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17
An Efficient Incremental Algorithm for Solving Systems of Linear Diophantine Equations
, 1994
"... In this paper, we describe an algorithm for solving systems of linear Diophantine equations based on a generalization of an algorithm for solving one equation due to Fortenbacher [3]. It can solve a system as a whole, or be used incrementally when the system is a sequential accumulation of several s ..."
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Cited by 35 (0 self)
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In this paper, we describe an algorithm for solving systems of linear Diophantine equations based on a generalization of an algorithm for solving one equation due to Fortenbacher [3]. It can solve a system as a whole, or be used incrementally when the system is a sequential accumulation of several subsystems. The proof of termination of the algorithm is difficult, whereas the proofs of completeness and correctness are straightforward generalizations of Fortenbacher's proof.
Normalised Rewriting and Normalised Completion
, 1994
"... We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algor ..."
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Cited by 19 (2 self)
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We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algorithm for normalised rewriting. It contains as an instance the usual AC completion algorithm, but also the wellknown Buchberger's algorithm for computing standard bases of polynomial ideals. We investigate the particular case of completion of ground equations, In this case we prove by a uniform method that completion modulo E terminates, for some interesting E. As a consequence, we obtain the decidability of the word problem for some classes of equational theories. We give implementation results which shows the efficiency of normalised completion with respect to completion modulo AC. 1 Introduction Equational axioms are very common in most sciences, including computer science. Equations can ...
Complete Solving of Linear Diophantine Equations and Inequations without Adding Variables
 PROC. OF 1ST INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 1995
"... In this paper, we present an algorithm for solving directly linear Diophantine systems of both equations and inequations. Here directly means without adding slack variables for encoding inequalities as equalities. This algorithm is an extension of the algorithm due to Contejean and Devie [9] for so ..."
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Cited by 13 (1 self)
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In this paper, we present an algorithm for solving directly linear Diophantine systems of both equations and inequations. Here directly means without adding slack variables for encoding inequalities as equalities. This algorithm is an extension of the algorithm due to Contejean and Devie [9] for solving linear Diophantine systems of equations, which is itself a generalization of the algorithm of Fortenbacher [6] for solving a single linear Diophantine equation. All the nice properties of the algorithm of Contejean and Devie are still satisfied by the new algorithm: it is complete, i.e. provides a (finite) description of the set of solutions, it can be implemented with a bounded stack, and it admits an incremental version. All of these characteristics enable its easy integration in the CLP paradigm.
Automated Deduction with Associative Commutative Operators
, 1993
"... We propose a new inference system for automated deduction with equality and associative commutative operators. This system is an extension of the ordered paramodulation strategy. However, rather than using associativity and commutativity as the other axioms, they are handled by the ACunification a ..."
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Cited by 11 (1 self)
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We propose a new inference system for automated deduction with equality and associative commutative operators. This system is an extension of the ordered paramodulation strategy. However, rather than using associativity and commutativity as the other axioms, they are handled by the ACunification algorithm and the inference rules. Moreover, we prove the refutational completeness of this system without needing the functional reflexive axioms or ACaxioms. Such a result is obtained by semantic tree techniques. We also show that the inference system is compatible with simplification rules.
On The Use Of Constraints In Automated Deduction
, 1995
"... . This paper presents three approaches dealing with constraints in automated deduction. Each of them illustrates a different point. The expression of strategies using constraints is shown through the example of a completion process using ordered and basic strategies. The schematization of complex un ..."
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Cited by 7 (1 self)
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. This paper presents three approaches dealing with constraints in automated deduction. Each of them illustrates a different point. The expression of strategies using constraints is shown through the example of a completion process using ordered and basic strategies. The schematization of complex unification problems through constraints is illustrated by the example of an equational theorem prover with associativity and commutativity axioms. The incorporation of builtin theories in a deduction process is done for a narrowing process which solves queries in theories defined by rewrite rules with builtin constraints. Advantages of using constraints in automated deduction are emphasized and new challenging problems in this area are pointed out. 1 Motivations Constraints have been introduced in automated deduction since about 1990, although one could find similar ideas in theory resolution [32] and in higherorder resolution [16]. The idea is to distinguish two levels of deduction and t...
Fast methods for solving linear diophantine equations
 Proceedings of the 6th Portuguese Conference on Artificial Intelligence, 727
, 1993
"... Abstract. We present some recent results from our research on methods for finding the minimal solutions to linear Diophantine equations over the naturals. We give an overview of a family of methods we developed and describe two of them, called Slopes algorithm and Rectangles algorithm. From empirica ..."
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Cited by 7 (4 self)
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Abstract. We present some recent results from our research on methods for finding the minimal solutions to linear Diophantine equations over the naturals. We give an overview of a family of methods we developed and describe two of them, called Slopes algorithm and Rectangles algorithm. From empirical evidence obtained by directly comparing our methods with others, and which is partly presented here, we are convinced that ours are the fastest known to date when the equation coefficients are not too small (ie., greater than 2 or 3). 1
A Methodological View of Constraint Solving
, 1996
"... Constraints have become very popular during the last decade. Constraints allow to define sets of data by means of logical formulae. Our goal here is to survey the notion of constraint system and to give examples of constraint systems operating on various domains, such as natural, rational or real nu ..."
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Cited by 6 (2 self)
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Constraints have become very popular during the last decade. Constraints allow to define sets of data by means of logical formulae. Our goal here is to survey the notion of constraint system and to give examples of constraint systems operating on various domains, such as natural, rational or real numbers, finite domains, and term domains. We classify the different methods used for solving constraints, syntactic methods based on transformations, semantic methods based on adequate representations of constraints, hybrid methods combining transformations and enumerations. Examples are used throughout the paper to illustrate the concepts and methods. We also discuss applications of constraints to various fields, such as programming, operations research, and theorem proving. y CNRS and LRI, Bat. 490, Universit'e de Paris Sud, 91405 ORSAY Cedex, France fcomon, jouannaudg@lri.lri.fr z COSYTEC, Parc Club Orsay Universit'e, 4 Rue Jean Rostand, 91893 Orsay Cedex, France dincbas@cosytec.fr x ...
Solving linear Diophantine equations using the geometric structure of the solution space
 PROC. RTA'97, LNCS 1232
, 1997
"... In the development of algorithms for finding the minimal solutions of systems of linear Diophantine equations, little use has been made (to our knowledge) of the results by Stanley using the geometric properties of the solution space. Building upon these results, we present a new algorithm, and we s ..."
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Cited by 3 (2 self)
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In the development of algorithms for finding the minimal solutions of systems of linear Diophantine equations, little use has been made (to our knowledge) of the results by Stanley using the geometric properties of the solution space. Building upon these results, we present a new algorithm, and we suggest the use of geometric properties of the solution space in finding bounds for searching solutions and in having a qualitative evaluation of the difficulty in solving a given system.
On the Complexity of Counting the Hilbert Basis of a Linear Diophantine System
, 1999
"... We investigate the computational complexity of counting the Hilbert basis of a homogeneous system of linear Diophantine equations. We establish lower and upper bounds on the complexity of this problem by showing that counting the Hilbert basis is #Phard and belongs to the class #NP. Moreover, we in ..."
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Cited by 2 (2 self)
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We investigate the computational complexity of counting the Hilbert basis of a homogeneous system of linear Diophantine equations. We establish lower and upper bounds on the complexity of this problem by showing that counting the Hilbert basis is #Phard and belongs to the class #NP. Moreover, we investigate the complexity of variants obtained by restricting the number of occurrences of the variables in the system.