Abstract

It is shown that for sets of Horn clauses saturated under basic paramodulation, the word and unifiability problems are in NP, and the number of minimal unifiers is simply exponential (i). For Horn sets saturated wrt. a special ordering under the more restrictive inference rule of basic superposition, the word and unifiability problems are still decidable and unification is finitary (ii). These two results are applied to the following languages. For shallow presentations (equations with variables at depth at most one) we show that the closure under paramodulation can be computed in polynomial time. Applying result (i), it follows that shallow unifiability is in NP, which is optimal since unifiability in ground theories is already NP-hard. The shallow word problem is even shown to be polynomial. Generalizing shallow theories to the Horn case, we obtain (two versions of) a language we call Catalog, a natural extension of Datalog to include functions and equality. The closure under paramo...

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