Small-step Operational Semantics Research Articles

Refocusing in Reduction Semantics

The evaluation function of a reduction semantics (i.e., a small-step operational semantics with an explicit representation of the reduction context) is canonically defined as the transitive closure of (1) decomposing a term into a reduction context and a redex, (2) contracting this redex, and (3) plugging the contractum in the context. Directly implementing this evaluation function therefore yields an interpreter with a worst-case overhead, for each step, that is linear in the size of the input term. <br /> <br />We present sufficient conditions over the constituents of a reduction semantics to circumvent this overhead, by replacing the composition of (3) plugging and (1) decomposing by a single ``refocus'' function mapping a contractum and a context into a new context and a new redex, if any. We also show how to construct such a refocus function, we prove the correctness of this construction, and we analyze the complexity of the resulting refocus function. <br /> <br />The refocused evaluation function of a reduction semantics implements the transitive closure of the refocus function, i.e., a ``pre-abstract machine.'' Fusing the refocus function with the trampoline function computing the transitive closure gives a state-transition function, i.e., an abstract machine. This abstract machine clearly separates between the transitions implementing the congruence rules of the reduction semantics and the transitions implementing its reduction rules. The construction of a refocus function therefore shows how to mechanically obtain an abstract machine out of a reduction semantics, which was done previously on a case-by-case basis. <br /> <br />We illustrate refocusing by mechanically constructing Felleisen et al.'s CK machine from a call-by-value reduction semantics of the lambda-calculus, and by constructing a substitution-based version of Krivine's machine from a call-by-name reduction semantics of the lambda-calculus. We also mechanically construct three one-pass CPS transformers from three quadratic context-based CPS transformers for the lambda-calculus.

Category theory for operational semantics

We use the concept of a distributive law of a monad over a copointed endofunctor to define and develop a reformulation and mild generalisation of Turi and Plotkin's notion of an abstract operational rule. We make our abstract definition and give a precise analysis of the relationship between it and Turi and Plotkin's definition. Following Turi and Plotkin, our definition, suitably restricted, agrees with the notion of a set of GSOS-rules, allowing one to construct both an operational model and a canonical, internally fully abstract denotational model. Going beyond Turi and Plotkin, we construct what might be seen as large-step operational semantics from small-step operational semantics and we show how our definition allows one to combine distributive laws, in particular accounting for the combination of operational semantics with congruences.

On the Representation of McCarthy's amb in the π-calculus

We study the encoding of ▪, the call by name λ-calculus enriched with McCarthy's amb operator, into the π-calculus. Semantically, amb is a challenging operator, for the fairness constraints that it expresses. We prove that, under a certain interpretation of divergence in the λ-calculus (weak divergence), a faithful encoding is impossible. However, with a different interpretation of divergence (strong divergence), the encoding is possible, and for this case we derive results and coinductive proof methods to reason about ▪ that are similar to those for the encoding of pure λ-calculi. We then use these methods to derive the most important laws concerning amb. We take bisimilarity as behavioural equivalence on the π-calculus, which sheds some light on the relationship between fairness and bisimilarity. As a spin-off result, we show that there is no small-step operational semantics for ▪ that preserves the branching structure of terms and yields the correct predicates of convergence and (weak) divergence

Syntactic Type Soundness Results for the Region Calculus

The region calculus of Tofte and Talpin is a polymorphically typed lambda calculus with annotations that make memory allocation and deallocation explicit. It is intended as an intermediate language for implementing Hindley-Milner typed functional languages such as ML without traditional trace-based garbage collection. Static region and effect inference can be used to annotate a statically typed ML program with memory management primitives. Soundness of the calculus with respect to the region and effect system is crucial to guarantee safe deallocation of regions, i.e., deallocation should only take place for objects which are provably dead. The original soundness proof by Tofte and Talpin requires a complex co-inductive safety relation. In this paper, we present two small-step operational semantics for the region calculus and prove their type soundness with respect to the region and effect system. Following the standard syntactic approach of Wright, Felleisen, and Harper, we obtain simple inductive proofs. The first semantics is store-less. It is simple and elegant and gives rise to perspicuous proofs. The second semantics provides a store-based model for the region calculus. Albeit slightly more complicated, its additional expressiveness allows us to model operations on references with destructive update. A pure fragment of both small-step semantics is then proven equivalent to the original big-step operational approach of Tofte and Talpin. This leads to an alternative soundness proof for their evaluation-style formulation.

