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Paramodulation with NonMonotonic Orderings
 In 14th IEEE Symposium on Logic in Computer Science (LICS
, 1999
"... All current completeness results for ordered paramodulation require the term ordering Ø to be wellfounded, monotonic and total(izable) on ground terms. Here we introduce a new proof technique where the only properties required for Ø are wellfoundedness and the subterm property 1 . The technique ..."
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Cited by 10 (8 self)
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All current completeness results for ordered paramodulation require the term ordering Ø to be wellfounded, monotonic and total(izable) on ground terms. Here we introduce a new proof technique where the only properties required for Ø are wellfoundedness and the subterm property 1 . The technique is a relatively simple and elegant application of some fundamental results on the termination and confluence of ground term rewrite systems (TRS). By a careful further analysis of our technique, we obtain the first KnuthBendix completion procedure that finds a convergent TRS for a given set of equations E and a (possibly nontotalizable) reduction ordering Ø whenever it exists 2 . Note that being a reduction ordering is the minimal possible requirement on Ø, since a TRS terminates if, and only if, it is contained in a reduction ordering. Keywords: term rewriting, automated deduction. 1 Introduction Deduction with equality is fundamental in mathematics, logics and many applications of ...
Paramodulation with BuiltIn Abelian Groups
 in `15th IEEE Symposium on Logic in Computer Science (LICS
, 2000
"... A new technique is presented for superposition with firstorder clauses with builtin abelian groups (AG). Compared with previous approaches, it is simpler, and no inferences with the AG axioms or abstraction rules are needed. Furthermore, AGunification is used instead of the computationally more ex ..."
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Cited by 6 (4 self)
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A new technique is presented for superposition with firstorder clauses with builtin abelian groups (AG). Compared with previous approaches, it is simpler, and no inferences with the AG axioms or abstraction rules are needed. Furthermore, AGunification is used instead of the computationally more expensive unification modulo associativity and commutativity. Due to the simplicity and restrictiveness of our inference system, its compatibility with redundancy notions and constraints, and the fact that standard term orderings like RPO can be used, we believe that our technique will become the method of choice for practice, as well as a basis for new theoretical developments like logicbased complexity and decidability analysis. Keywords: term rewriting, automated deduction. 1 Introduction It is crucial for the performance of a deduction system that it incorporates specialized techniques to work efficiently with standard algebraic theories, since a nave handling of some axioms (like assoc...
On the Completeness of Arbitrary Selection Strategies for Paramodulation
 In Proceedings of ICALP 2001
, 2001
"... A crucial way for reducing the search space in automated deduction are the socalled selection strategies: in each clause, the subset of selected literals are the only ones involved in inferences. For firstorder Horn clauses without equality, resolution is complete with an arbitrary selection o ..."
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Cited by 5 (1 self)
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A crucial way for reducing the search space in automated deduction are the socalled selection strategies: in each clause, the subset of selected literals are the only ones involved in inferences. For firstorder Horn clauses without equality, resolution is complete with an arbitrary selection of one single literal in each clause [dN96]. For Horn clauses with builtin equality, i.e., paramodulationbased inference systems, the situation is far more complex. Here we show that if a paramodulationbased inference system is complete with eager selection of negative equations and, moreover, it is compatible with equality constraint inheritance, then it is complete with arbitrary selection strategies. A first important application of this result is the one for paramodulation wrt. nonmonotonic orderings, which was left open in [BGNR99]. 1
Rewritebased Deduction and Symbolic Constraints
 In Proceedings of the 16th International Conference on Automated Deduction, volume 1632 of LNAI
, 1997
"... Introduction Building a stateoftheart theorem prover requires the combination of at least three main ingredients: good theory, clever heuristics, and the necessary engineering skills to implement it all in an efficient way. Progress in each of these ingredients interacts in different ways. On t ..."
