A rationale for conditional equational programming

AbstractConditional equations provide a paradigm of computation that combines the clean syntax and semantics of LISP-like functional programming with Prolog-like logic programming in a uniform manner. For functional programming, equations are used as rules for left-to-right rewriting; for logic programming, the same rules are used for conditional narrowing. Together, rewriting and narrowing provide increased expressive power. We discuss some aspects of the theory of conditional rewriting, and the reasons underlying certain choices in designing a language based on them. The most important correctness property a conditional rewriting program may posses is ground confluence; this ensures that at most one value can be computed from any given (variable-free) input term. We give criteria for confluence. Reasonable conditions for ensuring the completeness of narrowing as an operational mechanism for solving goals are provided; these results are then extended to handle rewriting with existentially quantified conditions and built-in predicates. Some termination issues are also considered, including the case of rewriting with higher-order terms

Datatype Laws Without Signatures

Using the well-known categorical notion of `functor' one may define the concept of datatype (algebra) without being forced to introduce a signature, that is, names and typings for the individual sorts (types) and operations involved. This has proved to be advantageous for those theory developments where one is not interested in the syntactic appearance of an algebra. The categorical notion of `transformer' developed in this paper allows the same approach to laws: without using signatures one can define the concept of law for datatypes (lawful algebras), and investigate the equational specification of datatypes in a syntax-free way. A transformer is a special kind of functor and also a natural transformation on the level of dialgebras. Transformers are quite expressive, satisfy several closure properties, and are related to naturality and Wadler's Theorems For Free. In fact, any colimit is an initial lawful algebra

On the confluence of lambda-calculus with conditional rewriting

The confluence of untyped \lambda-calculus with unconditional rewriting is now well un- derstood. In this paper, we investigate the confluence of \lambda-calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This extends results of M\"uller and Dougherty for unconditional rewriting. Two cases are considered, whether \beta-reduction is allowed or not in the evaluation of conditions. Moreover, Dougherty's result is improved from the assumption of strongly normalizing \beta-reduction to weakly normalizing \beta-reduction. We also provide examples showing that outside these conditions, modularity of confluence is difficult to achieve. Second, we go beyond the algebraic framework and get new confluence results using a restricted notion of orthogonality that takes advantage of the conditional part of rewrite rules

Searching for a Solution to Program Verification=Equation Solving in CCS

International audienceUnder non-exponential discounting, we develop a dynamic theory for stopping problems in continuous time. Our framework covers discount functions that induce decreasing impatience. Due to the inherent time inconsistency, we look for equilibrium stopping policies, formulated as fixed points of an operator. Under appropriate conditions, fixed-point iterations converge to equilibrium stopping policies. This iterative approach corresponds to the hierarchy of strategic reasoning in game theory and provides “agent-specific” results: it assigns one specific equilibrium stopping policy to each agent according to her initial behavior. In particular, it leads to a precise mathematical connection between the naive behavior and the sophisticated one. Our theory is illustrated in a real options model

Rn and Gn Logics

This paper proposes a simple, set-theoretic framework providingexpressive typing, higher-order functions and initial models atthe same time. Building upon Russell's ramified theory of types, we developthe theory of Rn-logics, which are axiomatisable by an order-sortedequational Horn logic with a membership predicate, and of Gn-logics,that provide in addition partial functions. The latter are therefore moreadapted to the use in the program specification domain, while sharing interesting properties, like existence of an initial model, with Rn-logics. Operational semantics of Rn-/Gn-logics presentations is obtained throughorder-sorted conditional rewriting