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Kripke semantics for higher-order type theory applied to constraint logic programming languages. (English) Zbl 1387.68059
Summary: We define a Kripke semantics for Intuitionistic Higher-Order Logic with constraints formulated within Church’s Theory of Types via the addition of a new constraint base type. We then define an executable fragment, $${hoHH}(\mathcal{C})$$, of the logic: a higher-order logic programming language with typed $$\lambda$$-abstraction, implication and universal quantification in goals and constraints, and give a modified model theory for this fragment. Both formal systems are shown sound and complete for their respective semantics.
We also solve the impredicativity problem in $$\lambda$$Prolog semantics, namely how to give a definition of truth without appealing to induction on subformula structure. In the last section we give a simple semantics-based conservative extension proof that the language $$hoHH(\mathcal{C})$$ satisfies a uniformity property along the lines of [D. Miller et al., Ann. Pure Appl. Logic 51, No. 1–2, 125–157 (1991; Zbl 0721.03037)].
##### MSC:
 68N17 Logic programming 03B15 Higher-order logic; type theory (MSC2010)
##### Software:
HiLog; CiaoPP; Hiord; Ciao; CoALP
Full Text:
##### References:
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