolic level, that an intruder actively attacking a protocol execution may use these collision algorithms in reasonable time during the attack. Our decision procedure relies on the reduction of constraint solving for an intruder exploiting the collision properties of hush functions to constraint solving for an intruder operating on words

The Finite Basis Problem for Kiselman Monoids

In an earlier paper, the second-named author has described the identities holding in the so-called Catalan monoids. Here we extend this description to a certain family of Hecke--Kiselman monoids including the Kiselman monoids $\mathcal{K}_n$. As a consequence, we conclude that the identities of $\mathcal{K}_n$ are nonfinitely based for every $n\ge 4$ and exhibit a finite identity basis for the identities of each of the monoids $\mathcal{K}_2$ and $\mathcal{K}_3$. In the third version a question left open in the initial submission has beed answered.Comment: 16 pages, 1 table, 1 figur

Second-Order Algebraic Theories

Fiore and Hur recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categorical algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding

Canonized Rewriting and Ground AC Completion Modulo Shostak Theories : Design and Implementation

AC-completion efficiently handles equality modulo associative and commutative function symbols. When the input is ground, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground AC-completion for deciding formulas in the combination of the theory of equality with user-defined AC symbols, uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our algorithm, called AC(X), is obtained by augmenting in a modular way ground AC-completion with the canonizer and solver present for the theory X. This integration rests on canonized rewriting, a new relation reminiscent to normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is proved sound, complete and terminating, and is implemented to extend the core of the Alt-Ergo theorem prover.Comment: 30 pages, full version of the paper TACAS'11 paper "Canonized Rewriting and Ground AC-Completion Modulo Shostak Theories" accepted for publication by LMCS (Logical Methods in Computer Science

Deduction modulo theory

This paper is a survey on Deduction modulo theor