Dag Normann

• Normann, Dag & Sanders, Sam (2026). The uncountability of the reals and the axiom of choice. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik. ISSN 9122-845X. 72, s. 425–436. doi: 10.60866/CAM.323. Fulltekst i vitenarkiv Vis sammendrag

  Abstract The uncountability of the reals was first established by Cantor in what was later heralded as the first paper on set theory. Since the latter constitutes the de facto foundations of mathematics, the logical study of the uncountability of the reals is a worthy endeavour for historical, foundational, and conceptual reasons. In this paper, we shall study the following principle: NIN[0,1]: there is no injection from the unit interval to the natural numbers. We show that relatively strong logical systems cannot prove NIN[0,1]. In particular, the former system implies second-order arithmetic and fragments of the axiom of choice, including dependent choice. We also study the latter choice fragments in Kohlenbach's higher-order reverse mathematics.

• Normann, Dag & Sanders, Sam (2025). On sequential theorems in Reverse Mathematics. Archive for Mathematical Logic. ISSN 0933-5846. 65, s. 275–295. doi: 10.1007/s00153-025-00991-4. Fulltekst i vitenarkiv Vis sammendrag

  Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence of solutions. The original and sequential theorem can often be proved via the same (or similar) proof and often have the same (or similar) logical properties, esp. if everything is formulated in the language of second-order arithmetic. In this paper, we identify basic theorems of third-order arithmetic, e.g. concerning semi-continuous functions, such that the sequential versions have very different logical properties. In particular, depending on the constructive status of the original theorem, very different and independent choice principles are needed. Despite these differences, the associated Reverse Mathematics, working in Kohlenbach’s higher-order framework, is rather elegant and is still based at the core on weak König’s lemma.

• Normann, Dag (2024). Computability and non-monotone induction. Computability. ISSN 2211-3568. 13(2), s. 161–196. doi: 10.3233/COM-200321. Fulltekst i vitenarkiv Vis sammendrag

  Non-monotone inductive definitions were studied in the late 1960’s and early 1970’s with the aim of understanding connections between the complexity of the formulas defining the induction steps and the ordinals measuring the rank of the inductive process. In general, any type 2 functional will generate an inductive process, and in this paper we will view non-monotone induction as a functional of type 3. We investigate the associated computation theory of this functional inherited from the Kleene schemes and we investigate the associated companion of sets with codes computable in the functional of non-monotone induction. The interest in this functional is motivated from observing that constructions via non-monotone induction appear to be natural in classical analysis. There are two groups of results: We establish strong closure properties of the least ordinal without a code computable in the functional of non-monotone induction, and we provide a characterisation of the class of functionals of type 3 computable from this functional, a characterisation in terms of sequential operators working in transfinite time. We will also see that the full computational power of non-monotone induction is required when this principle is used to construct functionals witnessing the compactness of the Cantor space and of closed, bounded intervals.

• Normann, Dag & Sanders, Sam (2024). On some computational properties of open sets. Journal of Logic and Computation. ISSN 0955-792X. 35(4). doi: 10.1093/logcom/exae048. Fulltekst i vitenarkiv Vis sammendrag

  Open sets are central to mathematics, especially analysis and topology, in ways few notions are. In most, if not all, computational approaches to mathematics, open sets are only studied indirectly via their ‘codes’ or ‘representations’. In this paper, we study how hard it is to compute, given an arbitrary open set of reals, the most common representation, i.e. a countable set of open intervals. We work in Kleene’s higher order computability theory, in particular its equivalent lambda calculus formulation due to Platek. We establish many computational equivalences between on one hand the ‘structure’ functional that converts open sets to the aforementioned representation, and on the other hand functionals arising from mainstream mathematics, like basic properties of semi-continuous functions, the Urysohn lemma and the Tietze extension theorem. We also compare these functionals with known operations on regulated and bounded variation functions, and the Lebesgue measure restricted to closed sets. We obtain a number of natural computational equivalences for the latter involving theorems from mainstream mathematics.

