What Is a Theory?

Deduction modulo is a way to express a theory using computation rules instead of axioms. We present in this paper an extension of deduction modulo, called Polarized deduction modulo, where some rules can only be used at positive occurrences, while others can only be used at negative ones. We show that all theories in propositional calculus can be expressed in this framework and that cuts can always be eliminated with such theories. Mathematical proofs are almost never built in pure logic, but besides the deduction rules and the logical axioms that express the meaning of the connectors and quantifiers, they use something else a theory that expresses the meaning of the other symbols of the language. Examples of theories are equational theories, arithmetic, type theory, set theory, ... The usual definition of a theory, as a set of axioms, is sufficient when one is interested in the provability relation, but, as well-known, it is not when one is interested in the structure of proofs and in the theorem proving process. For instance, we can define a theory with the axioms a = b and b = c (where a, b and c are individual symbols) and prove the proposition a = c. However, we may also define this theory by the computation rules a −→ b and c −→ b and then a proposition t = u is provable if t and u have the same normal form using these computation rules. The advantages of this presentation are numerous. – We know that all the symbols occurring in a proof of t = u must occur in t or in u or one of their reducts. For instance, the symbol d need not be used in a proof of a = c. We get this way analyticity results. – In automated theorem proving, we can use this kind of results to reduce the search space. In fact, in this case, we just need to reduce deterministically the terms and check the identity of their normal forms. We get this way decisions algorithms. – Since the normal form of the proposition a = d is b = d and b and d are distinct, the proposition a = d is not provable in this theory. We get this way independence results and, in particular, consistency results. – In an interactive theorem prover, we can reduce the proposition to be proved, before we display it to the user. This way, the user is relieved from doing trivial computations. To define a theory with computation rules, not any set of rules is convenient. For instance, if instead of taking the rules a −→ b, c −→ b we take the rules b −→ a, b −→ c, we lose the property that a proposition t = u is provable if t and u have a common reduct. To be convenient, a rewrite system must be confluent. Confluence, and sometimes also termination, are necessary to have analyticity results, completeness of proof search methods, independence results, ... When we have rules rewriting propositions directly, for instance x× y = 0 −→ x = 0 ∨ y = 0 confluence is not sufficient anymore to have these results, but cut elimination is also required [7, 4]. Confluence and cut elimination are related. For instance, with the non confluent system b −→ a, b −→ c, we can prove the proposition a = c introducing a cut on the proposition b = b, but, because the rewrite system is not confluent, this cut cannot be eliminated. Confluence can thus be seen as a special case of cut elimination when only terms are rewritten [6], but in the general case, confluence is not a sufficient condition for cut elimination. Computation rules are not the only alternative to axioms. Another one is to add non logical deduction rules to predicate logic either taking an introduction and elimination rule for the abstraction symbol in various formulations of set theory [15, 2, 10, 1, 3, 9] or interpreting logic programs or definitions as deduction rules [11, 16, 17, 13] or in a more general setting [14]. Non logical deduction rules and computation rules have some similarities, but we believe that computation rules have some advantages. For instance, non logical deduction rules may blur the notion of cut in natural deduction and extra proof reduction rules have to be added (see, for instance, [5]). Also with some non logical deduction rules, the contradiction ⊥ may have a cut free proof and thus consistency is not always a consequence of cut elimination. In contrast, the notion of cut, the proof reduction rules and the properties of cut free proofs remain the usual ones with computation rules. When a theory is given by a set of axioms, we sometimes want to find an alternative way to present it with computation rules, in such a way that cut elimination holds. From cut elimination, we can deduce analyticity results, consistency and various independence results, completeness of proof search methods and in some cases decision algorithms. Many theories have been presented in such a way, including various equational theories, several presentations of simple type theory (with combinators or lambda-calculus, with or without the axiom of infinity, ...), the theory of equality, arithmetic, ... However, a systematic way of transforming a set of axioms into a set of rewrite rules is still to be found. A step in this direction is Knuth-Bendix method [12] and its extensions, that permit to transform some equational theories into rewrite systems with the cut elimination property (i.e. with the confluence property). Another step in this direction is the result of S. Negri and J. Von Plato [14] that gives a way to transform some sets of axioms, in particular all quantifier free theories, into a set of non logical deduction rules in sequent calculus, preserving cut elimination. In this paper, we propose a way to transform any consistent quantifier free theory into a set of computation rules with the cut elimination property. Our first attempt was to use Deduction modulo [7, 8] or Asymmetric deduction Modulo [6] as a general framework where computation and deduction can be mixed. In Deduction modulo, the introduction rule of conjunction