Complete trace models of state and control

Abstract

We consider a hierarchy of four typed call-by-value languages with either higher-order or ground-type references and with either $$\mathrm {call/cc}$$ call / cc or no control operator.Our first result is a fully abstract trace model for the most expressive setting, featuring both higher-order references and $$\mathrm {call/cc}$$ call / cc , constructed in the spirit of operational game semantics. Next we examine the impact of suppressing higher-order references and callcc in contexts and provide an operational explanation for the game-semantic conditions known as visibility and bracketing respectively. This allows us to refine the original model to provide fully abstract trace models of interaction with contexts that need not use higher-order references or $$\mathrm {call/cc}$$ call / cc . Along the way, we discuss the relationship between error- and termination-based contextual testing in each case, and relate the two to trace and complete trace equivalence respectively.Overall, the paper provides a systematic development of operational game semantics for all four cases, which represent the state-based face of the so-called semantic cube.

Compositional relational reasoning via operational game semantics

“…will play an important role in the paper. As shown in [7], they correspond to trace equivalence in operational game semantics. We will review the connection in the next section and, using it as a foundation, develop a sound and complete technique for establishing ∼ = x(ciu) err (and, thus, ∼ = x err for x-terms).…”

“…Assuming countably infinite sets Loc (locations) and Var (variables), HOSC typing judgments take the form Σ; Γ M : τ , where Σ and Γ are finite partial functions that assign types to locations and variables respectively. All the typing rules are standard and can be found in the full version of [7]. In typing judgements, we often write Σ as shorthand for Σ; ∅ (closed) and Γ as shorthand for ∅; Γ (location-free).…”

“…The absence of control constructs in contexts turns out to correspond to a bracketing condition [20], while the absence of higher-order references corresponds to visibility [21]. An operational account of the correspondences has been presented in [7] through a series of labelled transition systems x[HOSC], which generate traces corresponding to interactions of HOSC terms with x-contexts. In this section, we give a unifying presentation of the results as a single LTS, called L x , which can be specialised to each case x through simple local adjustments.…”

“…M )}. We can then restate the full abstraction results from [7] as follows. They establish an exact correspondence between ciuequivalence and trace equivalence.…”

“…Game semantics can capture the resultant variations in the discriminating power by imposing restrictions on what moves O is allowed to make during play: the absence of general references in contexts (x = GOSC, GOS) corresponds to a condition called O-visibility, while the absence of control operators (x = HOS, GOS) corresponds to O-bracketing. In recent work [7], it was shown how to capture ∼ = x err in the spirit of operational game semantics: for each x, one can define an LTS, called x[HOSC], such that the induced notion of trace equivalence (strategy equality) coincides with ∼ = x err for terms of x. Indeed, the traces of x[HOSC] are those of HOSC[HOSC] that conform to restrictions relevant to x.…”