Theory of recursive functions and effective computability
Theory of recursive functions and effective computability
Introduction to higher order categorical logic
Introduction to higher order categorical logic
New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic
Information and Computation - Special issue: Selections from 1990 IEEE symposium on logic in computer science
Information and Computation - Special issue: Selections from 1990 IEEE symposium on logic in computer science
A type-theoretical alternative to ISWIM, CUCH, OWHY
Theoretical Computer Science - A collection of contributions in honour of Corrado Bo¨hm on the occasion of his 70th birthday
Computation and reasoning: a type theory for computer science
Computation and reasoning: a type theory for computer science
Synthetic Domain Theory in Type Theory: Another Logic of Computable Functions
TPHOLs '96 Proceedings of the 9th International Conference on Theorem Proving in Higher Order Logics
Categories and Effective Computations
Category Theory and Computer Science
General Synthetic Domain Theory - A Logical Approach
CTCS '97 Proceedings of the 7th International Conference on Category Theory and Computer Science
Type Theory via Exact Categories
LICS '98 Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science
A uniform approach to domain theory in realizability models
Mathematical Structures in Computer Science
Formalizing Synthetic Domain Theory
Journal of Automated Reasoning
Complete Axioms for Categorical Fixed-Point Operators
LICS '00 Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science
Realizability: a historical essay
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science
Variations on realizability: realizing the propositional axiom of choice
Mathematical Structures in Computer Science
Compactly generated domain theory
Mathematical Structures in Computer Science
A Convenient Category of Domains
Electronic Notes in Theoretical Computer Science (ENTCS)
Synthetic Domain Theory and Models of Linear Abadi & Plotkin Logic
Electronic Notes in Theoretical Computer Science (ENTCS)
Injective Convergence Spaces and Equilogical Spaces via Pretopological Spaces
Electronic Notes in Theoretical Computer Science (ENTCS)
Comparing free algebras in Topological and Classical Domain Theory
Theoretical Computer Science
Some reasons for generalising domain theory
Mathematical Structures in Computer Science
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Synthetic domain theory (SDT) is a version of Domain Theory where ‘all functions are continuous’. Following the original suggestion of Dana Scott, several approaches to SDT have been developed that are logical or categorical, axiomatic or model-oriented in character and that are either specialised towards Scott domains or aim at providing a general theory axiomatising the structure common to the various notions of domains studied so far.In Reus and Streicher (1993), Reus (1995) and Reus (1998), we have developed a logical and axiomatic version of SDT, which is special in the sense that it captures the essence of Domain Theory à la Scott but rules out, for example, Stable Domain Theory, as it requires order on function spaces to be pointwise. In this article we will give a logical and axiomatic account of a general SDT with the aim of grasping the structure common to all notions of domains.As in loc.cit., the underlying logic is a sufficiently expressive version of constructive type theory. We start with a few basic axioms giving rise to a core theory on top of which we study various notions of predomains (such as, for example, complete and well-complete S-spaces (Longley and Simpson 1997)), define the appropriate notion of domain and verify the usual induction principles of domain theory.Although each domain carries a logically definable ‘specialization order’, we avoid order-theoretic notions as much as possible in the formulation of axioms and theorems. The reason is that the order on function spaces cannot be required to be pointwise, as this would rule out the model of stable domains à la Berry.The consequent use of logical language – understood as the internal language of some categorical model of type theory – avoids the irritating coexistence of the internal and the external view pervading purely categorical approaches. Therefore, the paper is aimed at providing an elementary introduction to synthetic domain theory, albeit requiring some knowledge of basic type theory.