Dag Normann

• Normann, Dag (2024). Computability and non-monotone induction. Computability - The Journal of the Assosiation. ISSN 2211-3568. doi: 10.3233/COM-200332. Fulltekst i vitenarkiv
• Normann, Dag & Sanders, Sam (2023). The Biggest Five of Reverse Mathematics. Journal of Mathematical Logic. ISSN 0219-0613. doi: 10.1142/S0219061324500077. Fulltekst i vitenarkiv
• Normann, Dag & Sanders, Sam (2022). On the computational properties of basic mathematical notions. Journal of Logic and Computation. ISSN 0955-792X. 32(8), s. 1747–1795. doi: 10.1093/logcom/exac075.
• Normann, Dag & Sanders, Sam (2022). On Robust theorems due to Bolzano, Weierstrass, Jordan, and Cantor. Journal of Symbolic Logic (JSL). ISSN 0022-4812. doi: 10.1017/jsl.2022.71. Fulltekst i vitenarkiv
• Normann, Dag & Sanders, Sam (2022). On the uncountability of R. Journal of Symbolic Logic (JSL). ISSN 0022-4812. 87(4), s. 1–45. doi: 10.1017/jsl.2022.27.
• Normann, Dag & Sanders, Sam (2021). Betwixt Turing and Kleene. Lecture Notes in Computer Science (LNCS). ISSN 0302-9743. 13137, s. 236–252. doi: 10.1007/978-3-030-93100-1_15. Fulltekst i vitenarkiv
• Normann, Dag & Sanders, Sam (2021). The Axiom of Choice in computability theory and Reverse Mathematics with a cameo for the Continuum Hypothesis. Journal of Logic and Computation. ISSN 0955-792X. 31(1), s. 297–325. doi: 10.1093/logcom/exaa080.
• Normann, Dag & Sanders, Sam (2020). Open sets in computability theory and reverse mathematics. Journal of Logic and Computation. ISSN 0955-792X. 30(8), s. 1639–1679. doi: 10.1093/logcom/exaa049.
• Normann, Dag (2020). Measure-theoretic Uniformity and the Suslin Functional. Computability - The Journal of the Assosiation. ISSN 2211-3568. 10(2), s. 91–105. doi: 10.3233/COM-190248. Fulltekst i vitenarkiv
• Normann, Dag & Sanders, Sam (2020). Pincherle's theorem in reverse mathematics and computability theory. Annals of Pure and Applied Logic. ISSN 0168-0072. 171(5). doi: 10.1016/j.apal.2020.102788. Fulltekst i vitenarkiv
• Normann, Dag & Sanders, Sam (2019). Computability Theory, Nonstandard Analysis, and Their Connections. Journal of Symbolic Logic (JSL). ISSN 0022-4812. 84(4), s. 1422–1465. doi: 10.1017/jsl.2019.69. Fulltekst i vitenarkiv
• Normann, Dag & Sanders, Sam (2019). The strength of compactness in Computability Theory and Nonstandard Analysis. Annals of Pure and Applied Logic. ISSN 0168-0072. 170(11), s. 1–42. doi: 10.1016/j.apal.2019.05.007. Fulltekst i vitenarkiv
• Normann, Dag & Sanders, Sam (2018). On the mathematical and foundational significance of the uncountable. Journal of Mathematical Logic. ISSN 0219-0613. 18(2). doi: 10.1142/S0219061319500016. Fulltekst i vitenarkiv
• Normann, Dag (2018). Functionals of Type 3 as Realisers of Classical Theorems in Analysis. I Manea, Florin; Miller, Russel G. & Nowotka, Dirk (Red.), Sailing Routes in the World of Computation. Springer Nature. ISSN 978-3-319-94417-3. s. 318–327. doi: 10.1007/978-3-319-94418-0_32. Fulltekst i vitenarkiv
• Normann, Dag (2018). The sequential functionals of type $(\iota \rightarrow \iota)^n \rightarrow \iota$ form a dcpo for all $n \in {\mathbb N}$. Logical Methods in Computer Science. ISSN 1860-5974. 14(1). doi: 10.23638/LMCS-14(1:23)2018. Fulltekst i vitenarkiv
• Normann, Dag & Tait, William (2017). On the Computability of the Fan Functional. I Jäger, Gerhard & Sieg, Wilfried (Red.), Feferman on Foundations Logic, Mathematics, Philosophy. Springer Nature. ISSN 978-3-319-63332-9. s. 57–70. doi: 10.1007/978-3-319-63334-3_3.
