Least fixpoints as meanings of recursive definitions.

We explore some foundational issues in the development of a theory of intensional semantics. A programming language may be given a variety of semantics, differing in the level of abstraction; one generally chooses the semantics at an abstraction level appropriate for reasoning about a particular kind of program property. Extensional semantics are typically appropriate for proving properties such as partial correctness, but an intensional semantics at a lower abstraction level is required in order to reason about computation strategy and thereby support reasoning about intensional aspects of behavior such as order of evaluation and efficiency. It is obviously desirable to be able to establish sensible relationships between two semantics for the same language, and we seek a general category-theoretic framework that permits this. Beginning with an "extensional" category, whose morphisms we can think of as functions of some kind, we model a notion of computation as a comonad with certain e...

Lower, upper, sandwich, mixed, and convex power domains are isomorphic to domains of second order predicates mapping predicates on the ground domain to logical values in a semiring. The various power domains differ in the nature of the underlying semiring logic and in logical constraints on the second order predicates.

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Abstract. We redevelop and extend Dams’s results on over- and underapproximation with higher-order Galois connections: (1) We show how Galois connections are generated from U-GLB-L-LUBclosed binary relations, and we apply them to lower and upper powerset constructions, which are weaker forms of powerdomains appropriate for abstraction studies. (2) We use the powerset types within a family of logical relations, show when the logical relations preserve U-GLB-L-LUB-closure, and show that simulation is a logical relation. We use the logical relations to rebuild Dams’s most-precise simulations, revealing the inner structure of overand under-approximation. (3) We extract validation and refutation logics from the logical relations, state their resemblance to Hennessey-Milner logic and description logic, and obtain easy proofs of soundness and best precision. Almost all Galois-connection-based static analyses are over-approximating: For

. Two lower bag domain constructions are introduced: the initial construction which gives free lower monoids, and the final construction which is defined explicitly in terms of second order functions. The latter is analyzed closely. For sober dcpo's, the elements of the final lower bag domains can be described concretely as bags. For continuous domains, initial and final lower bag domains coincide. They are continuous again and can be described via a basis which is constructed from a basis of the argument domain. The lower bag domain construction preserves algebraicity and the properties I and M, but does not preserve bounded completeness, property L, or bifiniteness. 1 Introduction Power domain constructions [13, 15, 16] were introduced to describe the denotational semantics of non-deterministic programming languages. A power domain construction is a domain constructor P , which maps domains to domains, together with some families of continuous operations. If X is the semantic domain ...

Abstract. We study the underapproximation of the predicate transformers used to give semantics to the modalities in dynamic and temporal logic. Because predicate transformers operate on state sets, we define appropriate powerdomains for sound approximation. We study four such domains — two are based on “set inclusion ” approximation, and two are based on “quantification ” approximation — and we apply the domains to synthesize the most precise, underapproximating �pre and pre transformers, in the latter case, introducing a focus operation. We also show why the expected abstractions of post and �post are unsound, and we use the powerdomains to guide us to correct, sound underapproximations. 1

The initial lower and upper power domain constructions P and P commute under composition for all cpos. The common result P (P X) and P (P X) is the free frame over the cpo X. 1 Introduction In [FM90], the lower and upper power domain constructions were shown to commute on bounded complete algebraic cpos. For the proof, these cpos were represented by information systems. In this paper, we show that the two constructions P and P commute on all cpos X, and moreover, that the common result P (P X) ¸ = P (P X) is the free frame over X. Our proof differs completely from the proof of [FM90]. It uses algebraic methods and does not rely on any explicit representations of the power domains. 2 The used algebraic structures In this section, we define the categories of algebraic structures that will be used in our proof. We suppose the usual definitions of poset P = (P; ), directed set, greatest lower bound (meet), and least upper bound (join). Definition 2.1 (1) A cpo is a poset...

An R-module M is observable iff all its elements can be distinguished by observing them by means of linear morphisms from M to R. We show that free observable R-modules can be explicitly described as the cores of the final power domains with characteristic semiring R. Then, the general theory is applied to the cases of the lower and the upper semiring. All lower modules are observable, whereas there are non-observable upper modules. Accordingly, all known lower power constructions coincide, whereas there are at least three different upper power constructions. We show that they coincide for continuous ground domains, but differ on more general domains. 1 Introduction A power domain construction maps every domain X into a so-called power domain over X whose points represent sets of points of the ground domain. Power domain constructions were originally proposed to model the semantics of non-deterministic programming languages [Plo76, Smy78, HP79, Mai85]. Other motivations are the sema...

. We study relations between predicate transformers and multifunctions in a topological setting based on closure operators. We give topological definitions of safety and liveness predicates and using these predicates we define predicate transformers. State transformers are multifunctions with values in the collection of fixed points of a closure operator. We derive several isomorphisms between predicate transformers and multifunctions. By choosing different closure operators we obtain multifunctions based on the usual power set construction, on the Hoare, Smyth and Plotkin power domains, and based on the compact and closed metric power constructions. Moreover, they are all related by isomorphisms to the predicate transformers. 1 Introduction There are (at least) two different ways of assigning a denotational semantics to a programming language: forward or backward. A typical forward semantics is a semantics that models a program as a function from initial states to final states. In th...