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On the Foundations of Final Semantics: NonStandard Sets, Metric Spaces, Partial Orders
 PROCEEDINGS OF THE REX WORKSHOP ON SEMANTICS: FOUNDATIONS AND APPLICATIONS, VOLUME 666 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are ..."
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Cited by 48 (10 self)
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Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are also used here for giving a new comprehensive presentation of the (still) nonstandard theory of nonwellfounded sets (as nonstandard sets are usually called). This paper is meant to provide a basis to a more general project aiming at a full exploitation of the finality of the domains in the semantics of programming languages  concurrent ones among them. Such a final semantics enjoys uniformity and generality. For instance, semantic observational equivalences like bisimulation can be derived as instances of a single `coalgebraic' definition (introduced elsewhere), which is parametric of the functor appearing in the domain equation. Some properties of this general form of equivalence are also studied in this paper.
Categories of Containers
 In Proceedings of Foundations of Software Science and Computation Structures
, 2003
"... Abstract. We introduce the notion of containers as a mathematical formalisation of the idea that many important datatypes consist of templates where data is stored. We show that containers have good closure properties under a variety of constructions including the formation of initial algebras and f ..."
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Cited by 40 (7 self)
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Abstract. We introduce the notion of containers as a mathematical formalisation of the idea that many important datatypes consist of templates where data is stored. We show that containers have good closure properties under a variety of constructions including the formation of initial algebras and final coalgebras. We also show that containers include strictly positive types and shapely types but that there are containers which do not correspond to either of these. Further, we derive a representation result classifying the nature of polymorphic functions between containers. We finish this paper with an application to the theory of shapely types and refer to a forthcoming paper which applies this theory to differentiable types. 1
A power domain for countable nondeterminism
 in : Lecture
"... This paper proposes a general powerdomain for countable nDndeterminism and uses it to give the denotational semantics of a simple imperative programming language with a fair parallel construct. As already known from the simple case of a discrete cpo [AP] countable nondeterminism seems to force the c ..."
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Cited by 19 (1 self)
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This paper proposes a general powerdomain for countable nDndeterminism and uses it to give the denotational semantics of a simple imperative programming language with a fair parallel construct. As already known from the simple case of a discrete cpo [AP] countable nondeterminism seems to force the consideration of noncontinuous
Generic trace semantics via coinduction
 Logical Methods in Comp. Sci
, 2007
"... Abstract. Trace semantics has been defined for various kinds of statebased systems, notably with different forms of branching such as nondeterminism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these “trace ..."
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Cited by 17 (6 self)
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Abstract. Trace semantics has been defined for various kinds of statebased systems, notably with different forms of branching such as nondeterminism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these “trace
Generic trace theory
 International Workshop on Coalgebraic Methods in Computer Science (CMCS 2006), volume 164 of Elect. Notes in Theor. Comp. Sci
, 2006
"... Trace semantics has been defined for various nondeterministic systems with different input/output types, or with different types of “nondeterminism ” such as classical nondeterminism (with a set of possible choices) vs. probabilistic nondeterminism. In this paper we claim that these various forms ..."
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Cited by 8 (4 self)
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Trace semantics has been defined for various nondeterministic systems with different input/output types, or with different types of “nondeterminism ” such as classical nondeterminism (with a set of possible choices) vs. probabilistic nondeterminism. In this paper we claim that these various forms of “trace semantics” are instances of a single categorical construction, namely coinduction in a Kleisli category. This claim is based on our main technical result that an initial algebra in
Representing Nested Inductive Types Using Wtypes
"... We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive ..."
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Cited by 6 (3 self)
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We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive
Constructive Lattice Theory
, 1993
"... A notion of simulation of one datatype by another is defined as a constructive preorder. A calculus of datatype simulation is then developed by formulating constructive versions of leastfixedpoint theorems in lattice theory. The calculus is applied to the construction of several isomorphisms betwe ..."
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Cited by 3 (2 self)
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A notion of simulation of one datatype by another is defined as a constructive preorder. A calculus of datatype simulation is then developed by formulating constructive versions of leastfixedpoint theorems in lattice theory. The calculus is applied to the construction of several isomorphisms between classes of datatypes. In particular constructive adaptations of theorems in lattice theory about closure operators are shown to yield simulations and isomorphisms between monad structures, and constructive adaptations of theorems in regular algebra are shown to yield isomorphisms between list structures. A question to which any respectable theory of datatypes should provide immediate answers is when two datatypes are isomorphic, i.e. entirely equivalent modulo implementation details. A subsidiary question is when one datatype simulates another. This second question is of interest in its own right but is also important to answering the first question since isomorphism is frequently reduce...
Understanding Inductive Types in Constructions
, 1993
"... In this paper we extend the Calculus of Constructions with generalized inductive types. The extension is justified by showing that the usual set theoretical model can be effectivized. It is also pointed out that the model given in a published paper for a collection of inductive types in a different ..."
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Cited by 2 (1 self)
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In this paper we extend the Calculus of Constructions with generalized inductive types. The extension is justified by showing that the usual set theoretical model can be effectivized. It is also pointed out that the model given in a published paper for a collection of inductive types in a different style is wrong. Copyright c fl1993. All rights reserved. Reproduction of all or part of this work is permitted for educational or research purposes on condition that (1) this copyright notice is included, (2) proper attribution to the author or authors is made and (3) no commercial gain is involved. Technical Reports issued by the Department of Computer Science, Manchester University, are available by anonymous ftp from m1.cs.man.ac.uk (130.88.13.4) in the directory /pub/TR. The files are stored as PostScript, in compressed form, with the report number as filename. Alternatively, reports are available by post from The Computer Library, Department of Computer Science, The University, Oxford R...
semantics as developed by Scott and Strachey and their followers (Gordon [13],
, 1982
"... Abstract. Recursive specifications of domains plays a crucial role in denotational semantics as developed by Scott and Strachey and their followers. The purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place. T ..."
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Abstract. Recursive specifications of domains plays a crucial role in denotational semantics as developed by Scott and Strachey and their followers. The purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place. The idea is to follow the wellknown analogy between partial orders and categories, generalizing from least fixedpoints of continuous functions over cpos to initial ones of continuous functors over ocategories. To apply these general ideas we introduce Wand’s Ocategories where the morphismsets have a partial order structure and which include almost all the categories occurring in semantics. The idea is to find solutions in a derived category of embeddings and we give ordertheoretic conditions which are easy to verify and which imply the needed categorical ones. The main tool is a very general form of the limitcolimit coincidence remarked by Scott. In the concluding section we outline how compatibility considerations are to be included in the framework. A future paper will show how Scott’s universal domain method can be included too.