We consider the problem of solving linear equations over various semirings. In particular, solving of linear equations over polynomial rings with the additional restriction that the solutions must have only non-negati...
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We consider the problem of solving linear equations over various semirings. In particular, solving of linear equations over polynomial rings with the additional restriction that the solutions must have only non-negative coefficients is shown to be undecidable. Applications to undecidability proofs of several unification problems are illustrated, one of which, unification modulo one associative-commutative function and one endomorphism, has been a long-standing open problem. The problem of solving multiset constraints is also shown to be undecidable.
It is shown that unifiability is decidable in theories presented by a set of ground equations with several associative-communicative symbols (ground AC theories). This result applies, for instance, to finitely present...
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It is shown that unifiability is decidable in theories presented by a set of ground equations with several associative-communicative symbols (ground AC theories). This result applies, for instance, to finitely presented commutative semigroups, and it extends the authors' previous work (P. Narendran and M. Rusinwithch, 1991) where they gave an algorithm for solving the uniform word problem in ground AC theories.< >
There is increasing awareness that planning and model checking are closely related fields. Abstraction means to perform search in an over-approximation of the original problem instance, with a potentially much smaller...
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There is increasing awareness that planning and model checking are closely related fields. Abstraction means to perform search in an over-approximation of the original problem instance, with a potentially much smaller state space. This is the most essential method in model checking. One would expect that it can also be made successful in planning. We show, however, that this is likely to not be the case. The main reason is that, while in model checking one traditionally uses blind search to exhaust the state space and prove the absence of solutions, in planning informed search is used to find solutions. We give an exhaustive theoretical and practical account of the use of abstraction in planning. For all abstraction (over-approximation) methods known in planning, we prove that they cannot improve the best-case behavior of informed search. While this is easy to see for heuristic search, we were quite surprised to find that it also holds, in most cases, for the resolution-style proofs of unsolvability underlying SAT-based optimal planners. This result is potentially relevant also for model checking, where SAT-based techniques have recently been combined with abstraction. Exploring the issue in planning practice, we find that even hand-made abstractions do not tend to improve the performance of planners, unless the attacked task contains huge amounts of irrelevance. We relate these findings to the kinds of application domains that are typically addressed in model checking.
An algorithm for computing a complete set of unifiers for two terms involving associative-commutative function symbols is presented. It is based on a nondeterministic algorithm given by the authors in 1986 to show the...
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An algorithm for computing a complete set of unifiers for two terms involving associative-commutative function symbols is presented. It is based on a nondeterministic algorithm given by the authors in 1986 to show the NP-completeness of associative-commutative unifiability. The algorithm is easy to understand, and its termination can be easily established. Its complexity is easily analyzed and shown to be doubly exponential in the size of the input terms. The analysis also shows that there is a double-exponential upper bound on the size of a complete set of unifiers of two input terms. Since there is a family of simple associative-commutative unification problems which have complete sets of unifiers whose size is doubly exponential, the algorithm is optimal in its order of complexity in this sense.< >
We discuss a sequent style clause-based proof system that supports several important strategies in automatic theorem proving. The system has a goal-subgoal structure and supports back chaining with caching;it permits ...
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For finite convergent term-rewriting systems the equational unification problem is shown to be recursively independent of the equational matching problem, the word matching problem, and the (simultaneous) 2nd-order eq...
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A new decision procedure for the existential fragment of ordering constraints expressed using the recursive path ordering is presented. This procedure is nondeterministic and checks whether a set of constraints is sol...
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Path dissolution is an efficient generalization of the method of analytic tableaux. Both methods feature (in the propositional case) strong completeness, the lack of reliance upon conjunctive normal form (CNF), and th...
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Path dissolution is an efficient generalization of the method of analytic tableaux. Both methods feature (in the propositional case) strong completeness, the lack of reliance upon conjunctive normal form (CNF), and the ability to produce a list of essential models (satisfying interpretations) of a formula. Dissolution can speed up every step in a tableau deduction in classical logic. The authors consider means for adapting both techniques to multiple-valued logics, and show that the speed-up theorem applies in this more general setting. These results are pertinent for modeling uncertainty and commonsense reasoning.< >
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