We give a definition of Q-net, a generalization of Petri nets based on a Lawvere theory Q, for which many existing variants of Petri nets are a special case. This definition is functorial with respect to change in Lawvere theory, and we exploit this to explore the relationships between different kinds of Q-nets. To justify our definition of Q-net, we construct a family of adjunctions for each Lawvere theory explicating the way in which Q-nets present free models of Q in Cat. This gives a functorial description of the operational semantics for an arbitrary category of Q-nets. We show how this can be used to construct the semantics for Petri nets, pre-nets, integer nets, and elementary net systems.

Recursive definitions of predicates are usually interpreted either inductively or coinductively. Recently, a more powerful approach has been proposed, called flexible coinduction, to express a variety of intermediate interpretations, necessary in some cases to get the correct meaning. We provide a detailed formal account of an extension of logic programming supporting flexible coinduction. Syntactically, programs are enriched by coclauses, clauses with a special meaning used to tune the interpretation of predicates. As usual, the declarative semantics can be expressed as a fixed point which, however, is not necessarily the least, nor the greatest one, but is determined by the coclauses. Correspondingly, the operational semantics is a combination of standard SLD resolution and coSLD resolution. We prove that the operational semantics is sound and complete with respect to declarative semantics restricted to finite comodels.

The reachability semantics for Petri nets can be studied using open Petri nets. For us, an “open” Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category Open(Petri), which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category ${\mathbb O}$ pen(Petri). We describe two forms of semantics for open Petri nets using symmetric monoidal double functors out of ${\mathbb O}$ pen(Petri). The first, an operational semantics, gives for each open Petri net a category whose morphisms are the processes that this net can carry out. This is done in a compositional way, so that these categories can be computed on smaller subnets and then glued together. The second, a reachability semantics, simply says which markings of the outputs can be reached from a given marking of the inputs.

In this paper we present an operational semantics for the ‘call-by-name’ probabilistic λ-calculus, whose main feature is to use only deterministic relations and to have no constraint on the reduction strategy. The calculus enjoys similar properties to the usual λ-calculus. In particular we prove it to be confluent, and we prove a standardisation theorem.

Probabilistic operational semantics for a nondeterministic extension of pure λ-calculus is studied. In this semantics, a term evaluates to a (finite or infinite) distribution of values. Small-step and big-step semantics, inductively and coinductively defined, are given. Moreover, small-step and big-step semantics are shown to produce identical outcomes, both in call-by-value and in call-by-name. Plotkin’s CPS translation is extended to accommodate the choice operator and shown correct with respect to the operational semantics. Finally, the expressive power of the obtained system is studied: the calculus is shown to be sound and complete with respect to computable probability distributions.

The paper presents a simple and concise proof of correctness of the magic transformation. We believe that it may provide a useful example of formal reasoning about logic programs. The correctness property concerns the declarative semantics. The proof, however, refers to the operational semantics (LD-resolution) of the source programs. Its conciseness is due to applying a suitable proof method.

We observe that the various formulations of the operational semantics of Constraint Handling Rules proposed over the years fall into a spectrum ranging from the analytical to the pragmatic. While existing analytical formulations facilitate program analysis and formal proofs of program properties, they cannot be implemented as is. We propose a novel operational semantics ω!, which has a strong analytical foundation, while featuring a terminating execution model. We prove its soundness and completeness with respect to existing analytical formulations and we provide an implementation in the form of a source-to-source transformation to CHR with rule priorities.

We investigate the usage of rule dependency graphs and their colorings for characterizing and computing answer sets of logic programs. This approach provides us with insights into the interplay between rules when inducing answer sets. We start with different characterizations of answer sets in terms of totally colored dependency graphs that differ in graph-theoretical aspects. We then develop a series of operational characterizations of answer sets in terms of operators on partial colorings. In analogy to the notion of a derivation in proof theory, our operational characterizations are expressed as (non-deterministically formed) sequences of colorings, turning an uncolored graph into a totally colored one. In this way, we obtain an operational framework in which different combinations of operators result in different formal properties. Among others, we identify the basic strategy employed by the noMoRe system and justify its algorithmic approach. Furthermore, we distinguish operations corresponding to Fitting's operator as well as to well-founded semantics.

The semantics of the Prolog ‘cut’ construct is explored in the context of some desirable properties of logic programming systems, referred to as the witness properties. The witness properties concern the operational consistency of responses to queries. A generalization of Prolog with negation as failure and cut is described, and shown not to have the witness properties. A restriction of the system is then described, which preserves the choice and first-solution behaviour of cut but allows the system to have the witness properties. The notion of cut in the restricted system is more restricted than the Prolog hard cut, but retains the useful first-solution behaviour of hard cut, not retained by other proposed cuts such as the ‘soft cut’. It is argued that the restricted system achieves a good compromise between the power and utility of the Prolog cut and the need for internal consistency in logic programming systems. The restricted system is given an abstract semantics, which depends on the witness properties; this semantics suggests that the restricted system has a deeper connection to logic than simply permitting some computations which are logical. Parts of this paper appeared previously in a different form in the Proceedings of the 1995 International Logic Programming Symposium (Andrews, 1995).