• Normann, Dag & Sanders, Sam (2024). On the logical and computational properties of the Vitali covering theorem. Annals of Pure and Applied Logic. ISSN 0168-0072. 176(1). doi: 10.1016/j.apal.2024.103505. 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 Vis sammendrag

  The aim of Reverse Mathematics (RM for short) is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of only four systems; these five systems together are called the ‘Big Five’. The aim of this paper is to greatly extend the Big Five phenomenon as follows: there are two supposedly fundamentally different approaches to RM where the main difference is whether the language is restricted to second-order objects or if one allows third-order objects. In this paper, we unite these two strands of RM by establishing numerous equivalences involving the second-order Big Five systems on one hand, and well-known third-order theorems from analysis about (possibly) discontinuous functions on the other hand. We both study relatively tame notions, like cadlag or Baire 1, and potentially wild ones, like quasi-continuity. We also show that slight generalizations and variations of the aforementioned third-order theorems fall far outside of the Big Five.

• 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. Fulltekst i vitenarkiv
• Normann, Dag & Sanders, Sam (2022). On Robust theorems due to Bolzano, Weierstrass, Jordan, and Cantor. Journal of Symbolic Logic (JSL). ISSN 0022-4812. 89(3), s. 1077–1127. doi: 10.1017/jsl.2022.71. Fulltekst i vitenarkiv Vis sammendrag

  Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics where the aim is to identify the minimal axioms needed to prove a given theorem from ordinary, i.e., non-set theoretic, mathematics. This program has unveiled surprising regularities: the minimal axioms are very often equivalent to the theorem over the base theory, a weak system of ‘computable mathematics’, while most theorems are either provable in this base theory, or equivalent to one of only four logical systems. The latter plus the base theory are called the ‘Big Five’ and the associated equivalences are robust following Montalbán, i.e., stable under small variations of the theorems at hand. Working in Kohlenbach’s higher-order RM, we obtain two new and long series of equivalences based on theorems due to Bolzano, Weierstrass, Jordan, and Cantor; these equivalences are extremely robust and have no counterpart among the Big Five systems. Thus, higher-order RM is much richer than its second-order cousin, boasting at least two extra ‘Big’ systems.

• 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. 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. Fulltekst i vitenarkiv
• 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 Vis sammendrag

  Turing’s famous ‘machine’ model constitutes the first intuitively convincing framework for computing with real numbers. Kleene’s computation schemes S1–S9 extend Turing’s approach and provide a framework for computing with objects of any finite type. Various research programs have been proposed in which higher-order objects, like functions on the real numbers, are represented/coded as real numbers, so as to make them amenable to the Turing framework. It is then a natural question whether there is any significant difference between the Kleene approach or the Turing-approach-via-codes. Continuous functions being well-studied in this context, we study functions of bounded variation, which have at most countably many points of discontinuity. A central result is the Jordan decomposition theorem that a function of bounded variation on [0, 1] equals the difference of two monotone functions. We show that for this theorem and related results, the difference between the Kleene approach and the Turing-approach-via-codes is huge, in that full second-order arithmetic readily comes to the fore in Kleene’s approach, in the guise of Kleene’s quantifier ∃3.

• 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. 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 Vis sammendrag

  We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first ‘local-to-global’ principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness, but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms are necessary to prove Pincherle's theorem, does not have an unique or unambiguous answer, in contrast to compactness. We establish similar differences for the computational properties of compactness and Pincherle's theorem. We establish the same differences for other local-to-global principles, even going back to Weierstrass. We also greatly sharpen the known computational power of compactness, for the most shared with Pincherle's theorem however. Finally, countable choice plays an important role in the previous, we therefore study this axiom together with the intimately related Lindelöf lemma.

• Normann, Dag (2020). Measure-theoretic Uniformity and the Suslin Functional. Computability. ISSN 2211-3568. 10(2), s. 91–105. doi: 10.3233/COM-190248. Fulltekst i vitenarkiv Vis sammendrag

  Given a set A in the unit interval and the associated Lebesgue measure λ, it is a natural question whether we may (in some sense) compute the measure λ(A) in terms of the set A. Under the moniker measure theoretic uniformity, Tanaka and Sacks have (independently) provided a positive answer for the well-known class of hyperarithmetical sets of reals, and provided a basis theorem for such sets of positive measure. The hyperarithmetical sets are exactly the sets computable in terms of the functional 2E, in the sense of Kleene’s S1–S9. In turn, Kleene’s 2E essentially corresponds to arithmetical comprehension as in ACA0. In this paper, we generalise the aforementioned results to the ‘next level’, namely Π11-CA0, in the form of the Suslin functional, or the equivalent hyperjump. We also generalise the Tanaka-Sacks basis theorem to sets of positive measure that are semi-computability relative to the Suslin functional. Finally, we discuss similar generalisations for infinite time Turing machines.

