Light Dialectica revisited

Dialectica Interpretation with Marked Counterexamples

“…The presented variant of the Dialectica interpretation interleaves extracted programs with additional information, which is utilised during evaluation in order to omit redundant calculations. Other kinds of redundancies can be avoided by means of uniform annotations, as described in [8,12]. It can be argued that extensions of such technical nature may obscure the obtained computational content.…”

Modal Functional (Dialectica) Interpretation

“…We build upon functional arithmetical systems NA and (the light annotated) NA l from [17] . While the verifying system NA basically is the Arithmetic Z of Berger, Buchholz and Schwichtenberg [4] in a slightly different presentation which is more suitable for light functional synthesis and features full classical logic (without strong existence) and full extensionality 4 , its light counterpart NA l is only partly classical.…”

“…Moreover, the input system NA l is weakly extensional and its contraction (and hence also induction) rule is restricted for soundness of the (light) functional interpretation of NA l into NA . In computing terms, the program synthesis algorithm provided by the light Dialectica (of [17], as inherited from the one 5 of [13]) produces correct output only modulo the above-mentioned restrictions on Extensionality and Contraction 6 . If not for the weak extensionality, NA l were a conservative extension of NA .…”

“…The present work supersedes the functional synthesis technique outlined in our previous paper [17] by adding a useful device for (homogeneously) combining the effect of previous optimizations by partly and fully uniform quantifiers in a compact discharger of constructive potential, namely the modal operator (and its weak co-modality ≡ ¬ ¬ ). Proofs which are not necessarily constructive may yet potentially contain constructive content; in order to make use of this constructive 'charge' contained in a (non-constructive) proof, various 'discharge' instruments have been created over the past decades.…”

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We adapt our light Dialectica interpretation [17] 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 Gödel's T and classical S 4 . 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 S 4 .

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