• Normann, Dag (2016). On the Cantor-Bendixson rank of a set that is searchable in Gödel's T. Computability - The Journal of the Assosiation. ISSN 2211-3568. 5(1), s. 61–74. doi: 10.3233/COM-150043. Fulltekst i vitenarkiv
• Normann, Dag (2015). The extensional realizability model of continuous functionals and three weakly non-constructive classical theorems. Logical Methods in Computer Science. ISSN 1860-5974. 11(1). doi: 10.2168/LMCS-11(1:8)2015.
• Normann, Dag (2014). Higher Generalizations of the Turing Model. I Downey, Rod (Red.), Turing's Legacy: Developments from Turing's Ideas in Logic. Cambridge University Press. ISSN 978-1-107-04348-0. s. 397–433. doi: 10.1017/cbo9781107338579.012.
• Normann, Dag (2012). The Continuous Functionals as Limit Spaces. I Berger, Ulrich; Diener, Hannes; Schuster, Peter & Seisenberger, Monika (Red.), Logic, Construction, Computation. Ontos Verlag. ISSN 978-3-86838-158-0. s. 353–379. doi: 10.1515/9783110324921.353.
• Normann, Dag & Sazonov, Vladimir Yu. (2012). The extensional ordering of the sequential functionals. Annals of Pure and Applied Logic. ISSN 0168-0072. 163(5), s. 575–603. doi: 10.1016/j.apal.2011.06.013.
• Normann, Dag (2011). Banach spaces as data types. Logical Methods in Computer Science. ISSN 1860-5974. 7(2). doi: 10.2168/LMCS-7(2:11)2011.
• Normann, Dag (2011). Experiments on an Internal Approach to Typed Algorithms in Analysis. I Cooper, Barry & Sorbi, Andrea (Red.), Computability in Context. Computation and Logic in the Real World. Imperial College Press. ISSN 978-1-84816-245-7. s. 297–327. doi: 10.1142/9781848162778_0009.
• Normann, Dag (2009). A rich hierarchy of functionals of finite types. Logical Methods in Computer Science. ISSN 1860-5974. 5(3). doi: 10.2168/LMCS-5(3:11)2009.
• Normann, Dag (2008). Applications of the Kleene-Kreisel Density Theorem to Theoretical Computer Science. I Cooper, S. Barry; Løwe, Benedikt & Sorbi, Andrea (Red.), New Computational Paradigms; Changing Conceptions of What is Computable. Springer Publishing Company. ISSN 978-0-387-36033-1. s. 119–138.
• Normann, Dag (2008). Internal Density Theorems for Hierarchies of Continuous Functionals. I Beckmann, Arnold*; Dimitracopoulos, Costas & Löwe, Benedikt (Red.), Logic and Theory of Algorithms. Springer. ISSN 978-3-540-69405-2. s. 467–475.
• Normann, Dag (2006). Definability and Reducibility in Higher Types over the Reals. I Stoltenberg-Hansen, Viggo & Väänänen, Jouko (Red.), LOGIC COLLOQUIUM´03. Association for Symbolic Logic. ISSN 1-56881-293-0. s. 200–220.
• Normann, Dag (2006). Computing with functionals - Computability theory or computer science? Bulletin of Symbolic Logic. ISSN 1079-8986. 12(1), s. 43–59.
• Normann, Dag (2006). On sequential functionals of type 3. Mathematical Structures in Computer Science. ISSN 0960-1295. 16, s. 279–289.
• Normann, Dag (2005). Comparing hierarchies of total functionals. Logical Methods in Computer Science. ISSN 1860-5974. 1(2). doi: 10.2168/LMCS-1(2)2005. Vis sammendrag

  We show that if the Kleene-Kreisel continuous functionals of each type is zero-dimensional, then the intensional and extensional typed hierarchies of total continuous functionals over the reals coincide. As a tool of independent interest, we show that the Kleene-Kreisel functionals may be embedded into the extensional hierarchy.

