Jan Willem Klop

ACP τ : a universal axiom system for process specification

Starting with Basic Process Algebra (BPA), an axiom system for alternative composition (+) and sequential composition (·) of processes, we give a presentation in several intermediate stages leading to ACPτ, Algebra of Communicating Processes with abstraction. At each successive stage an example is given showing that the specification power is increased. Also some graph models for the respective axiom systems are informally presented. We conclude with the Finite Specification Theorem for ACPτ, stating that each finitely branching, effectively presented process (as an element of the graph model) can be specified in ACPτ by means of a finite system of guarded recursion equations.

On equal μ -terms

Abstract We consider the rewrite system R μ with μ x . M → μ M [ x : = μ x . M ] as its single rewrite rule, where the signature consists of the variable binding operator commonly denoted by μ , first-order symbols, which in this paper are restricted to a binary function symbol F , and possibly some constant symbols. This kernel system denoting recursively defined objects occurs in several contexts, e.g. it is the framework of recursive types, with F as the function type constructor. For general signatures, this rewriting system is widely used to represent and manipulate infinite regular trees. The main concern of this paper is the convertibility relation for μ -terms as given by the μ -rule, in particular, its decidability. This relation is sometimes called weak μ -equality, in contrast with strong μ -equality, which is given by the equality of the possibly infinite tree unwinding of μ -terms. While strong equality has received much attention, the opposite is the case for weak μ -equality. We present three alternative proofs for decidability of weak μ -equality. The first two proofs build upon an ingenious proof method of Cardone and Coppo. Prior to that, we prepare the way by an analysis of α -conversion. We then give a decidability proof in an ‘ α -free’ manner, essentially treating μ -terms as first-order terms, and next, a proof in higher-order style, employing α -equivalence classes and viewing R μ as a higher-order rewriting system. The third decidability proof is also derived in an α -free manner, exploiting the regular nature of the set of μ -reducts, enabling an appeal to the theory of tree automata. We conclude with additional results concerning decidability of reachability, and upward-joinability of μ -reduction, and of convertibility by α -free μ -reduction.

A formalized proof system for total correctness of while programs

We introduce datatype specifications based on schemes, a slight generalization of first order specifications. For a schematic specification (Σ, \(\mathbb{E}\)), Hoares Logic HL (Σ, \(\mathbb{E}\)) for partial correctness is defined as usual and on top of it a proof system (Σ, \(\mathbb{E}\)) ⊢ p → S ↓ for termination assertions is defined. The system is first order in nature, but we prove it sound and complete w.r.t. a second order semantics. We provide a translation of a standard proof system HLT(A) for total correctness on a structure A into our format.