Skip to main content

Nonflatness and totality


We interpret finite types as domains over nonflat inductive base types in order to bring out the finitary core that seems to be inherent in the concept of totality. We prove a strong version of the Kreisel density theorem by providing a total compact element as a witness, a result that we cannot hope to have if we work with flat base types. To this end, we develop tools that deal adequately with possibly inconsistent finite sets of information. The classical density theorem is reestablished via a ‘finite density theorem,’ and corollaries are obtained, among them Berger's separation property.

Hide All
Bauer, A. and Birkedal, L. (2000). Continuous functionals of dependent types and equilogical spaces. In: Proceedings of the 14th EASCL Annual Conference on Computer Science Logic (CSL '2000), Fischbachau, Germany, August 21–26, Springer, 202–216.
Berger, U. (1990). Totale Objekte und Mengen in der Bereichstheorie (Total objects and sets in domain theory). Univ. München, Fak. für Mathematik. ii, 122 S.
Berger, U. (1993). Total sets and objects in domain theory. Annals of Pure and Applied Logic 60 (2) 91117.
Berger, U. (1999a). Continuous functionals of dependent and transfinite types. In: Models and Computability. Invited Papers from the Logic Colloquium '97, European Meeting of the Association for Symbolic Logic, Leeds, UK, July 6–13, Cambridge University Press, 122.
Berger, U. (1999b). Effectivity and density in domains: A survey. In: A Tutorial Workshop on Realizability Semantics and Applications. A Workshop Associated to the Federated Logic Conference, Trento, Italy, June 30–July 1, Elsevier, 13.
Berger, U. (2002). Computability and totality in domains. Mathematical Structures in Computer Science 12 (3) 281294.
Berger, U. (2009). Realisability and adequacy for (co)induction. In: Proceedings of the 6th International Conference on Computability and Complexity in Analysis (CCA'09), August 18–22, Ljubljana, Slovenia, Schloss Dagstuhl – Leibniz Zentrum für Informatik, 12.
Berger, U. (2011). From coinductive proofs to exact real arithmetic: Theory and applications. Logical Methods in Computer Science 7 (1) 24.
Berger, U., Miyamoto, K., Schwichtenberg, H. and Seisenberger, M. (2011). Minlog–-A tool for program extraction supporting algebras and coalgebras. In: Algebra and Coalgebra in Computer Science – Proceedings of the 4th International Conference (CALCO '11), Winchester, UK, August 30–September 2, 393–399.
Berger, U., Miyamoto, K., Schwichtenberg, H. and Tsuiki, H. (2016). Logic for gray-code computation. In Dieter, P. and Peter, S. (eds.): Concepts of Proof in Mathematics, Philosophy, and Computer Science. De Gruyter.
Berger, U. and Seisenberger, M. (2012). Proofs, programs, processes. Theory of Computing Systems 51 (3) 313329.
Ershov, Y.L. (1975a). Maximal and everywhere-defined functionals. Algebra Logic 13 210225.
Ershov, Y.L. (1975b). Theorie der Numerierungen. II (Theory of numberings. II) Übersetzung aus dem Russischen: H.-D. Hecker. Wissenschaftliche Redaktion: G. Asser. Berlin: VEB Deutscher Verlag der Wissenschaften. M 15.00 (1976). Sonderdruck aus Z. math. Logik Grundl. Math. 21 473–584 (1975).
Ershov, Y.L. (1977). Model ℂ of partial continuous functionals. In: Logic Colloquium 76, Proceedings of a Conference held at Oxford in 1976, Studies in Logic and the Foundations of Mathematics, vol. 87, 455–467.
Escardó, M.H. (2007). Infinite sets that admit fast exhaustive search. In: Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science (LICS '07), Washington, DC, USA, IEEE Computer Society, 443–452.
Escardó, M.H. (2008). Exhaustible sets in higher-type computation. Logical Methods in Computer Science 4 (3) 37.
Ghani, N., Hancock, P.G. and Pattinson, D. (2009). Continuous functions on final coalgebras. In: Proceedings of the 25th Conference on the Mathematical Foundations of Programming Semantics (MFPS '09), Oxford, UK, April 3–7, Elsevier, 3–18.
Hancock, P.G., Ghani, N. and Pattinson, D. (2009). Representations of stream processors using nested fixed points. Logical Methods in Computer Science 5 (3) 17.
Huber, S. (2010). On the computational content of choice axioms. Master's thesis, Mathematisches Institut, LMU.
Huber, S., Karádais, B.A. and Schwichtenberg, H. (2010). Towards a formal theory of computability. In: Ways of Proof Theory (Pohler's Festschrift), Ontos Verlag, 257282.
