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Complete Axioms for Categorical Fixedpoint Operators
 In Proceedings of 15th Annual Symposium on Logic in Computer Science
, 2000
"... We give an axiomatic treatment of fixedpoint operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the fre ..."
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Cited by 29 (6 self)
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We give an axiomatic treatment of fixedpoint operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the free iteration theory. We then show how iteration operators arise in axiomatic domain theory. One result derives them from the existence of sufficiently many bifree algebras (exploiting the universal property Freyd introduced in his notion of algebraic compactness) . Another result shows that, in the presence of a parameterized natural numbers object and an equational lifting monad, any uniform fixedpoint operator is necessarily an iteration operator. 1. Introduction Fixed points play a central role in domain theory. Traditionally, one works with a category such as Cppo, the category of !continuous functions between !complete pointed partial orders. This possesses a leastfixedpoint oper...
Comparing functional paradigms for exact realnumber computation
 in Proceedings ICALP 2002, Springer LNCS 2380
, 2002
"... Abstract. We compare the definability of total functionals over the reals in two functionalprogramming approaches to exact realnumber datatype of real numbers; and the intensional approach, in which one encodes real numbers using ordinary datatypes. We show that the type hierarchies coincide up to ..."
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Cited by 15 (3 self)
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Abstract. We compare the definability of total functionals over the reals in two functionalprogramming approaches to exact realnumber datatype of real numbers; and the intensional approach, in which one encodes real numbers using ordinary datatypes. We show that the type hierarchies coincide up to secondorder types, and we relate this fact to an analogous comparison of type hierarchies over the external and internal real numbers in Dana Scott’s category of equilogical spaces. We do not know whether similar coincidences hold at thirdorder types. However, we relate this question to a purely topological conjecture about the KleeneKreisel continuous functionals over the natural numbers. Finally, although it is known that, in the extensional approach, parallel primitives are necessary for programming total firstorder functions, we demonstrate that, in the intensional approach, such primitives are not needed for secondorder types and below. 1
A universal characterization of the closed euclidean interval (Extended Abstract)
 PROC. OF 16TH ANN. IEEE SYMP. ON LOGIC IN COMPUTER SCIENCE, LICS'01
, 2001
"... We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basi ..."
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Cited by 10 (0 self)
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We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest. In the
On the nonsequential nature of the intervaldomain model of realnumber computation
 Mathematical Structures in Computer Science
"... of realnumber computation ..."
A Universal Characterisation of the Closed Euclidean Interval
 in: Proceedings of 16th Annual IEEE Symposium on Logic in Computer Science
"... We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. We test the notion in categories of interest. In the category of sets, any closed and bounded interval of real numbers is an interval object. In the ca ..."
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Cited by 3 (3 self)
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We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. We test the notion in categories of interest. In the category of sets, any closed and bounded interval of real numbers is an interval object. In the category of topological spaces, the interval objects are closed and bounded intervals with the Euclidean topology. We also prove that an interval object exists in any elementary topos with natural numbers object. The universal property of an interval object provides a mechanism for defining functions on the interval. We use this to define basic arithmetic operations, and to verify equations between them. It also allows us to develop an analogue of the primitive recursive functions, yielding a natural class of computable functions on the interval. Contents 1
in a highertype setting
"... We show that, in a fairly general setting including highertypes, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a nondeterministic prog ..."
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We show that, in a fairly general setting including highertypes, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a nondeterministic program. The other two involve existential quantification and integration. We also perform first steps towards the semidecidability of similar tests under the simultaneous presence of nondeterministic and probabilistic choice. Keywords: Nondeterministic and probabilistic computation, highertype computability theory and exhaustible sets, may and must testing, operational and denotational semantics, powerdomains. 1