Abstract
We adapt our light Dialectica interpretation to usual and light modal formulas (with universal quantification on boolean and natural variables) and prove it sound for a non-standard modal arithmetic based on Goedel's T and classical S4. The range of this light modal Dialectica is the usual (non-modal) classical Arithmetic in all finite types (with booleans); the propositional kernel of its domain is Boolean and not S4. The `heavy' modal Dialectica interpretation is a new technique, as it cannot be simulated within our previous light Dialectica. The synthesized functionals are at least as good as before, while the translation process is improved. Through our modal Dialectica, the existence of a realizer for the defining axiom of classical S5 reduces to the Drinking Principle (cf. Smullyan).
Highlights
The present work supersedes the functional synthesis technique outlined in our previous paper [HT10] by adding a useful device for combining the effect of previous optimizations by partly and fully uniform quantifiers in a compact releaser of constructive potential, namely the modal operator
We adapt our light Dialectica interpretation to usual and light modal formulas and prove it sound for a non-standard modal arithmetic based on Godel’s T and classical S4
We design two non-standard modal arithmetics, NAm ⊂ NAml, for functional program synthesis. The soundness of these input systems is syntactically given via our modal functional interpretation by the target system, namely classical decidable-predicate Arithmetic with higher-type functionals, in a Natural Deduction presentation.[3]
Summary
The flag apparatus for decorating[2] both quantifiers and implications (throughout the input proofs) tends to become too complex for human operators (so that Oliva’s detour to the linear logic substructure [Oli12] may seem a better alternative). We propose a middle path between removing computational content of (‘computationally correct’) proofs via the second author’s “deep annotation” mechanism and Oliva’s “shallow annotation” equivalent approach (cf Section 6 of [Tri09]). Cannot be simulated within our previous light Dialectica ( is a strict addition to our previous light Arithmetic), it certainly is implementable within either of Trifonov’s or Oliva’s systems. The purpose of our approach has been the rapid implementation in the actual Minlog system (cf [Sea] and Chapter 7 of [SW11], in particular Section 7.4). Our modal systems are normal according to the definition from [Fit07], and non-standard since the normality scheme AxK is (syntactically) derivable from the axiom scheme AxT
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