Karádais, B.A. (2013). Towards an Arithmetic with Approximations. Ph.D. thesis, Mathematisches Institut, LMU.
Karádais, B.A. (2016). Atomicity, coherence of information, and point-free structures. Annals of Pure and Applied Logic 167 (9) 753769.
Karádais, B.A. (2018). Normal forms, linearity, and prime algebraicity over nonflat domains. Mathematical Logic Quarterly, to appear.
Kleene, S.C. (1959). Countable functionals. In: Heyting, A. (ed.) Constructivity in Mathematics: Proceedings of the Colloquium held at Amsterdam in 1957, Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co, 81–100.
Kreisel, G. (1959). Interpretation of analysis by means of constructive functionals of finite types. In: Heyting, A. (eds.) Constructivity in Mathematics: Proceedings of the Colloquium held at Amsterdam, 1957, Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co, 101–128.
Longley, J. and Normann, D. (2015). Higher-Order Computability, Springer.
Miyamoto, K., Forsberg, F.N. and Schwichtenberg, H. (2013). Program extraction from nested definitions. In: Interactive Theorem Proving – Proceedings of the 4th International Conference (ITP '13), Rennes, France, July 22–26, 370–385.
Miyamoto, K. and Schwichtenberg, H. (2015). Program extraction in exact real arithmetic. Mathematical Structures in Computer Science 25 (8) 16921704.
Normann, D. (1981). Countable functionals and the projective hierarchy. Journal of Symbolic Logic 46 209215.
Normann, D. (1989). Kleene-spaces. In: Logic Colloqium '88, Proceedings of the Colloqium held at Padova/Italy in 1988, Studies in Logic and the Foundations of Mathematics, vol. 127 91–109.
Normann, D. (1996). A hierarchy of domains with totality, but without density. In: Computability, Enumerability, Unsolvability. Directions in Recursion Theory, Cambridge University Press, 233257.
Normann, D. (1997). Closing the gap between the continuous functionals and recursion in 3E. Archive for Mathematical Logic 36 (4–5) 269287.
Normann, D. (1999). The continuous functionals. In: Handbook of Computability Theory, Elsevier, 251275.
Normann, D. (2000a). Computability over the partial continuous functionals. Journal of Symbolic Logic 65 (3) 11331142.
Normann, D. (2000b). The Cook–Berger problem. A guide to the solution. In: Proceedings of the Domains IVth Workshop, Haus Humboldtstein, Remagen-Rolandseck, Germany, October 2–4, Elsevier, 9.
Normann, D. (2009). Applications of the Kleene-Kreisel density theorem to theoretical computer science. In: New Computational Paradigms. Changing Conceptions of What is Computable, Springer, 119138.
Plotkin, G.D. (1978). Tω as a universal domain. Journal of Computer and System Sciences 17 (2) 209236.
Plotkin, G.D. (1999). Full abstraction, totality and PCF. Mathematical Structures in Computer Science 9 (1) 120.
Rinaldi, D. (2014). Formal methods in the theories of rings and domains. Ph.D. thesis, Mathematisches Institut, LMU.
Rutten, J.J.M.M. (2000). Universal coalgebra: A theory of systems. Theoretical Computer Science 249 (1) 380.
Sambin, G. (1987). Intuitionistic formal spaces-–A first communication. In Skordev, D. G. (ed.): Mathematical Logic and its Applications (Druzhba, 1986), Plenum, 187204.
Schwichtenberg, H. (1996). Density and choice for total continuous functionals. In Odifreddi, P. (ed.): Kreiseliana: About and Around Georg Kreisel, A K Peters, 335362.
Schwichtenberg, H. (2007). Recursion on the partial continuous functionals. In: Dimitracopoulos, C., Newelski, L., Normann, D. and Steel, J. (eds.), Logic Colloquium ('05), Lecture Notes in Logic, vol. 28, Association for Symbolic Logic, 173201.
Schwichtenberg, H. and Wainer, S.S. (2012). Proofs and Computations, Perspectives in Logic, Cambridge University Press.
Scott, D.S. (1982). Domains for denotational semantics. In: Automata, Languages and Programming (Aarhus, 1982), Lecture Notes in Computer Science, vol. 140, Springer, 577613.
Stoltenberg-Hansen, V., Lindström, I. and Griffor, E.R. (1994). Mathematical Theory of Domains, Cambridge Tracts in Theoretical Computer Science, vol. 22, Cambridge University Press.
Winskel, G. and Larsen, K.G. (1984). Using information systems to solve recursive domain equations effectively. In Kahn, G., MacQueen, D. B., Plotkin, G. D. (eds.): Semantics of Data Types, Springer-Verlag, 109129.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed