Results 1 
8 of
8
Operational domain theory and topology of a sequential language
 In Proceedings of the 20th Annual IEEE Symposium on Logic In Computer Science
, 2005
"... A number of authors have exported domaintheoretic techniques from denotational semantics to the operational study of contextual equivalence and order. We further develop this, and, moreover, we additionally export topological techniques. In particular, we work with an operational notion of compact ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
(Show Context)
A number of authors have exported domaintheoretic techniques from denotational semantics to the operational study of contextual equivalence and order. We further develop this, and, moreover, we additionally export topological techniques. In particular, we work with an operational notion of compact set and show that total programs with values on certain types are uniformly continuous on compact sets of total elements. We apply this and other conclusions to prove the correctness of nontrivial programs that manipulate infinite data. What is interesting is that the development applies to sequential programming languages, in addition to languages with parallel features. 1
Inductive Definition and Domain Theoretic Properties of Fully Abstract Models for PCF and PCF+
 LOGICAL METHODS IN COMPUTER SCIENCE 3(3:7), 1–50 (2007)
, 2007
"... A construction of fully abstract typed models for PCF and PCF+ (i.e., PCF+ “parallel conditional function”), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent nondeterministic strategies introduced by the author in the sevent ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
A construction of fully abstract typed models for PCF and PCF+ (i.e., PCF+ “parallel conditional function”), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent nondeterministic strategies introduced by the author in the seventies. Although these notions of strategies are old, the definition of the fully abstract models is new, in that it is given levelbylevel in the finite type hierarchy. To prove full abstraction and nondcpo domain theoretic properties of these models, a theory of computational strategies is developed. This is also an alternative and, in a sense, an analogue to the later game strategy semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong; and Nickau. In both cases of PCF and PCF+ there are definable universal (surjective) functionals from numerical functions to any given type, respectively, which also makes each of these models unique up to isomorphism. Although such models are nonomegacomplete and therefore not continuous in the traditional terminology, they are also proved to be sequentially complete (a weakened form of omegacompleteness), “naturally” continuous (with respect to existing directed “pointwise”, or “natural” lubs) and also “naturally” omegaalgebraic and “naturally” bounded complete—appropriate generalisation of the ordinary notions of domain theory to the case of nondcpos.
External Examiner
, 2006
"... The results reported in Part III consist of joint work with Martín Escardó [14]. All the other results reported in this thesis are due to the author, except for background results, which are clearly stated as such. Some of the results in Part IV have already appeared as [28]. Note This version of th ..."
Abstract
 Add to MetaCart
(Show Context)
The results reported in Part III consist of joint work with Martín Escardó [14]. All the other results reported in this thesis are due to the author, except for background results, which are clearly stated as such. Some of the results in Part IV have already appeared as [28]. Note This version of the thesis, produced on October 31, 2006, is the result of completing all the minor modifications as suggested by both the examiners in the viva report (Ref: CLM/AC/497773). We develop an operational domain theory to reason about programs in sequential functional languages. The central idea is to export domaintheoretic techniques of the Scott denotational semantics directly to the study of contextual preorder and equivalence. We investigate to what extent this can be done for two deterministic functional programming languages: PCF (Programminglanguage for Computable Functionals) and FPC (Fixed Point Calculus).
On Natural Nondcpo Domains
"... in his 87th year whose influence on me and help cannot be overstated. model for PCF of hereditarily sequential functionals is not ωcomplete and therefore not continuous in the traditional terminology (in contrast to the old fully abstract continuous dcpo model of Milner). This is also applicable to ..."
