Bisimilarity as a Theory of Functional Programming

AbstractMorris-style contextual equivalence — invariance of termination under any context of ground type — is the usual notion of operational equivalence for deterministic functional languages such as FPC (PCF plus sums, products and recursive types). Contextual equivalence is hard to establish directly. Instead we define a labelled transition system for call-by-name FPC (and variants) and prove that CCS-style bisimilarity equals contextual equivalence — a form of operational extensionality. Using co-induction we establish equational laws for FPC. By considering variations of Milner's ‘bisimulations up to ∼’ we obtain a second co-inductive characterisation of contextual equivalence in terms of reduction behaviour and production of values. Hence we use co-inductive proofs to establish contextual equivalence in a series of stream-processing examples. Finally, we consider a form of Milner's original context lemma for FPC, but conclude that our form of bisimilarity supports simpler co-inductive proofs

A Generalization of Short-Cut Fusion and Its Correctness Proof

Short-cut fusion is a program transformation technique that uses a single local transformation - called the foldr build rule - to remove certain intermediate lists from modularly constructed functional programs. Arguments that short-cut fusion is correct typical appeal either to intuition or to "free theorems" - even though the latter have not been known to hold for the languages supporting higher-order polymorphic functions and fixed point recursion in which short-cut fusion is usually applied. In this paper we use Pitts' recent demonstration that contextual equivalence in such languages is relationally parametric to prove that programs in them which have undergone short-cut fusion are contextually equivalent to their unfused counterparts. For each algebraic data type we then define a generalization of build which constructs substitution instances of its associated data structures, and use Pitts' techniques to prove the correctness of a contextual equivalence-preserving fusion rule which generalizes short-cut fusion. These rules optimize compositions of functions that uniformly consume algebraic data structures with functions that uniformly produces substitution instances of those data structures

A complete proof of the safety of Nöcker's strictness analysis

This paper proves correctness of Nöcker's method of strictness analysis, implemented in the Clean compiler, which is an effective way for strictness analysis in lazy functional languages based on their operational semantics. We improve upon the work of Clark, Hankin and Hunt did on the correctness of the abstract reduction rules. Our method fully considers the cycle detection rules, which are the main strength of Nöcker's strictness analysis. Our algorithm SAL is a reformulation of Nöcker's strictness analysis algorithm in a higher-order call-by-need lambda-calculus with case, constructors, letrec, and seq, extended by set constants like Top or Inf, denoting sets of expressions. It is also possible to define new set constants by recursive equations with a greatest fixpoint semantics. The operational semantics is a small-step semantics. Equality of expressions is defined by a contextual semantics that observes termination of expressions. Basically, SAL is a non-termination checker. The proof of its correctness and hence of Nöcker's strictness analysis is based mainly on an exact analysis of the lengths of normal order reduction sequences. The main measure being the number of 'essential' reductions in a normal order reduction sequence. Our tools and results provide new insights into call-by-need lambda-calculi, the role of sharing in functional programming languages, and into strictness analysis in general. The correctness result provides a foundation for Nöcker's strictness analysis in Clean, and also for its use in Haskell

Basic Action Theory

Action semantics is a semantic description framework with very goodpragmatic properties but until now a rather weak theory for reasoningabout programs. A strong action theory would have a great practicalpotential, as it would facilitate reasoning about the large class ofprogramming languages that can be described in action semantics.This report develops the foundations for a richer action theory, bybringing together concepts and techniques from process theory andfrom work on operational reasoning about functional programs. Semanticpreorders and equivalences in the action semantics setting arestudied and useful operational techniques for establishing contextualequivalences are presented. These techniques are applied to establishequational and inequational action laws and an induction rule

FUNDIO: a lambda-calculus with letrec, case, constructors, and an IO-interface : approaching a theory of unsafePerformIO

This paper proposes a non-standard way to combine lazy functional languages with I/O. In order to demonstrate the usefulness of the approach, a tiny lazy functional core language FUNDIO , which is also a call-by-need lambda calculus, is investigated. The syntax of FUNDIO has case, letrec, constructors and an IO-interface: its operational semantics is described by small-step reductions. A contextual approximation and equivalence depending on the input-output behavior of normal order reduction sequences is defined and a context lemma is proved. This enables to study a semantics of FUNDIO and its semantic properties. The paper demonstrates that the technique of complete reduction diagrams enables to show a considerable set of program transformations to be correct. Several optimizations of evaluation are given, including strictness optimizations and an abstract machine, and shown to be correct w.r.t. contextual equivalence. Correctness of strictness optimizations also justifies correctness of parallel evaluation. Thus this calculus has a potential to integrate non-strict functional programming with a non-deterministic approach to input-output and also to provide a useful semantics for this combination. It is argued that monadic IO and unsafePerformIO can be combined in Haskell, and that the result is reliable, if all reductions and transformations are correct w.r.t. to the FUNDIO-semantics. Of course, we do not address the typing problems the are involved in the usage of Haskell s unsafePerformIO. The semantics can also be used as a novel semantics for strict functional languages with IO, where the sequence of IOs is not fixed

From Operational Semantics to Domain Theory

This paper builds domain theoretic concepts upon an operational foundation. The basic operational theory consists of a single step reduction system from which an operational ordering and equivalence on programs are defined. The theory is then extended to include concepts from domain theory, including the notions of directed set, least upper bound, complete partial order, monotonicity, continuity, finite element, !-algebraicity, full abstraction, and least fixed point properties. We conclude by using these concepts to construct a (strongly) fully abstract continuous model for our language. In addition we generalize a result of Milner and prove the uniqueness of such models. Contents 1 Introduction 2 1.1 Related Work : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 The Syntax and Semantics 3 2.1 Syntax : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 Semantics : : : : : : : : : : : : : : : : : : : : : : ..