Abstract
The relationship between program testing and Martin-Löf’s meaning explanations for intuitionistic type theory is investigated. The judgements of intuitionistic type theory are viewed as conjectures which can be tested in order to be corroborated or refuted. This point of view provides a new perspective on the meaning of hypothetical judgements, since tests for such judgements need methods for generating inputs. Among other things, we need to generate function input. The continuity principle is invoked and the impredicativity of types of functionals is rejected. Furthermore, we provide testing semantics only for decidable identity types. At the end we turn to impredicative type theories, and discuss possible testing semantics for such theories. In particular we propose that testing for impredicative type theory should be based on the evaluation of open expressions. This is in contrast to our testing semantics for Martin-Löf’s predicative intuitionistic type theory which is based on the evaluation of closed expressions.
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Acknowledgements
I would like to express my deep gratitude to Per Martin-Löf for his profound ideas and for many discussions and much help over the years. The present paper owes a lot to these discussionson, for example, on the nature of the meaning explanations, on the distinction between the pre-mathematical and the meta-mathematical, and on the meaning of induction.
I would also like to thank Erik Palmgren and an anonymous referee for useful feedback on a preliminary version of this paper. The paper is based on a talk given several times. I am grateful for interesting comments by many people who attended these talks, for example, Andreas Abel, Peter Aczel, Pierre Clairambault, Thierry Coquand, Peter Hancock, Bengt Nordström, Andrew Pitts, Gordon Plotkin, Michael Rathjen, Dag Prawitz, Peter Schroeder-Heister, Helmut Schwichtenberg, Anton Setzer, and Sören Stenlund.
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Dybjer, P. (2012). Program Testing and the Meaning Explanations of Intuitionistic Type Theory. In: Dybjer, P., Lindström, S., Palmgren, E., Sundholm, G. (eds) Epistemology versus Ontology. Logic, Epistemology, and the Unity of Science, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4435-6_11
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