• 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 9783319944173. s. 318–327. doi: 10.1007/978-3-319-94418-0_32. Fulltekst i vitenarkiv Vis sammendrag

  We investigate the relative computational strength of combinations of four higher order functionals, the jump and hyperjump seen as functionals of type 2 and realisers for the compactness of Cantor space and the Lindelöf property of Baire space seen as functionals of type 3. We compare them with the closure operator for non-monotone inductive definitions of sets of integers, also seen as a functional of type 3.

• 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. 14(1). doi: 10.23638/23)2018. Fulltekst i vitenarkiv Vis sammendrag

  We prove that the sequential functionals of some fixed types at type level 2, taking finite sequences of unary functions as arguments, do form a directed complete partial ordering. This gives a full characterisation of for which types the partially ordered set of sequential functionals has this property. As a tool, we prove a normal form theorem for the finite sequential functionals of the types in question,

• 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 9783319633329. s. 57–70. doi: 10.1007/978-3-319-63334-3_3. Fulltekst i vitenarkiv
• Normann, Dag (2016). On the Cantor-Bendixson rank of a set that is searchable in Gödel's T. Computability. ISSN 2211-3568. 5(1), s. 61–74. doi: 10.3233/COM-150043. Fulltekst i vitenarkiv Vis sammendrag

  We prove that if a closed subset X of the Baire space is searchable in the sense of Escardó [Logical Methods in Computer Science 4(3) (2008), 3], and by a functional definable in Gödel’s T, then X is countable, and the Cantor–Bendixson rank of X is bounded by the ordinal ε0. To this end we introduce evaluation trees, well founded decorated trees that induce operators from the full set-theoretical class NN→N to NN. We prove that when a search operator for a set X is induced from an evaluation tree T, then the Cantor–Bendixson rank of X is bounded by the Kleene–Brouwer order type of T. Further we use a theorem due to Howard [Journal of Symbolic Logic 45(3) (1980), 493–504], estimating the ordinal complexity of the reduction tree of a term in system T, to show that all functionals of type (NN→N)→NN definable in T can be computed using an evaluation tree of ordinal rank below ε0. © 2016 – IOS Press and the authors. All rights reserved

• Normann, Dag (2015). The extensional realizability model of continuous functionals and three weakly non-constructive classical theorems. Logical Methods in Computer Science. 11(1). doi: 10.2168/8)2015. Fulltekst i vitenarkiv
• 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 9781107043480. s. 397–433. doi: 10.1017/cbo9781107338579.012. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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 9783868381580. s. 353–379. doi: 10.1515/9783110324921.353. Fulltekst i vitenarkiv
• 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 9781848162457. s. 297–327. doi: 10.1142/9781848162778_0009. Fulltekst i vitenarkiv
• Normann, Dag (2011). Banach spaces as data types. Logical Methods in Computer Science. 7(2). doi: 10.2168/11)2011. Fulltekst i vitenarkiv
• Normann, Dag (2009). A rich hierarchy of functionals of finite types. Logical Methods in Computer Science. 5(3). doi: 10.2168/11)2009. Fulltekst i vitenarkiv
• 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 9780387360331. s. 119–138. Fulltekst i vitenarkiv
• 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 9783540694052. s. 467–475. Fulltekst i vitenarkiv
• 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 9781568812939. s. 200–220. Fulltekst i vitenarkiv
• Normann, Dag (2006). On sequential functionals of type 3. Mathematical Structures in Computer Science. ISSN 0960-1295. 16, s. 279–289. Fulltekst i vitenarkiv
• Normann, Dag (2006). Computing with functionals - Computability theory or computer science? Bulletin of Symbolic Logic. ISSN 1079-8986. 12(1), s. 43–59. Fulltekst i vitenarkiv
• Normann, Dag (2005). Comparing hierarchies of total functionals. Logical Methods in Computer Science. 1(2). doi: 10.2168/LMCS-1(2)2005. Fulltekst i vitenarkiv 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). A Nonstandard Characterisation of the Type-structure of Continuous Functionals Over the Reals. Electronical Notes in Theoretical Computer Science. 73, s. 141–147. Fulltekst i vitenarkiv 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 (2004). Hierarchies of total functionals over the reals. Theoretical Computer Science. ISSN 0304-3975. 316, s. 137–151. Fulltekst i vitenarkiv
• Normann, Dag (2002). Exact real number computations relative to hereditarily total functionals. Theoretical Computer Science. ISSN 0304-3975. 284, s. 437–453. Fulltekst i vitenarkiv 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 (2002). Om å velge den rette modell for beregnbarhet over de reelle tallene. Normat. ISSN 0801-3500. 49(3), s. 97–104. Fulltekst i vitenarkiv Vis sammendrag

