A Coinductive Equational Characterisation of Trace Inclusion for Regular Processes

Abstract

In 1966 Arto Salomaa gave a complete axiomatisation of regular expressions. It can be viewed as a sound and complete proof system for regular processes with respect to the behavioural equivalence called language equivalence. This proof system consists of a finite set of axioms together with one inductive proof rule.

We show that the behavioural preorder called language containment or trace inclusion can be characterised in a similar manner, but using a coinductive rather than an inductive proof rule.

This work was supported with the financial support of the Science Foundation Ireland grant 13/RC/2094 and co-funded under the European Regional Development Fund through the Southern & Eastern Regional Operational Programme to Lero – the Irish Software Research Centre.

Some Proofs

Guarded Terms: A variable x is guarded in the term t if each free occurrence of x in t occurs underneath a prefix $a.-$. A recursion $\textsc {rec} x. t$ is guarded if x is guarded in t. Finally a term u is a guarded term if every sub-term of the form $\textsc {rec} x. t$ is guarded.

It will be convenient to have an inductive principle for guarded processes, that is closed terms which are guarded.

Definition 8

Let $\Downarrow$ be the least predicate over processes which satisfies the following rules:

• (a) $\mathop {\textsf {0}}\Downarrow , a.p \Downarrow$

• (b) $p \Downarrow , q \Downarrow$ implies $\tau .p \Downarrow$ and $(p + q) \Downarrow$

• (c) $t \{ {x}\mapsto { \textsc {rec} x. t } \} \Downarrow$ implies $\textsc {rec} x. t \Downarrow$.

Lemma 3

Suppose x is guarded in t for every $x \in \mathop {\textsc {fv}( t)}$. Then $t\rho \, \Downarrow$, for any substitution $\rho$ such that $\mathop {\textsc {dom}}( \rho ) \subseteq \mathop {\textsc {fv}( t)}$.

Proof

By structural induction on t.    $\square$

Proposition 7

If p is guarded then $p \Downarrow$.

Proof

By structural induction on p. The only non-trivial case is when it has the form $\textsc {rec} x. t$, where we know that x is guarded in t. By the previous lemma this means that $t \{ {x}\mapsto { \textsc {rec} x. t } \} \Downarrow$, and therefore employing rule (c) from Definition 8 we can conclude that $\textsc {rec} x. t \Downarrow$.    $\square$

In the remainder of this appendix we will assume that all processes are guarded; this assumption is also used throughout the paper.

Proposition 8

(Proposition 1 ). For every process p there exists a head normal form $\mathop {\textsc {hnf}( p)}$ such that $p =_{ineq} \mathop {\textsc {hnf}( p)}$.

Proof

By induction on $p \Downarrow$. We proceed by an analysis of the structure of p.

• If p has the form a.q, or $\mathop {\textsf {0}}$ then it is already a hnf.

• If p is $\textsc {rec} x. t$ then by induction on $\Downarrow$ we know that there is some hnf h such that $t \{ {x}\mapsto {\textsc {rec} x. t} \} =_{ineq} h$. Using the $\textsc {(Ufd)}$ and $\textsc {(Fld)}$ rules we obtain $\textsc {rec} x. t =_{ineq} h$.

• If p is $\tau .q$ again the result follows by induction, using the axiom $\tau .X = X$.

• Finally suppose p has the form $p_1 + p_2$. By induction $p_i =_{ineq} h_i$ for some hnfs $h_1,h_2$. Suppose these have the form $\sum _{a \in A_i} a.p_a^i$, for $i = 1,2$. Then one can show that

  $$\begin{aligned} p&=_{ineq} \sum _{a \in (A_1 - A_2)} a.p_a^1 \,+\, \sum _{a \in (A_2 - A_1)} a.p_a^2 \,+\, \sum _{a \in (A_2 \cap A_1)} a.(p_a^1 + p_a^2) \end{aligned}$$

  which is in hnf.    $\square$

Corollary 3

(Proposition 3 ). If then .

Proof

The proof proceeds by induction on $p \Downarrow$ and a case analysis on the construction of $\mathop {\textsc {hnf}( p)}$ as outlined in the previous proposition.    $\square$