We review the origins of structural operational semantics. The main publication ‘A Structural Approach to Operational Semantics, ’ also known as the ‘Aarhus Notes, ’ appeared in 1981 [G.D. Plotkin, A structural approach to operational semantics, DAIMI FN-19, Computer Science Department, Aarhus University, 1981]. The development of the ideas dates back to the early 1970s, involving many people and building on previous work on programming languages and logic. The former included abstract syntax, the SECD machine, and the abstract interpreting machines of the Vienna school; the latter included the λ-calculus and formal systems. The initial development of structural operational semantics was for simple functional languages, more or less variations of the λ-calculus; after that the ideas were gradually extended to include languages with parallel features, such as Milner’s CCS. This experience set the ground for a more systematic exposition, the subject of an invited course of lectures at Aarhus University; some of these appeared in print as the 1981 Notes. We discuss the content of these lectures and some related considerations such as ‘small state’ versus ‘grand state, ’ structural versus compositional semantics, the influence of the Scott–Strachey approach to denotational semantics, the treatment of recursion and jumps, and static semantics. We next discuss relations with other work and some immediate further development. We conclude with an account of an old, previously unpublished, idea: an alternative, perhaps more readable, graphical presentation of systems of rules for operational semantics.

this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSL--TR--95--664, Stanford University

We investigate the power of Kahn-style dataflow networks, with processes that may exhibit indeterminate behavior. Our main result is a theorem about networks of "monotone" processes, which shows: (1) that the input/output relation of such a network is a total and monotone relation; and (2) every relation that is total, monotone, and continuous in a certain sense, is the input/output relation of such a network. Now, the class of monotone networks includes networks that compute arbitrary continuous input/output functions, an "angelic merge" network, and an "infinity-fair merge" network that exhibits countably indeterminate branching. Since the "fair merge" relation is neither monotone nor continuous, a corollary of our main result is the impossibility of implementing fair merge in terms of continuous functions, angelic merge, and infinity-fair merge. Our results are established by applying the powerful technique of "residuals" to the computations of a network. Residuals, which have previ...

Both pre-orders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' retraction pairs between them has a limit in the category of complete domains/spaces with retraction pairs as morphisms. The pre-order version was discovered first by Scott in 1969, and is referred to as Scott's inverse limit theorem. The metric version was mainly developed by de Bakker and Zucker and refined and generalized by America and Rutten. The theorem in both its versions provides the main tool for solving recursive domain equations. The proofs of the two versions of the theorem look astonishingly similar, but until now the preconditions for the pre-order and the metric versions have seemed to be fundamentally different. In this thesis we establish a more general theory of domains based on the noti...

. An extension of Dijkstra's guarded command language is studied, including unbounded demonic choice and a backtrack operator. We consider three orderings on this language: a refinement ordering defined by Back, a new deadlock ordering, and an approximation ordering of Nelson. The deadlock ordering is in between the two other orderings. All operators are monotonic in Nelson's ordering, but backtracking is not monotonic in Back's ordering and sequential composition is not monotonic for the deadlock ordering. At first sight recursion can only be added using Nelson's ordering. We show that, under certain circumstances, least fixed points for non-monotonic functions can be obtained by iteration from the least element. This permits the addition of recursion even using Back's ordering or the deadlock ordering in a fully compositional way. In order to give a semantic characterization of the three orderings that relates initial states to possible outcomes of the computation, the relation betwe...

We consider the following questions: What kind of logic has a natural semantics in multi-player (rather than 2-player) games? How can we express branching quantifiers, and other partial-information constructs, with a properly compositional syntax and semantics? We develop a logic in answer to these questions, with a formal semantics based on multiple concurrent strategies, formalized as closure operators on Kahn-Plotkin concrete domains. Partial information constraints are represented as co-closure operators. We address the syntactic issues by treating syntactic constituents, including quantifiers, as arrows in a category, with arities and co-arities. This enables a fully compositional account of a wide

Refinement calculi for imperative programs provide an integrated framework for programs and specifications and allow one to develop programs from specifications in a systematic fashion. The semantics of these calculi has traditionally been de ned in terms of predicate transformers and poses several challenges in defining a state transformer semantics in the denotational style. We de ne a novel semantics in terms of sets of state transformers and prove it to be isomorphic to positively multiplicative predicate transformers. This semantics disagrees with the traditional semantics in some places and the consequences of the disagreement are analyzed.

An extension of Dijkstra's guarded command language is studied, including sequential composition, demonic choice and a backtrack operator. We consider three orderings on this language: a refinement ordering defined by Back, a new deadlock ordering, and an approximation ordering of Nelson. The deadlock ordering is in between the two other orderings. All operators are monotonic in Nelson's ordering, but backtracking is not monotonic in Back's ordering and sequential composition is not monotonic for the deadlock ordering. At first sight recursion can only be added using Nelson's ordering. By extending the theory of fixed points in partial orderings we show that, under certain circumstances, least fixed points for non monotonic functions can be obtained by iteration from the least element. This permits us the addition of recursion even using Back's ordering or the deadlock ordering. In order to give a semantic characterization of the three orderings that relates initial states to possible outcomes of the computation, the relations between predicate transformers and discrete powerdomains is studied. Three powerdomains are considered: two versions of the Smyth powerdomain and the Egli-Milner powerdomain. For each of them an isomorphism is proved with a suitable domain of predicate transformers. 1991 Mathematics Subject Classification: 68Q55, 68Q10, 68Q60, 06A06. 1991 CR Categories: D.3.1, F.1.2, F.3.1, F.3.2. Keywords and Phrases: weakest preconditions, predicate transformers, refinement, deadlock, backtracking, fixed points, fixed point transformations, Smyth powerdomain, EgliMilner powerdomain, recursion, denotational semantics. Note: The research of Marcello Bonsangue was initially supported by a grant of the Universita' degli Studi di Milano, Italy, and later by a grant o...

We consider the following questions: What kind of logic has a natural semantics in multi-player (rather than 2-player) games? How can we express branching quantifiers, and other partial-information constructs, with a properly compositional syntax and semantics? We develop a logic in answer to these questions, with a formal semantics based on multiple concurrent strategies, formalized as closure operators on Kahn-Plotkin concrete domains. Partial information constraints are represented as co-closure operators. We address the syntactic issues by treating syntactic constituents, including quantifiers, as arrows in a category, with arities and co-arities. This enables a fully compositional account of a wide

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