Abstract
 Add to MetaCart
(Show Context)
in his 87th year whose influence on me and help cannot be overstated. model for PCF of hereditarily sequential functionals is not ωcomplete and therefore not continuous in the traditional terminology (in contrast to the old fully abstract continuous dcpo model of Milner). This is also applicable to a wider class of models such as the recently constructed by the author fully abstract (universal) model for PCF + = PCF + parallel if. Here we will present an outline of a general approach to this kind of “natural ” domains which, although being nondcpos, allow considering “naturally ” continuous functions (with respect to existing directed “pointwise”, or “natural ” least upper bounds) and also have appropriate version of “naturally ” algebraic and “naturally ” bounded complete “natural ” domains. This is the nondcpo analogue of the wellknown concept of Scott domains, or equivalently, the complete fspaces of Ershov. In fact, the latter version of natural domains, if considered under “natural ” Scott topology, exactly corresponds to the class of fspaces, not necessarily complete. 1
Natural nondcpo Domains and fSpaces Abstract
"... hereditarilysequential functionals is not ωcomplete (in contrast to the old fully abstract continuous dcpo model of Milner). This is also applicable to a potentially (universal) model for PCF + = PCF + pif (parallel if). Here we will present an outline of a general approach to this kind of ‘natura ..."
Abstract
 Add to MetaCart
(Show Context)
hereditarilysequential functionals is not ωcomplete (in contrast to the old fully abstract continuous dcpo model of Milner). This is also applicable to a potentially (universal) model for PCF + = PCF + pif (parallel if). Here we will present an outline of a general approach to this kind of ‘natural ’ domains which, although being nondcpos, allow considering ‘naturally ’ continuous functions (with respect to existing directed ‘pointwise’, or ‘natural ’ least upper bounds). There is also an appropriate version of ‘naturally ’ algebraic and ‘naturally ’ bounded complete ‘natural’ domains which serves as the nondcpo analogue of the wellknown concept of Scott domains, or equivalently, the complete fspaces of Ershov. It is shown that this special version of ‘natural ’ domains, if considered under ‘natural ’ Scott topology, exactly corresponds to the class of fspaces, not necessarily complete. Key words: domain theory, dcpo and nondcpo domains, Scott topology,
Declaration
, 2006
"... The results reported in Part III consist of joint work with Mart́ın Escardo ́ [14]. All the other results reported in this thesis are due to the author, except for background results, which are clearly stated as such. Some of the results in Part IV have already appeared as [28]. Note This version of ..."
Abstract
 Add to MetaCart
(Show Context)
The results reported in Part III consist of joint work with Mart́ın Escardo ́ [14]. All the other results reported in this thesis are due to the author, except for background results, which are clearly stated as such. Some of the results in Part IV have already appeared as [28]. Note This version of the thesis, produced on October 31, 2006, is the result of completing all the minor modifications as suggested by both the examiners in the viva report (Ref: CLM/AC/497773). i We develop an operational domain theory to reason about programs in sequential functional languages. The central idea is to export domaintheoretic techniques of the Scott denotational semantics directly to the study of contextual preorder and equivalence. We investigate to what extent this can be done for two deterministic functional programming languages: PCF (Programminglanguage for Computable Functionals) and FPC (Fixed Point
FILTER CONVERGENCE STRUCTURES ON POSETS
"... Abstract. In this paper, we introduce two convergence structures on each poset and thus embed the category of posets and Scott continuous maps into the category of convergence spaces which is cartesianclosed. More specifically, for each poset P, we define two convergence spaces (P, ↓d) and (P, ↓c). ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. In this paper, we introduce two convergence structures on each poset and thus embed the category of posets and Scott continuous maps into the category of convergence spaces which is cartesianclosed. More specifically, for each poset P, we define two convergence spaces (P, ↓d) and (P, ↓c). The convergence space (P, ↓c) was first constructed by Heckmann for directed complete posets. The main results of our investigation are: (i) (P, ↓d) induces the Scott topology on P; (ii) if P is a continuous poset, then ↓c=↓d but in general they are different; (iii) (P, ↓d) is topological iff P is a continuous poset; (iv) (P, ↓d) is topological iff it is pretopological; (v) if Y is a topological space then the function space [Pd → Y] is topological, where Pd = (P, ↓d); (vi) (P ×Q)d is homeomorphic to Pd × Qd for any posets P and Q; (vii) the meetcontinuity and strongly meetcontinuity are equivalent for directedcomplete meet semilattices; (viii) for a meetsemilattice P, (P, ↓d) is a convergence meetsemilattice iff (P,≤) is meetcontinuous. 1