Skeletal semantics and their interpretations

Abstract

The development of mechanised language specification based on structured operational semantics, with applications to verified compilers and sound program analysis, requires huge effort. General theory and frameworks have been proposed to help with this effort. However, none of this work provides a systematic way of developing concrete and abstract semantics, connected together by a general consistency result. We introduce a skeletal semantics of a language, where each skeleton describes the complete semantic behaviour of a language construct. We define a general notion of interpretation, which provides a systematic and language-independent way of deriving semantic judgements from the skeletal semantics. We explore four generic interpretations: a simple well-formedness interpretation; a concrete interpretation; an abstract interpretation; and a constraint generator for flow-sensitive analysis. We prove general consistency results between interpretations, depending only on simple language-dependent lemmas. We illustrate our ideas using a simple While language.(sound) with respect to a concrete semantics, and even suggests a methodology for how to construct consistent abstract semantics from concrete semantics [Cousot 1999;Midtgaard and Jensen 2008;Van Horn and Might 2011]. We focus on abstract semantics arising from concrete operational semantics. A prominent example can be found in the Verasco project [Jourdan et al. 2015] which provides a Coq-certified static analyser based on abstract interpretation, specifically targeting Com-pCert's mechanised C specification. Schmidt [Schmidt 1995[Schmidt , 1997a has demonstrated how to build abstract derivations from concrete derivations arising from an operational semantics, illustrating a close connection between the abstract and concrete semantics. The concepts are general, but the work does not attempt to be systematic. Inspired by Schmidt, Bodin et al. [Bodin et al. 2015] have identified a general rule format that can be systematically instantiated to both concrete and abstract semantics, with a general consistency result. However, their general rule format is based on a non-standard style of operational semantics, called pretty-big-step operational semantics [Charguéraud 2013], introduced to provide a Coq-mechanised specification of JavaScript [Bodin et al. 2014]. It does not provide a general systematic approach for constructing an abstract semantics from a standard operational semantics.A general framework provides a unifying meta-language for writing operational inference rules, in order to develop general environments for analysis [Harper et al. 1987;Jung et al. 2017;Pfenning and Schürmann 1999;Roşu and Şerbănuţă 2010]. Much of the work on frameworks does not aim to describe abstract analysis. One notable exception is the Iris framework [Jung et al. 2017] for reasoning about concurrent programs. Iris provides a systematic method for building a concurrent program logic from concrete operational semantics, proving a general consistency result. It starts from a conc...

A Cellular Howe Theorem

“…However, a difference is that we do not insist that transitions be stable under substitution. Links with other relevant work, e.g., Bodin et al [6], though desirable, remain unclear, perhaps because of the very different methods used. Furthermore, the cellularity used here is close to but different from the ∨ -familiality of [21].…”

Gillian, part i: a multi-language platform for symbolic execution

“…We can coarsely divide the existing parametric abstract interpretation frameworks into two groups: those based on small-step semantics [17,28,29,37,59] and those based on co-inductive big-step semantics [5,58]. Small-step frameworks follow a general methodology, first proposed in [37], for deriving sound and computable abstract interpreters from a given concrete interpreter written in small-step style.…”

Gillian, Part I: A Multi-language Platform for Symbolic Execution

“…A state model S = ⟨|S |, V, A, ea⟩ is proper if and only if its set of actions, A, includes the following distinguished actions and families of actions: (1) {setVar x } x ∈X , for assigning a given value v to a given variable x in the store of a given state σ , pretty-printed σ .setVar x (v); (2) setStore, for replacing the entire store of a given state σ with a new store ρ, pretty-printed σ .setStore(ρ) 2 ; (3) getStore, for obtaining the store of the given state σ , pretty-printed σ .getStore(−); (4) {eval e } e ∈E , for evaluating the expression e in a given state σ , pretty-printed σ .eval e (−), (5) assume, for extending the given state with the information denoted by a given value v, pretty-printed σ .assume(v); and (6) uSym and iSym, for generating new uninterpreted and interpreted symbols, and pretty-printed σ .uSym(v) and σ .iSym(v), respectively. From now on, we work with proper state models.…”

“…We can coarsely divide the existing parametric abstract interpretation frameworks into two groups: those based on small-step semantics [17,28,29,37,59] and those based on co-inductive big-step semantics [5,58]. Small-step frameworks follow a general methodology, first proposed in [37], for deriving sound and computable abstract interpreters from a given concrete interpreter written in small-step style.…”

“…Big-Step frameworks follow an alternative approach, first outlined in [58], which introduces a methodology for designing abstract interpreters based on co-inductively defined big-step semantics. Following similar ideas, Bodin et al [5] developed a general skeletal semantics framework in Coq, for creating concrete and abstract big-step semantics connected with a general consistency result, leaving the user to prove only a number of language-dependent lemmas. This work, however, has only been applied to a simple While language, making its broader applicability difficult to assess.…”