311 research outputs found
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
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