On a New Notion of Partial Refinement

Formal specification techniques allow expressing idealized specifications, which abstract from restrictions that may arise in implementations. However, partial implementations are universal in software development due to practical limitations. Our goal is to contribute to a method of program refinement that allows for partial implementations. For programs with a normal and an exceptional exit, we propose a new notion of partial refinement which allows an implementation to terminate exceptionally if the desired results cannot be achieved, provided the initial state is maintained. Partial refinement leads to a systematic method of developing programs with exception handling.Comment: In Proceedings Refine 2013, arXiv:1305.563

Computable decision making on the reals and other spaces via partiality and nondeterminism

Though many safety-critical software systems use floating point to represent real-world input and output, programmers usually have idealized versions in mind that compute with real numbers. Significant deviations from the ideal can cause errors and jeopardize safety. Some programming systems implement exact real arithmetic, which resolves this matter but complicates others, such as decision making. In these systems, it is impossible to compute (total and deterministic) discrete decisions based on connected spaces such as $\mathbb{R}$. We present programming-language semantics based on constructive topology with variants allowing nondeterminism and/or partiality. Either nondeterminism or partiality suffices to allow computable decision making on connected spaces such as $\mathbb{R}$. We then introduce pattern matching on spaces, a language construct for creating programs on spaces, generalizing pattern matching in functional programming, where patterns need not represent decidable predicates and also may overlap or be inexhaustive, giving rise to nondeterminism or partiality, respectively. Nondeterminism and/or partiality also yield formal logics for constructing approximate decision procedures. We implemented these constructs in the Marshall language for exact real arithmetic.Comment: This is an extended version of a paper due to appear in the proceedings of the ACM/IEEE Symposium on Logic in Computer Science (LICS) in July 201

Strategic polymorphism requires just two combinators!

In previous work, we introduced the notion of functional strategies: first-class generic functions that can traverse terms of any type while mixing uniform and type-specific behaviour. Functional strategies transpose the notion of term rewriting strategies (with coverage of traversal) to the functional programming paradigm. Meanwhile, a number of Haskell-based models and combinator suites were proposed to support generic programming with functional strategies. In the present paper, we provide a compact and matured reconstruction of functional strategies. We capture strategic polymorphism by just two primitive combinators. This is done without commitment to a specific functional language. We analyse the design space for implementational models of functional strategies. For completeness, we also provide an operational reference model for implementing functional strategies (in Haskell). We demonstrate the generality of our approach by reconstructing representative fragments of the Strafunski library for functional strategies.Comment: A preliminary version of this paper was presented at IFL 2002, and included in the informal preproceedings of the worksho

Rewriting and Well-Definedness within a Proof System

Term rewriting has a significant presence in various areas, not least in automated theorem proving where it is used as a proof technique. Many theorem provers employ specialised proof tactics for rewriting. This results in an interleaving between deduction and computation (i.e., rewriting) steps. If the logic of reasoning supports partial functions, it is necessary that rewriting copes with potentially ill-defined terms. In this paper, we provide a basis for integrating rewriting with a deductive proof system that deals with well-definedness. The definitions and theorems presented in this paper are the theoretical foundations for an extensible rewriting-based prover that has been implemented for the set theoretical formalism Event-B.Comment: In Proceedings PAR 2010, arXiv:1012.455