• Normann, Dag (2004). Hierarchies of total functionals over the reals. Theoretical Computer Science. ISSN 0304-3975. 316, s. 137–151.
• Normann, Dag (2004). A Nonstandard Characterisation of the Type-structure of Continuous Functionals Over the Reals. Electronical Notes in Theoretical Computer Science. ISSN 1571-0661. 73, s. 141–147. Vis sammendrag

  We prove that one of the two standard hierarchies of total continuous functionals over the reals from domain theory can be characterised as equivalence classes of the near-standard objects in the hierarchy of hyperfinite functionals over a hyperfinite discretisation of the real line.

• Normann, Dag & Rørdam, Christian (2002). The Computational Power of M^omega. Mathematical logic quarterly. ISSN 0942-5616. 48(1), s. 117–124. Vis sammendrag

  By showing that the mu-comptability of a total object is equivalent if seen over the partial or hereditarily total continuous functionals, we show that Niggl's M^omega is strictly weaker than Plotkin's PCF + PA

• Normann, Dag (2002). Exact real number computations relative to hereditarily total functionals. Theoretical Computer Science. ISSN 0304-3975. 284, s. 437–453. Vis sammendrag

  We show that the continuous existential quantifier is not definable in Real PCF from all functionals equivalent to a given total one in a uniform way. We show that there is a maximal real valued computable functional of type 2.

• Normann, Dag & Geir, Waagbø (2002). Limit spaces and transfinite types. Archive for Mathematical Logic. ISSN 0933-5846. 41, s. 525–539. Vis sammendrag

  We give a characterisation of an extension of the Kleene-Kreisel continuous functionals to objects of transfinite type using limit spaces of transfinite types.

• Normann, Dag (2002). Om å velge den rette modell for beregnbarhet over de reelle tallene. Normat. ISSN 0801-3500. 49(3), s. 97–104. Vis sammendrag

  Vi ser på alternative matematiske modeller for å studere typet beregnbarhet over de reelle tallene.

• Normann, Dag (2002). Representation theorems for transfinite computability and definability. Archive for Mathematical Logic. ISSN 0933-5846. 41, s. 721–741. Vis sammendrag

  We show how Kreisel's representation theorem for sets in the analytical hierarchy can be generalized to sets defined by positive induction and use this to estimate the complexity of constructions in the theory of domains with totality.

• Normann, Dag (2002). Continuity, proof systems and the theory of transfinite computations. Archive for Mathematical Logic. ISSN 0933-5846. 41, s. 765–788. Vis sammendrag

  We use the theory of domains with totality to construct some logics generalizing omega-logic and beta-logic and we prove a completenes theorem for these logics. We give an aplication in Higher computability theory.

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• Longley, John & Normann, Dag (2015). Higher-Order Computability. Springer. ISBN 978-3-662-47991-9. 585 s.
• Normann, Dag; Dimitracopoulos, Costas; Newelski, Ludomir & Steel, John R. (2008). Logic Colloquium 2005. Cambridge University Press. ISBN 9780521884259. 288 s.

Se alle arbeider i Cristin

• Normann, Dag (2023). A higher order perspective on computational analysis.
• Normann, Dag (2022). A recent development in the computability theory for functionals of type 3.
• Normann, Dag (2022). Typed lambda calculus and operators of mainstream mathematics.
• Normann, Dag (2022). The complexity of operators constructed in mainstream mathematics.
• Normann, Dag & Sanders, Sam (2022). Betwixed Turing and Kleene.
• Normann, Dag (2021). An alternative perspective on Reverse Mathematics.
• Normann, Dag (2019). The Computational Complexity of the Heine Borel Theorem.
• Normann, Dag (2018). Functionals of Type 3 as Realisers of Classical Theorems in Analysis.
• Normann, Dag (2018). David Marker, Lectures on Infinitary Model Theory. Studia Logica: An International Journal for Symbolic Logic. ISSN 0039-3215. doi: 10.1007/s11225-018-9827-3. Fulltekst i vitenarkiv
• Normann, Dag (2018). A comparative study of the Heine-Borel theorem, the Lindelöf Lemma and non-monotone inductive definitions of sets of integers.
• Normann, Dag (2016). Revisiting Transfinite Types.
• Normann, Dag (2015). Alan Turing - mer enn en kodebryter. Vis sammendrag

