On Operational Semantics of Congruence Relation Defined in Algebraic Language ASL/

An algebraic specification (or text) specifies a congruence relation on a set of expressions. In algebraic language $ASL/*$ , a pair $(G, AX)$ is called a text, where $G$ is a context-free grammar and $AX$ is a set of axioms. A text $t=(G, AX)$ specifies the set $E_{G}$ of expressions generated by $G$ and the least congruence relation on $E_{G}$ satisfying all the axioms in $AX$ , where ‘congruency’ is defined based on the syntax (phrase structure) of the expressions. In general, for a text $t$ in $ASL/*$ , the condition, (A) $e$ is congruent with $e’$ in $t$ , is not logically equivalent to the condition, (B) $e’$ is obtained from $e$ by rewriting $e$ when the axioms in $t$ are regarded as ‘bidirectional‘ rewrite rules. We present a sufficient condition for a text $t$ under which (A) and (B) are equivalent for any pair of expressions $e$ and $e’$ , which means that the congruence relation specified by $t$ is simply defined operationally.