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

• Normann, Dag & Geir, Waagbø (2002). Limit spaces and transfinite types. Archive for Mathematical Logic. ISSN 0933-5846. 41, s. 525–539. Fulltekst i vitenarkiv 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 & Rørdam, Christian (2002). The Computational Power of M^omega. Mathematical logic quarterly. ISSN 0942-5616. 48(1), s. 117–124. Fulltekst i vitenarkiv 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). Continuity, proof systems and the theory of transfinite computations. Archive for Mathematical Logic. ISSN 0933-5846. 41, s. 765–788. Fulltekst i vitenarkiv 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.

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

Se alle arbeider i NVA

• Longley, John & Normann, Dag (2015). Higher-Order Computability. Springer. ISBN 9783662479919. 585 s. Fulltekst i vitenarkiv
• Normann, Dag; Dimitracopoulos, Costas; Newelski, Ludomir & Steel, John R. (2008). Logic Colloquium 2005. Cambridge University Press. ISBN 9780521884259. 288 s. Fulltekst i vitenarkiv

Se alle arbeider i NVA

• Normann, Dag (2023). A higher order perspective on computational analysis. Fulltekst i vitenarkiv
• Normann, Dag (2022). A recent development in the computability theory for functionals of type 3. Fulltekst i vitenarkiv
• Normann, Dag & Sanders, Sam (2022). Betwixed Turing and Kleene. Fulltekst i vitenarkiv
• Normann, Dag (2022). Typed lambda calculus and operators of mainstream mathematics. Fulltekst i vitenarkiv
• Normann, Dag (2022). The complexity of operators constructed in mainstream mathematics. Fulltekst i vitenarkiv
• Normann, Dag (2021). An alternative perspective on Reverse Mathematics. Fulltekst i vitenarkiv
• Normann, Dag (2019). The Computational Complexity of the Heine Borel Theorem. Fulltekst i vitenarkiv
• Normann, Dag (2018). Functionals of Type 3 as Realisers of Classical Theorems in Analysis. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• Normann, Dag (2016). Revisiting Transfinite Types. Fulltekst i vitenarkiv
• Normann, Dag (2015). Alan Turing - mer enn en kodebryter. Fulltekst i vitenarkiv Vis sammendrag

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

• Normann, Dag (2015). Abels tårn. [Radio]. NRK. Fulltekst i vitenarkiv Vis sammendrag