  Alan Turing løste Hilberts "Entscheidungsproblem", og ga med det et grunnlag for forståelsen av mekaniske beregninger og kunstig intelligens.

• Normann, Dag (2015). Var Turing oppfinneren av datamaskinen?
• Normann, Dag (2015). Abels tårn. [Radio]. NRK. Vis sammendrag

  Paneldiskusjon om Alan Turing

• Buss, Sam; Lowe, Benedikt; Normann, Dag & Soskov, Ivan (2013). Computability in Europe 2011 Preface. Annals of Pure and Applied Logic. ISSN 0168-0072. 164(5), s. 509–510. doi: 10.1016/j.apal.2013.02.001.
• Normann, Dag (2012). Sequential vs. Continuous Functionals of Finite types.
• Normann, Dag (2012). Internal or external, abstract or concrete? What does this mean for computing in a Banach space?
• Normann, Dag (2011). Sequential versus Continuous Models of Finite Types.
• Normann, Dag (2011). Fixed point computability of total functionals.
• Normann, Dag (2011). Banach spaces as data types.
• Normann, Dag (2011). Om Urysohns universelle metriske rom - og litt om Urysohn selv.
• Normann, Dag (2009). The sequential functionals are far from being algebraic domains. Vis sammendrag

  It is shown that there will only be special cases where an increasing sequence of sequential functionals of a fixed type will have a sequential pointwise least upper bound.

• Normann, Dag (2009). Sequential Functionals at Type level 2.
• Normann, Dag (2009). Finite Sequential Procedures with Observational Ordering.
• Normann, Dag (2009). Functionals of finite types - bridging logic and computer science.
• Normann, Dag (2009). Continuous vs. Sequential Functionals - Harmony or Disharmony?
• Normann, Dag (2009). Computations and functionals of finite types.
• Normann, Dag (2008). 50 years of continuous functionals.
• Normann, Dag (2008). Om kontinuerlige operatorer på datasstrømmer og et åpent problem i topologi.
• Normann, Dag (2008). Trenger vi uendelige mengder i diskret matematikk?
• Normann, Dag (2007). External and internal density theorems for limspace-interpretations of some types.
• Normann, Dag (2007). Hvor vanskelig kan det være? Om bondesjakk og det å finne maksimumspunktet til en kontinuerlig funksjon.
• Normann, Dag (2007). Kan datamaskiner spille bondesjakk.
• Normann, Dag (2007). Kan datamaskiner spille bondesjakk.
• Cooper, Barry; Löwe, Benedikt & Normann, Dag (2006). Mathematics of computing at CiE 2005. Mathematical Structures in Computer Science. ISSN 0960-1295. 16(5), s. 735–736.
• Normann, Dag (2006). Domain theory based hierarchies of total functionals.
• Normann, Dag (2006). The coincidense problem and its topological counterpart.
• Normann, Dag (2006). On sequential functionals of type 3.
• Normann, Dag (2006). The Continuous Functionals and Aspects of Computability.
• Normann, Dag (2005). Regning ved hjelp av tallerkenstabler - en innfallsvinkel til programmeringsteori.
• Normann, Dag (2005). Minesveiper - hva har det med matematikk å gjøre.
• Normann, Dag (2005). Om mulige og tilsynelatende umulige programmeringsoppgaver. Normat. ISSN 0801-3500. 53(1), s. 21–27.
• Normann, Dag (2005). Sequential functionals of type 3.
• Normann, Dag (2004). Minesveiper - hva har det med matematikk å gjøre? Vis sammendrag

  Vi ser på sammenhengen mellom minesveiper-spillet og problemet 'P = NP' .

