A Logic for the Specification and Proof of Controllable Processes of CCS

Abstract

This work has been motivated by the following general problem: find logics for the specification and proof of non deterministic programs. We suppose that programs belong to the set of terms T of an algebra with a congruence relation ~. The operators of the algebra correspond to program constructors and the relation ~ defines a concept of equivalence which is supposed to be adequate for the comparison of programs. A requirement for a logic with set of formulas F. to be a satisfactory tool for the specification and proof of such programs Is.

$$\forall {{\text{t}}_{\text{1}}}{\text{,}}{{\text{t}}_{\text{2}}} \in {\text{T }}\left( {{{\text{t}}_{\text{1}}} \sim {{\text{t}}_{\text{2}}}{\text{ iff }}\forall {\text{f}} \in {\text{F }}\left( {{{\text{t}}_{\text{1}}}{\text{l = f iff }}{{\text{t}}_{\text{2}}}{\text{l = f }}} \right)} \right)$$

i.e. the congruence ~ and the equivalence relation induced by the logic on programs, agree [GS]. Furthermore, the use of a logic as a tool for syntax directed proofs, requires that for any n-ary operator * of the algebra there exists an operator ⊗ on the logic such that

$$\forall {{\text{t}}_{\text{i}}} \in {\text{T, }}\forall {{\text{f}}_{\text{i}}} \in {\text{F }}{{\text{t}}_{\text{i}}}{\text{l = }}{{\text{f}}_{\text{i}}}{\text{ i = I}} \ldots {\text{n implies *}}\left( {{{\text{t}}_{\text{1}}}{\text{ \ldots }}{{\text{t}}_{\text{n}}}} \right)1 = \otimes \left( {{{\text{f}}_{\text{1}}}{\text{ \ldots }}{{\text{f}}_{\text{n}}}} \right)$$

and this is the strongest assertion one can deduce from the validity of t_i⊨f_i for i=1,..n.