Extraction of structured programs from specification proofs

We present a method using an extended logical system for obtaining programs from specifications written in a sublanguage of CASL. These programs are “correct” in the sense that they satisfy their specifications. The technique we use is to extract programs from proofs in formal logic by techniques due to Curry and Howard. The logical calculus, however, is novel because it adds structural rules corresponding to the standard ways of modifying specifications: translating (renaming), taking unions, and hiding signatures. Although programs extracted by the Curry-Howard process can be very cumbersome, we use a number of simplifications that ensure that the programs extracted are in a language close to a standard high-level programming language. We use this to produce an executable refinement of a given specification and we then provide a method for producing a program module that maximally respects the original structure of the specification. Throughout the paper we demonstrate the technique with a simple example.

Monodic temporal resolution

Until recently, First-Order Temporal Logic (FOTL) has been only partially understood. While it is well known that the full logic has no finite axiomatisation, a more detailed analysis of fragments of the logic was not previously available. However, a breakthrough by Hodkinson et al., identifying a finitely axiomatisable fragment, termed the monodic fragment, has led to improved understanding of FOTL. Yet, in order to utilise these theoretical advances, it is important to have appropriate proof techniques for this monodic fragment.In this paper, we modify and extend the clausal temporal resolution technique, originally developed for propositional temporal logics, to enable its use in such monodic fragments. We develop a specific normal form for monodic formulae in FOTL, and provide a complete resolution calculus for formulae in this form. Not only is this clausal resolution technique useful as a practical proof technique for certain monodic classes, but the use of this approach provides us with increased understanding of the monodic fragment. In particular, we here show how several features of monodic FOTL can be established as corollaries of the completeness result for the clausal temporal resolution method. These include definitions of new decidable monodic classes, simplification of existing monodic classes by reductions, and completeness of clausal temporal resolution in the case of monodic logics with expanding domains, a case with much significance in both theory and practice.

Mechanising first-order temporal resolution

First-order temporal logic is a concise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic first-order temporal logics has identified important enumerable and even decidable fragments. Although a complete and correct resolution-style calculus has already been suggested for this specific fragment, this calculus involves constructions too complex to be of practical value. In this paper, we develop a machine-oriented clausal resolution method which features radically simplified proof search. We first define a normal form for monodic formulae and then introduce a novel resolution calculus that can be applied to formulae in this normal form. By careful encoding, parts of the calculus can be implemented using classical first-order resolution and can, thus, be efficiently implemented. We prove correctness and completeness results for the calculus and illustrate it on a comprehensive example. An implementation of the method is briefly discussed.

Monodic Temporal Resolution

First-order temporal logic is a coincise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic first-order temporal logics have identified important enumerable and even decidable fragments. In this paper we present the first resolution-based calculus for monodic first-order temporal logic. Although the main focus of the paper is on establishing completeness result, we also consider implementation issues and define a basic loop-search algorithm that may be used to guide the temporal resolution system.

Stratified resolution

We introduce a calculus of stratified resolution, in which special attention is paid to clauses that "define" relations. If such clauses are discovered in the initial set of clauses, they are treated using the rule of definition unfolding, i.e. the rule that replaces defined relations by their definitions. Stratified resolution comes with a powerful notion of redundancy: a clause to which definition unfolding has been applied can be removed from the search space. To prove the completeness of stratified resolution with redundancies, we use a novel combination of Bachmair and Ganzingerâ??s model construction technique and a hierarchical construction of orderings and least fixpoints.

On the EA-style integrated processing of self-contained mathematical texts

A sound and complete first-order goal-oriented sequent-type calculus is developed with ``large-block'' inference rules. In particular, the calculus contains formal analogues of such natural proof-search techniques as handling definitions and applying auxiliary propositions.

Quantitative Information Flow, Relations and Polymorphic Types

This paper uses Shannon's information theory to give a quantitative definition of information flow in systems that transform inputs to outputs. For deterministic systems, the definition is shown to specialise to a simpler form when the information source and the known inputs jointly determine the inputs. For this special case, the definition is related to the classical security condition of non-interference and an equivalence is established between non-interference and independence of random variables. Quantitative information flow for deterministic systems is then presented in relational form. With this presentation, it is shown how relational parametricity can be used to derive upper and lower bounds on information flows through families of functions defined in the second order lambda calculus.