Effective Operators and Continuity Revisited

In programming language semantics different kinds of semantical domains are used, among them Scott domains and metric spaces. D. Scott raised the problem of finding a suitable class of spaces which should include Scott domains and metric spaces such that effective mappings between these spaces are continuous. It is well known that between spaces like effectively given Scott domains or constructive metric spaces such operators are effectively continuous and vice versa. But, as an example of Friedberg shows, effective mappings from metric spaces into Scott domains are not continuous in general.

[1]  Pierre America,et al.  Solving Reflexive Domain Equations in a Category of Complete Metric Spaces , 1987, J. Comput. Syst. Sci..

[2]  Claude E. Shannon,et al.  Computers and Automata , 1953, Proceedings of the IRE.

[3]  Alistair H. Lachlan,et al.  Effective operations in a general setting , 1964, Journal of Symbolic Logic.

[4]  Y. Ershov.  Model of Partial Continuous Functionals , 1977 .

[5]  Dana S. Scott,et al.  Some Domain Theory and Denotational Semantics in Coq , 2009, TPHOLs.

[6]  A. W. Roscoe,et al.  Metric Spaces as Models for Real-Time Concurrency , 1987, MFPS.

[7]  P. Young,et al.  Effective operators in a topological setting , 1984 .

[8]  Christopher Strachey,et al.  Toward a mathematical semantics for computer languages , 1971 .

[9]  Peter Lucas,et al.  Formal Semantics of Programming Languages: VDL , 1981, IBM J. Res. Dev..

[10]  Dana S. Scott,et al.  Data Types as Lattices , 1976, SIAM J. Comput..

[11]  D. Scott.  Models for Various Type-Free Calculi , 1973 .

[12]  L. S. Pontryagin,et al.  General Topology I , 1990 .

[13]  John C. Shepherdson,et al.  Effective operations on partial recursive functions , 1955 .

[14]  Michael B. Smyth,et al.  Quasi Uniformities: Reconciling Domains with Metric Spaces , 1987, MFPS.

[15]  Marian Boykan Pour-El.  A Comparison of Five “Computable” Operators† , 1960 .

[16]  A. W. Roscoe,et al.  Topology and category theory in computer science , 1991 .

[17]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[18]  Michael B. Smyth,et al.  Effectively given Domains , 1977, Theor. Comput. Sci..

[19]  Patrick Suppes,et al.  Logic, Methodology and Philosophy of Science , 1963 .

[20]  Robert L. Constable,et al.  Computability concepts for programming language semantics , 1975, Theor. Comput. Sci..

[21]  Y. Moschovakis.  Recursive metric spaces , 1964 .

[22]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[23]  William J. Collins,et al.  Discontinuities of Provably Correct Operators on the Provably Recursive Real Numbers , 1983, J. Symb. Log..

[24]  J. Helm.  On Effectively Computable Operators , 1971 .

[25]  J. W. de Bakker,et al.  Processes and the Denotational Semantics of Concurrency , 1982, Inf. Control..

[26]  Paul R. Young,et al.  AN EFFECTIVE OPERATOR, CONTINUOUS BUT NOT PARTIAL RECURSIVE , 1968 .