Explicit representation of terms defined by counter examples
Journal of Automated Reasoning
Equational problems anddisunification
Journal of Symbolic Computation
The resolution calculus
Extending Resolution for Model Construction
JELIA '90 Proceedings of the European Workshop on Logics in AI
Comparing Computational Representations of Herbrand Models
KGC '97 Proceedings of the 5th Kurt Gödel Colloquium on Computational Logic and Proof Theory
Decision Procedures and Model Building, or How to Improve Logical Information in Automated Deduction
Selected Papers from Automated Deduction in Classical and Non-Classical Logics
Solving Equational Problems Efficiently
CADE-16 Proceedings of the 16th International Conference on Automated Deduction: Automated Deduction
MACE4 and SEM: a comparison of finite model generators
Automated Reasoning and Mathematics
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The importance of models within automated deduction is generally acknowledged both in constructing countermodels (rather than just giving the answer "NO", if a given formula is found to be not a theorem) and in speeding up the deduction process itself (e.g. by semantic resolution refinement). However, little attention has been paid so far to the efficiency of algorithms to actually work with models. There are two fundamental decision problems as far as models are concerned, namely: the equivalence of 2 models and the truth evaluation of an arbitrary clause within a given model. This paper focuses on the efficiency of algorithms for these problems in case of Herbrand models given through atomic representations. Both problems have been shown to be coNP-hard in [Got 97], so there is a certain limit to the efficiency that we can possibly expect. Nevertheless, what we can do is find out the real "source" of complexity and make use of this theoretical result for devising an algorithm which, in general, has a considerably smaller upper bound on the complexity than previously known algorithms, e.g.: the partial saturation method in [FL 96] and the transformation to equational problems in [CZ 91]. The main result of this paper are algorithms for these two decision problems, where the complexity depends non-polynomially on the number of atoms (rather than on the total size) of the input model equivalence problem or clause evaluation problem, respectively. Hence, in contrast to the above mentioned algorithms, the complexity of the expressions involved (e.g.: the arity of the predicate symbols and, in particular, the term depth of the arguments) only has polynomial influence on the overall complexity of the algorithms.