- HD and Hajime Ishihara (to appear). Bishop Style Constructive Reverse Mathematics. .
In Vasco Brattka and Peter Hertling, Handbook of Computability and Complexity in Analysis, Springer.
- HD and Matthew Hendtlass (2018). Bishop's Lemma. Mathematical Logic Quarterly, 64(1-2), pp. 49–54.
Abstract Bishop's Lemma is a centrepiece in the development of constructive analysis. We show that 1.its proof requires some form of the axiom of choice; and that 2.the completeness requirement in Bishop's Lemma can be weakened and that there is a vast class of nonâcomplete spaces that Bishop's Lemma applies to.
- HD and Robert Lubarsky (2018). Notions of Cauchyness and Metastability. , pp. 140–153.
In Artemov, Sergei and Nerode, Anil, Logical Foundations of Computer Science, Springer International Publishing.
We show that several weakenings of the Cauchy condition are all equivalent under the assumption of countable choice, and investigate to what extent choice is necessary. We also show that the syntactically reminiscent notion of metastability allows similar variations, but is empty in terms of its constructive content.
- HD and Maarten McKubre-Jordens (2016). Paradoxes of material implication in minimal logic. .
In Christiansen, Henning and López, M. Dolores Jiménez and Roussanka Loukanova and Larry Moss, Partiality and Underspecification in Information, Languages, and Knowledge, Cambridge Scholars Publishing.
- HD (2015). Variations on a theme by Ishihara. Mathematical Structures in Computer Science, 25, pp. 1569–1577.
- Robert Lubarsky and HD (2014). Separating the Fan Theorem and Its Weakenings. The Journal of Symbolic Logic, 79(3), pp. 792–813.
- Robert Lubarsky and HD (2014). Principles weaker than BD-N. Journal of Symbolic Logic, 78(3), pp. 873–885.
- Robert Lubarsky and HD (2013). Separating the Fan Theorem and Its Weakenings. , 7734, pp. 280–295.
In Artemov, Sergei and Nerode, Anil, Logical Foundations of Computer Science, Springer Berlin Heidelberg.
- HD (2013). Weak König's Lemma Implies the Uniform Continuity Theorem. Computability, 2(1), pp. 9–13.
As the title suggests we are going to show that Weak König's Lemma implies the uniform continuity theorem; to be more precise the uniform continuity theorem for functions $[0, 1] \to \mathbb R$. This improves upon a result by J. Berger who has proven this implication for functions $[0,1] \to \mathbb N$.
- HD (2012). Reclassifying the antithesis of Specker's theorem. Archive for Mathematical Logic, 51, pp. 687–693.
- HD and Anton Hedin (2012). The Vitali covering theorem in constructive mathematics. Journal of Logic and Analysis, 4.
- HD and Peter Schuster (2011). Uniqueness, continuity and the existence of implicit functions in constructive analysis. LMS Journal of Computation and Mathematics, 14, pp. 127–136.
- HD and Iris Loeb (2011). Constructive reverse investigations into differential equations. Journal of Logic and Analysis, 3.
- HD and Peter Schuster (2010). On Choice Principles and Fan Theorems. Journal of Universal Computer Science, 16(18), pp. 2556–2562.
Veldman proved that the contrapositive of countable binary choice is a theorem of full-fledged intuitionism, to which end he used a principle of continuous choice and the fan theorem. It has turned out that continuous choice is unnecessary in this context, and that a weak form of the fan theorem suffices which holds in the presence of countable choice. In particular, the contrapositive of countable binary choice is valid in Bishop-style constructive mathematics. We further discuss a generalisation of this result and link it to Ishihara's boundedness principle BD-N.
- Douglas S. Bridges and HD (2010). The anti-Specker property, positivity, and total boundedness. Mathematical Logic Quarterly, 56(4), pp. 434–441.
- HD and Peter Schuster (2009). Uniqueness, Continuity, and Existence of Implicit Functions in Constructive Analysis. .
In Andrej Bauer and Peter Hertling and Ker-I Ko, 6th Int'l Conf. on Computability and Complexity in Analysis, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany.
- HD and Iris Loeb (2009). Sequences of Real Functions on $[0, 1]$ in Constructive Reverse Mathematics. Annals of Pure and Applied Logic, 157, pp. 50–61.
- HD (2008). Generalising compactness. Mathematical Logic Quarterly, 54(1), pp. 49–57.
- Douglas S. Bridges and HD (2007). The pseudocompactness of $[0,1]$ is equivalent to the uniform continuity theorem. Journal of Symbolic Logic, 72(4), pp. 1379–1384.
- Douglas S. Bridges and HD (2006). A constructive treatment of Urysohn's Lemma in an apartness space. Mathematical Logic Quarterly, 52(5), pp. 464–469.