Logic with Equality: Partisan Corroboration and Shifted Pairing
Information and Computation
Herbrand's theorem plays a fundamental role in automated theorem proving methods based on tableaux. The crucial step in procedures based on such methods can be described as the corroboration problem or the Herbrand skeleton problem, where, given a positive integer m, called multiplicity, and a quantifier free formula, one seeks a valid disjunction of m instantiations of that formula. In the presence of equality, which is the case in this paper, this problem was recently shown to be undecidable.
... The main contributions of this paper are two theorems. The first, the Partisan Corroboration Theorem, relates corroboration problems with different multiplicities. The second, the Shifted Pairing Theorem, is a finite tree automata formalization of a technique for proving undecidability results through direct encodings of valid Turing machine computations. These theorems are used in the paper to explain and sharpen several recent undecidability results related to the corroboration problem, the simultaneous rigid E-unification problem and the prenex fragment of intuitionistic logic with equality. ] 1999 Academic Press Article ID inco.1999.2797, available online at http:ÂÂwww.idealibrary.com on 205 0890-5401Â99 30.00 Any syntactic object is ground if it contains no variables. A substitution is ground if its range is ground, and it is said to be in a given language if the terms in its range are in that language. A set of substitutions is ground if each member is ground. Given a positive integer m, a set of m ground substitutions [% 1 , ..., % m ] is an m-corroborator for . if the disjunction .% 1 6 } } } 6 .% m is provable. A ground substitution % corroborates . if [%] 1-corroborates .; such a % is called a corroborator for .. One popular form of the classical Herbrand theorem (e.g., Herbrand, 1972) is this: An existential formula _x.(x) is provable if and only if there exists a positive integer m and m-corroborator for . in the language of .. The minimal appropriate number m will be called the minimum multiplicity for .. The minimum multiplicity for a formula may exceed one. Here is a formula for which the minimum multiplicity is two, suggested by Erik Palmgren in a different but similar context; we use"r" for the formal equality sign: The Herbrand theorem plays a fundamental role in automated theorem proving methods known as the rigid variable methods (Voronkov, 1997) . We can identify the following procedure underlying such methods. Let _x.(x) be a closed formula that we wish to prove. The principal procedure of rigid variable methods Step I. Choose a positive integer m. Step II. Check if there exists an m-corroborator for .. Step III. If Step II succeeds then _x.(x) is provable, otherwise increase m and return to Step II. The kernel of the principal procedure is of course Step II or The Corroboration Problem. Instance: A quantifier free formula . and a positive integer m. Question: Is the minimum multiplicity for . bounded by m? Corroboration for a fixed m is called m-corroboration. A detailed discussion of corroboration and related problems is given by Degtyarev, Gurevich, and Voronkov (1996) . It is important to us here that corroboration is intimately related to existential intuitionistic provability and simultaneous rigid E-unification (Gallier, Raatz, and Snyder, 1987) . The first of these problems is easy to formulate: The Existential Intuitionistic Provability Problem. Instance: An existential formula _x.(x). Question: Is the formula provable in intuitionistic logic with equality? 206 GUREVICH AND VEANES