Semantics for Algebraic Operations

Given a category C with finite products and a strong monad T on C, we investigate axioms under which an ObC-indexed family of operations of the form αx:(Tx)n → Tx provides a definitive semantics for algebraic operations added to the computational λ-calculus. We recall a definition for which we have elsewhere given adequacy results for both big and small step operational semantics, and we show that it is equivalent to a range of other possible natural definitions of algebraic operation. We outline examples and non-examples and we show that our definition is equivalent to one for call-by-name languages with effects, too.

Syntactic Type Soundness for the Region Calculus

The region calculus of Tofte and Talpin is an annotated polymorphically typed lambda calculus which makes memory allocation and deallocation explicit. It is intended as an intermediate language in a compiler for ML-like languages. The region annotations are obtained by static region and effect inference, which makes it an attractive alternative for garbage collection. Soundness of the region and effect system is crucial to guarantee safe deallocation of regions, i.e. deallocation should only take place for objects which are provable dead.Tofte and Talpin have proved type soundness of the region calculus using rule-based co-induction. This proof is quite complicated and not very intuitive. Much of the problem lies in the low-level big-step operational semantics which involves manipulations of an explicit store and which has a co-inductive definition. In this paper, we present a small-step operational semantics for the region calculus, based on syntactic rewriting. We prove type soundness following the approach of Wright and Felleisen, leading to very simple inductive proofs.

Interprocedural analyses: a comparison

We present a framework for program analysis of languages with procedures which is general enough to allow for a comparison of various approaches to interprocedural analysis. Our framework is based on a small-step operational semantics and subsumes both frameworks for imperative and for logic languages. We consider reachability analysis, that is, the problem of approximating the sets of program states reaching program points. We use our framework in order to clarify the impact of several independent design decisions on the precision of the analysis. Thus, we compare intraprocedural forward accumulation with intraprocedural backward accumulation. Furthermore, we consider both relational and functional approaches. While for relational analysis the accumulation strategy makes no difference in precision, we prove for functional analysis that forward accumulation may lose precision against backward accumulation. Concerning the relative precision of relational analyses and corresponding functional analyses, we exhibit scenarios where functional analysis does not lose precision. Finally, we explain why even an enhancement of functional analysis through disjunctive completion of the underlying abstract domain may sometimes lose precision against relational analysis.

Systematic realisation of control flow analyses for CML

We present a methodology for the systematic realisation of control flow analyses and illustrate it for Concurrent ML. We start with an abstract specification of the analysis that is next proved semantically sound with respect to a traditional small-step operational semantics; this result holds for terminating w well as non-terminating programs. The analysis is defined coinductively and it is shown that all programs have a least analysis result (that is indeed the best one). To realise the analysis we massage the specification in three stages: (i) to explicitly record reachability of subexpressions, (ii) to be defined in a syntax-directed manner, and (iii) to generate a set of constraints that subsequently can be solved by standard techniques. We prove equivalence results between the different versions of the analysis; in particular it follows that the least solution to the constraints generated will be the least analysis result also to the initial specification.

Polymorphic Subtyping for Effect Analysis: The Semantics

We study an annotated type and effect system that integrates <kbd>let</kbd>-polymorphism, effects, and subtyping into an annotated type and effect system for a fragment of Concurrent ML. First a small step operational semantics is defined for concurrent ML and next the annotated type and effect system is proved semantically sound. This provides insights into the rule for generalisation in the annotated type and effect system.