Program Extraction from Nested Definitions

Abstract

Abstract. Minlog is a proof assistant which automatically extracts computational content in an extension of Gödel's T from formalized proofs. We report on extending Minlog to deal with predicates defined using a particular combination of induction and coinduction, via so-called nested definitions. In order to increase the efficiency of the extracted programs, we have also implemented a feature to translate terms into Haskell programs. To illustrate our theory and implementation, a formalisation of a theory of uniformly continuous functions due to Berger is presented.

A realizability interpretation of Church's simple theory of types

“…In CST, the underlying space of real numbers can be modelled as a base type ι with suitable arithmetic operations with I : ι → o, similar to our earlier Example 3.1. In Berger (2011) and Miyamoto et al (2013) the same example is expressed in versions of second-order logic with considerable more effort. In type-theoretic systems like Agda and Coq this example cannot be formalized directly since the formal strict positivity and guardedness requirements of these systems are not met.…”

Optimized Program Extraction for Induction and Coinduction

“…Particular attention is paid to a special treatment of Harrop formulas (which have trivial realizers) leading to optimized programs. Similar work on this topic has been done in [18,13,2,14,15] and to a large extent implemented in the Minlog system [5]. Related methods of optimized program extraction can be found in [16] and in the systems Coq [9] and Nuprl [11].…”

Nonflatness and totality

“…Towards a common study of totality and cototality. Recently, 'cototal ideals,' that is, total ideals together with infinities like ∞ at type N, have been used to model stream-like objects at base types arising from initial algebras, offering an alternative to versions of semantics simultaneously based on initial algebras and final coalgebras (Rutten 2000;Hancock et al 2009;Ghani et al 2009); for this line of work, rooted in Berger (2009Berger ( , 2011 and Berger and Seisenberger (2012), see Berger et al (2011Berger et al ( , 2016, Schwichtenberg et al (2012), Miyamoto et al (2013) and Miyamoto and Schwichtenberg (2015). In view of the mismatch between transitivity and totality in a nonflat setting, which we described in Section 5.2, it looks like a refinement is possible, where totality should feature an increased degree of finiteness and should be studied hand in hand with an appropriate notion of cototality: beside more or less obvious differences of the two at base types (based on the proof of Lemma 5.13, for example, one could expect continuous 'cototalisations' to exist), their interplay at higher types remains terra incognita at the time of this writing.…”