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Cited by 5 (2 self)
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Introduction Building a stateoftheart theorem prover requires the combination of at least three main ingredients: good theory, clever heuristics, and the necessary engineering skills to implement it all in an efficient way. Progress in each of these ingredients interacts in different ways. On the one hand, new theoretical insights replace heuristics by more precise and effective techniques. For example, the completeness proof of basic paramodulation [NR95,BGLS95] shows why no inferences below Skolem functions are needed, as conjectured by McCune in [McC90]. Regarding implementation techniques, adhoc algorithms for procedures like demodulation or subsumption are replaced by efficient, reusable, generalpurpose indexing data structures for which the time and space requirements are wellknown. But, on the other hand, theory also advances in other directions, producing new ideas for which the development of implementation techniques and heuristics that make
Classes of Term Rewrite Systems with Polynomial Confluence Problems
 ACM Transactions on Computational Logic
, 2002
"... this article, we have assumed that all terms and the TRS R are built over a fixed signature F , which is not part of the input of the confluence problem. Indeed, if the arities of the input symbols are not bounded, we have the following result ..."
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Cited by 3 (1 self)
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this article, we have assumed that all terms and the TRS R are built over a fixed signature F , which is not part of the input of the confluence problem. Indeed, if the arities of the input symbols are not bounded, we have the following result
Superposition with Completely Builtin Abelian Groups
"... A new technique is presented for superposition with firstorder clauses with builtin abelian groups (AG). Compared with previous approaches, it is simpler, and AGunification is used instead of the computationally more expensive unification modulo associativity and commutativity. Furthermore, n ..."
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Cited by 3 (0 self)
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A new technique is presented for superposition with firstorder clauses with builtin abelian groups (AG). Compared with previous approaches, it is simpler, and AGunification is used instead of the computationally more expensive unification modulo associativity and commutativity. Furthermore, no inferences with the AG axioms or abstraction rules are needed; in this sense this is the first approach where AG is completely built in. 1.
Paramodulation and KnuthBendix Completion with Nontotal and Nonmonotonic Orderings
, 2001
"... Up to now, all existing completeness results for ordered paramodulation and KnuthBendix completion require the term ordering to be wellfounded, monotonic and total(izable) on ground terms. For several applications, these requirements are too strong, and hence weakening them has been a wellknown ..."
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Cited by 2 (1 self)
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Up to now, all existing completeness results for ordered paramodulation and KnuthBendix completion require the term ordering to be wellfounded, monotonic and total(izable) on ground terms. For several applications, these requirements are too strong, and hence weakening them has been a wellknown research challenge. Here we
A Polynomial Algorithm for . . . Shallow Term Rewrite Systems
, 2010
"... Term rewrite systems are useful in many areas of computer science. Two especially important areas are decision procedures for the word problem of some algebraic systems and rulebased programming. One of the most studied properties of rewrite systems is confluence, and one of the primary benefits o ..."
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Term rewrite systems are useful in many areas of computer science. Two especially important areas are decision procedures for the word problem of some algebraic systems and rulebased programming. One of the most studied properties of rewrite systems is confluence, and one of the primary benefits of having a confluent rewrite system is that the system also has uniqueness of normal forms. However, uniqueness of normal forms is an interesting property in its own right and well studied. Also, confluence can be too strong a requirement for applications. In this paper, we study the decidability of uniqueness of normal forms. Uniqueness of normal forms is decidable for ground rewrite systems, but is undecidable in general. This paper shows that the uniqueness of normal forms problem is decidable for the class of linear shallow term rewrite systems, and gives a decision procedure that is polynomial as long as the arities of the function symbols are bounded or the signature is fixed.
Inversion Strategies
"... . We consider the problem of inversion, for functions dened by groundconvergent rewrite systems. That is, we are looking for an algorithm for nding most general solutions to problems of the form t = ? G where G is a ground term. We rst reformulate the inversion procedure of [DM99] as goal rew ..."
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. We consider the problem of inversion, for functions dened by groundconvergent rewrite systems. That is, we are looking for an algorithm for nding most general solutions to problems of the form t = ? G where G is a ground term. We rst reformulate the inversion procedure of [DM99] as goal rewriting rules directed by strategies. Then, for the typical case of suciently dened specications, we present new rules and strategies for solving the inversion problem. 1 Introduction In rulebased languages like ELAN [BKK + 98] or Claire [CL96] it would be very useful to have an automatic inversion capability which automatically provides inverses of functions dened by the user. We design several new inference systems for solving inversion problems which are geared particularly to suciently complete rewrite systems, including a new failure rule based on an appropriate measure of distance to normal form. We emphasize the explicit description of rule application strategies, making...