Axioms in Definitional Calculi
نویسنده
چکیده
This paper introduces a restricted form of the axiom rule in calculi of Partial Inductive Definitions (PID). The paper argues that in calculi of PIDs the distinction between atomic and non-atomic formulae is not as clear as in traditional sequent calculi. Therefore the common restriction of the axiom rule to the atomic case is not adequate for this type of calculi. A novel proviso for the axiom rule and corresponding provisos for the left and right definition rules are introduced with an accompanying discussion and suggestions for possible applications in the domain of declarative specification of operational behaviour of logic programs. 1 Background and motivation In sequent calculi of Partial Inductive Definitions (PID) (see e.g. [1, 2, 3]) the axiom (Initial Sequent) rule has been modelled on the “Axiom schema” of the traditional (logical) sequent calculus (see e.g. [4]). However, in calculi of PIDs defined atoms are in many respects more properly understood as complex expressions. We will try to clarify this point with an example but first let us recall some basic notions of the theory of partial inductive definitions. Let the definiens of an atom a with respect to a set of definitional clauses D be the following set of conditions:
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