  Paneldiskusjon om Alan Turing

• Normann, Dag (2015). Var Turing oppfinneren av datamaskinen? Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• Normann, Dag (2012). Sequential vs. Continuous Functionals of Finite types. Fulltekst i vitenarkiv
• Normann, Dag (2012). Internal or external, abstract or concrete? What does this mean for computing in a Banach space? Fulltekst i vitenarkiv
• Normann, Dag (2011). Fixed point computability of total functionals. Fulltekst i vitenarkiv
• Normann, Dag (2011). Sequential versus Continuous Models of Finite Types. Fulltekst i vitenarkiv
• Normann, Dag (2011). Banach spaces as data types. Fulltekst i vitenarkiv
• Normann, Dag (2011). Om Urysohns universelle metriske rom - og litt om Urysohn selv. Fulltekst i vitenarkiv
• Normann, Dag (2009). Continuous vs. Sequential Functionals - Harmony or Disharmony? Fulltekst i vitenarkiv
• Normann, Dag (2009). Functionals of finite types - bridging logic and computer science. Fulltekst i vitenarkiv
• Normann, Dag (2009). Finite Sequential Procedures with Observational Ordering. Fulltekst i vitenarkiv
• Normann, Dag (2009). The sequential functionals are far from being algebraic domains. Fulltekst i vitenarkiv 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. Fulltekst i vitenarkiv
• Normann, Dag (2009). Computations and functionals of finite types. Fulltekst i vitenarkiv
• Normann, Dag (2008). Om kontinuerlige operatorer på datasstrømmer og et åpent problem i topologi. Fulltekst i vitenarkiv
• Normann, Dag (2008). 50 years of continuous functionals. Fulltekst i vitenarkiv
• Normann, Dag (2008). Trenger vi uendelige mengder i diskret matematikk? Fulltekst i vitenarkiv
• Normann, Dag (2007). Kan datamaskiner spille bondesjakk. Fulltekst i vitenarkiv
• Normann, Dag (2007). External and internal density theorems for limspace-interpretations of some types. Fulltekst i vitenarkiv
• Normann, Dag (2007). Hvor vanskelig kan det være? Om bondesjakk og det å finne maksimumspunktet til en kontinuerlig funksjon. Fulltekst i vitenarkiv
• Normann, Dag (2007). Kan datamaskiner spille bondesjakk. Fulltekst i vitenarkiv
• Normann, Dag (2006). The coincidense problem and its topological counterpart. Fulltekst i vitenarkiv
• Normann, Dag (2006). On sequential functionals of type 3. Fulltekst i vitenarkiv
• Normann, Dag (2006). Domain theory based hierarchies of total functionals. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• Normann, Dag (2006). The Continuous Functionals and Aspects of Computability. Fulltekst i vitenarkiv
• Normann, Dag (2005). Om mulige og tilsynelatende umulige programmeringsoppgaver. Normat. ISSN 0801-3500. 53(1), s. 21–27. Fulltekst i vitenarkiv
• Normann, Dag (2005). Regning ved hjelp av tallerkenstabler - en innfallsvinkel til programmeringsteori. Fulltekst i vitenarkiv
• Normann, Dag (2005). Sequential functionals of type 3. Fulltekst i vitenarkiv
• Normann, Dag (2005). Minesveiper - hva har det med matematikk å gjøre. Fulltekst i vitenarkiv
• Normann, Dag (2004). Regning ved hjelp av tallerkenstabler - en innfallsvinkel til programmeringsteori. Fulltekst i vitenarkiv 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. Fulltekst i vitenarkiv
• Normann, Dag (2004). Er du en kløpper i Minesveiper og kan bevise det, kan du bli en million dollar rikere. Fulltekst i vitenarkiv 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. Fulltekst i vitenarkiv 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). Er du en kløpper i Minesveiper og kan bevise det, kan du bli en million dollar rikere. Fulltekst i vitenarkiv
• Normann, Dag (2004). Turings verden - lag dine egne datamaskiner. Fulltekst i vitenarkiv Vis sammendrag

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

• Normann, Dag (2004). Minesveiper - hva har det med matematikk å gjøre? Fulltekst i vitenarkiv Vis sammendrag

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

• Normann, Dag (2004). Typed Computability over discrete and continuous base types. Fulltekst i vitenarkiv
• Normann, Dag (2003). Computability in higher type functionals: History, challenges and results, Fulltekst i vitenarkiv 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. Fulltekst i vitenarkiv 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 (2003). Er du en kløpper i minesveiper, og kan bevise det, kan du bli en million dollar rikere. Fulltekst i vitenarkiv Vis sammendrag

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

• Normann, Dag (2002). Hierarchies of total functionals over the reals. Fulltekst i vitenarkiv 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 (2002). Om deterministiske og ikke-deterministiske beregninger. Fulltekst i vitenarkiv Vis sammendrag

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

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