• Normann, Dag (2004). Turings verden - lag dine egne datamaskiner. Vis sammendrag

  Vi ga en kort innføring i hva Turingmaskiner er og i hvordan de virker.

• Normann, Dag (2004). Er du en kløpper i Minesveiper og kan bevise det, kan du bli en million dollar rikere.
• Normann, Dag (2004). Regning ved hjelp av tallerkenstabler - en innfallsvinkel til programmeringsteori. Vis sammendrag

  Presenterer et enkelt programmeringsspråk for regning med tall-registre og gir eksempler på hvordan enkelte funksjoner kan programmeres. Redegjør for videre egenskaper til dette enkle språket.

• Normann, Dag (2004). Minesveiper -hva har det med matematikk å gjøre.
• Normann, Dag (2004). Er du en kløpper i Minesveiper og kan bevise det, kan du bli en million dollar rikere. Vis sammendrag

  Vi ser på hvordan logiske kretser kan simuleres av situasjoner på et minesveiperbrett, og bruker det som en innføring i P = NP-problemet.

• Normann, Dag (2004). Minesveiper er NP-komplett. Vis sammendrag

  Logiske kretser kan simuleres av stillinger i minesveiperstillinger i polynomisk tid. Konsekvensen er at minesveiperproblemet er NP-komplett. Resultatet skyldes Richard Kaye.

• Normann, Dag (2004). Typed Computability over discrete and continuous base types.
• Normann, Dag (2003). Er du en kløpper i minesveiper, og kan bevise det, kan du bli en million dollar rikere. Vis sammendrag

  Vi diskuterer kompleksitetsklassen NP - og i hvilken forstand minesveiperspillet er NP-komplett.

• Normann, Dag (2003). Computability in higher type functionals: History, challenges and results, Vis sammendrag

  We present some of the development of higher type recursion theory over the last two decenniums, and a proof of the Cook-Berger Conjecture.

• Normann, Dag (2003). The continuous functionals in the perspective of domain theory. Vis sammendrag

  We discuss the influence from logic and theoretical computer science on a now unified theory of computability in functionals of higher types.

• Normann, Dag (2002). Om deterministiske og ikke-deterministiske beregninger. Vis sammendrag

  I foredraget gav jeg en introduksjon til P=NP-problemet, med definisjoner og eksempler, men uten bevis.

• Normann, Dag (2002). Hierarchies of total functionals over the reals. Vis sammendrag

  We use a standard representation of the reals as data-streams and domains to represent the higher type objects. We prove a density theorem for this hierarchy, and use it to compare this hierarchy with the one constructed from the standard domain representation of the reals.

• Normann, Dag (2001). Sammenhengen mellom klassiske paradokser og datamaskiners lunefullhet. Vis sammendrag

  Populærvitenskapelig foredrag som trekker linjene fra noen klassiske papadokser til Gödels og Turings behandling av bvisbarhet og beregnbarhet, med de konsekvenser dette har for matematikkens og informasjonsteknologiens grunnlag.

• Normann, Dag (2001). Definability of total objects in PCF and related calculi. Vis sammendrag

  Foredraget ga en oversikt over forfatterens bidrag til semantikk for typet lambda-kalkyle de siste fire årene. Dette omfattet bviset for Cook-Berger-formodningen, en analyse av de totale funksjonalene i typehierarkiet med reelle tall, naturlige tall og booleske verdier som grunntyper, et resultat om partielle funksjoner uniformt definerbare fra totale funksjoner og et fellesarbeid med Cand. Scient Christian Rørdam om sammenhengen mellom totalitet og typet my-rekursjon.

• Normann, Dag (2001). Definability and totality in typed lambda-calculi. Vis sammendrag

  We consider three typed lambda-calculi, PCF due to Scott/Plotkin, Real PCF due to Escardo and M-omega due to Niggl. For all three calculi we consider the semantics based on Scott-domains and adress problems related to the definability of, and relative to, hereditarily total objects.

• Normann, Dag (2000). Spillteori.

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