The notion of simultaneous rigid E-unification was introduced in 1987 in the area of automated theorem proving with equality in sequent-based methods, for example the connection method or the tableau method. Recently, simultaneous rigid E-unification was shown undecidable. Despite the importance of this notion, for example in theorem proving in intuitionistic logic, very little is known of its decidable fragments. We prove decidability results for fragments of monadic simultaneous rigid E-unification and show the connections between this notion and some algorithmic problems of logic and computer science. 1 Introduction Simultaneous rigid E-unification plays a crucial role in automatic proof methods for first-order logic with equality based on sequent calculi, such as semantic tableaux [13], the connection method [6] (also known as the mating method [1]), model elimination [26] and a dozen other procedures. All these methods are based on the Herbrand theorem and express the idea that ...

The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of these problems is the so-called simultaneous rigid E-unification problem. In the second part, we survey recent result on the simultaneous rigid E-unification problem.

The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of these problems is the so-called simultaneous rigid E-unification problem. In the second part, we survey recent result on the simultaneous rigid E-unification problem. 1 Problems This simultaneous rigid E-unification : : : It has too procedural a definition. Vaughan Pratt, private communication. We start with some decision problems having non-procedural definitions. Most of these problems naturally stem from the famous Herbrand theorem. 1.1 Herbrand's theorem and six decision problems The Herbrand theorem [28] asserts that an existential first-order formula 9¯x'(¯x) is provable (in classical first-order logic) if and only if a particular disjunction '( ¯ t 1 ) : : : '( ¯ t n ) is provab...

Simultaneous rigid E-unification has been introduced in the area of theorem proving with equality. It is used in extension procedures, like the tableau method or the connection method. Many articles in this area tacitly assume the existence of an algorithm for simultaneous rigid E-unification. There were several faulty proofs of the decidability of this problem. In this article we prove several results about the simultaneous rigid E-unification. Two results are reductions of known problems to simultaneous rigid E-unification. Both these problems are very hard. The word equation solving (unification under associativity) is reduced to the monadic case of simultaneous rigid E-unification. The variable-bounded semi-unification problem is reduced to the general simultaneous rigid E-unification. The word equation problem used in the first reduction is known to be decidable, but the decidability result is extremely non-trivial. As for the variablebounded semi-unification, its decidability is ...

The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of these problems is the so-called simultaneous rigid E-unification problem. In the second part, we survey recent result on the simultaneous rigid problem.

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