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OrderEnrichment for Categories of Partial Maps
, 1993
"... Introduction In (Plotkin 1985) a revitalised approach to domain theory was initiated. Roughly, the idea was to eliminate the bottom from the domains and to keep the functions partially defined. Thus replacing Cppo (the category of small cppos posets with a least element and closed under lubs of ..."
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Introduction In (Plotkin 1985) a revitalised approach to domain theory was initiated. Roughly, the idea was to eliminate the bottom from the domains and to keep the functions partially defined. Thus replacing Cppo (the category of small cppos posets with a least element and closed under lubs of !chains and continuous functions) with pCpo (the category of small cpos posets closed under lubs of !chains and partial continuous functions  see Subsection 3.1). One important point in the reformulation is the recognition of pCpo Research partially supported by Fundaci'on Antorchas and The British Council grant ARG 2281/14/6, and SERC grant RR30735. as a category of partial maps as, for example, such presentation fits better with standard formulations of recursion theory and it allows a categorical description of data types (via partial cartesian closed categories (Longo and Moggi 1984) with finite coproducts) in the presence of fixedpoint operators. Following the main moti
ReInterpreting the Modal µCalculus
 MODAL LOGIC AND PROCESS ALGEBRA
, 1995
"... We reexamine the modal µcalculus in the light of some classical theory of Boolean algebras and recent results on duality theory for a modal logic with fixed points. We propose interpreting formulas into a field of subsets of states instead of the full power set lattice used by Kozen. Under this in ..."
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We reexamine the modal µcalculus in the light of some classical theory of Boolean algebras and recent results on duality theory for a modal logic with fixed points. We propose interpreting formulas into a field of subsets of states instead of the full power set lattice used by Kozen. Under this interpretation we relate image compact modal frames with Scott continuity of the box modality, msaturated transition systems and descriptive modal frames. Also, it is shown that the class of image compact modal frames satisfies the HennessyMilner property. We conclude by showing that for descriptive modal µframes the standard interpretation coincides with the one we proposed.
Gray Code Representation of Exact Real Numbers
, 1999
"... We propose a new representation of exact real numbers using Gray code. This representation is injective and with a modied Type 2 machine induces the same computable structure on real numbers as admissible representations. This means that real numbers with the standard computable structure can be ..."
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We propose a new representation of exact real numbers using Gray code. This representation is injective and with a modied Type 2 machine induces the same computable structure on real numbers as admissible representations. This means that real numbers with the standard computable structure can be embedded into the space of innite words by considering a dierent sort of Type 2 machine. We also show that basic algorithms can be expressed naturally with respect to this representation. 1 Introduction One of the ways of dening computability of real functions is by representing a real number as an innite sequence and dening the computability of a function by the existence of a machine which inputs and outputs the representations oneway from left to right. This kind of machine with innite input and output is called a Type 2 machine. We can also dene the notion of a computable real number as a special case with no input. This notion of computability dates back to Turing[Turing 1...
Comment.Math.Univ.Carolin. 50,2 (2009) 297–314 297 Lattices of Scottclosed sets
"... Abstract. A dcpo P is continuous if and only if the lattice C(P) of all Scottclosed subsets of P is completely distributive. However, in the case where P is a noncontinuous dcpo, little is known about the order structure of C(P). In this paper, we study the ordertheoretic properties of C(P) for g ..."
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Abstract. A dcpo P is continuous if and only if the lattice C(P) of all Scottclosed subsets of P is completely distributive. However, in the case where P is a noncontinuous dcpo, little is known about the order structure of C(P). In this paper, we study the ordertheoretic properties of C(P) for general dcpo’s P. The main results are: (i) every C(P) is Ccontinuous; (ii) a complete lattice L is isomorphic to C(P) for a complete semilattice P if and only if L is weakstably Calgebraic; (iii) for any two complete semilattices P and Q, P and Q are isomorphic if and only if C(P) and C(Q) are isomorphic. In addition, we extend the function P 7 → C(P) to a left adjoint functor from the category DCPO of