Complete Sets of Reductions Modulo A Class of Equational Theories which Generate Infinite Congruence Classes

In this paper we present a generalization of the Knuth-Bendix procedure for generating a complete set of reductions modulo an equational theory. Previous such completion procedures have been restricted to equational theories which generate finite congruence classes. The distinguishing feature of this work is that we are able to generate complete sets of reductions for some equational theories which generate infinite congruence classes. In particular, we are able to handle the class of equational theories which contain the associative, commutative, and identity laws for one or more operators. We first generalize the notion of rewriting modulo an equational theory to include a special form of conditional reduction. We are able to show that this conditional rewriting relation restores the finite termination property which is often lost when rewriting in the presence of infinite congruence classes. We then develop Church-Rosser tests based on the conditional rewriting relation and set forth a completion procedure incorporating these tests. Finally, we describe a computer program which implements the theory and give the results of several experiments using the program

Smart matching

One of the most annoying aspects in the formalization of mathematics is the need of transforming notions to match a given, existing result. This kind of transformations, often based on a conspicuous background knowledge in the given scientific domain (mostly expressed in the form of equalities or isomorphisms), are usually implicit in the mathematical discourse, and it would be highly desirable to obtain a similar behavior in interactive provers. The paper describes the superposition-based implementation of this feature inside the Matita interactive theorem prover, focusing in particular on the so called smart application tactic, supporting smart matching between a goal and a given result.Comment: To appear in The 9th International Conference on Mathematical Knowledge Management: MKM 201

A formally verified proof of the prime number theorem

The prime number theorem, established by Hadamard and de la Vall'ee Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1 / ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erd"os in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.Comment: 23 page

Modularity and Combination of Associative Commutative Congruence Closure Algorithms enriched with Semantic Properties

Algorithms for computing congruence closure of ground equations over uninterpreted symbols and interpreted symbols satisfying associativity and commutativity (AC) properties are proposed. The algorithms are based on a framework for computing a congruence closure by abstracting nonflat terms by constants as proposed first in Kapur's congruence closure algorithm (RTA97). The framework is general, flexible, and has been extended also to develop congruence closure algorithms for the cases when associative-commutative function symbols can have additional properties including idempotency, nilpotency, identities, cancellativity and group properties as well as their various combinations. Algorithms are modular; their correctness and termination proofs are simple, exploiting modularity. Unlike earlier algorithms, the proposed algorithms neither rely on complex AC compatible well-founded orderings on nonvariable terms nor need to use the associative-commutative unification and extension rules in completion for generating canonical rewrite systems for congruence closures. They are particularly suited for integrating into the Satisfiability modulo Theories (SMT) solvers. A new way to view Groebner basis algorithm for polynomial ideals with integer coefficients as a combination of the congruence closures over the AC symbol * with the identity 1 and the congruence closure over an Abelian group with + is outlined

A System for the Diagnosis of Faults using a First Principles Approach

One of the primary areas of application of Artificial Intelligence is diagnosis. Diagnosis from first principles is a diagnostic technique which uses knowledge of the designed structure and function of a device to determine the possible causes of the malfunction. This work builds on the foundation of a theory of diagnosis by implementing and extending the theory. A correction to the algorithm which defines the theory is presented. The theory is extended for multiple sets of observations of the system and measurement data. A fundamental problem in diagnosis is selecting the measurement which will be of the most benefit in reducing the number of competing diagnoses for a system. A heuristic which selects a component whose measurement is likely to be beneficial in isolating the actual diagnosis is also presented

Superposition as a logical glue

The typical mathematical language systematically exploits notational and logical abuses whose resolution requires not just the knowledge of domain specific notation and conventions, but not trivial skills in the given mathematical discipline. A large part of this background knowledge is expressed in form of equalities and isomorphisms, allowing mathematicians to freely move between different incarnations of the same entity without even mentioning the transformation. Providing ITP-systems with similar capabilities seems to be a major way to improve their intelligence, and to ease the communication between the user and the machine. The present paper discusses our experience of integration of a superposition calculus within the Matita interactive prover, providing in particular a very flexible, "smart" application tactic, and a simple, innovative approach to automation.Comment: In Proceedings TYPES 2009, arXiv:1103.311

Automated Circuit Diagnosis using First Order Logic Tools

While Numerous Diagnostic Expert Systems Have Been Successfully Developed in Recent Years, They Are Almost Uniformly based on Heuristic Reasoning Techniques (I.e., Shallow Knowledge) in the Form of Rules. This Paper Reports on an Automated Circuit Diagnostic Tool based on Reiter\u27s Theory of Diagnosis. in Particular, this is a Theory of Diagnosis based on Deep Knowledge (I.e., Knowledge based on Certain Design Information) and using First Order Logic as the Representation Language. the Inference Mechanism Which is Incorporated as Part of the Diagnostic Tool is a Refutation based Theorem Prover using Rewriting Systems for Boolean Algebra Developed by Hsiang. Consequently, the Diagnostic Reasoning Tool is Broadly based on Reiter\u27s Model but Incorporates Complete Sets of Reductions for Boolean Algebra to Reason over Equa-Tional Descriptions of the Circuits to Be Analyzed. the Refutational Theorem Prover Uses an Associative Commutative Identity Unification Algorithm Described by Hsiang but Requires Additional Focusing Techniques in Order to Be Appropriate for Diagnosing Circuits. a Prototype Version of the Mainline Diagnostic Program Has Been Developed and Has Been Successfully Demonstrated on Several Small but Nontrivial Combinational Circuit Examples