On the complexity of Gödel's proof predicate

On Kripke-style Semantics for the Provability Logic of Gödel's Proof Predicate with Quantifiers on Proofs

Kripke-style semantics is suggested for the provability logic with quantifiers on proofs corresponding to the standard Gödel proof predicate. It is proved that the set of valid formulas is decidable. The arithmetical completeness is still an open issue.

Higher-Order Modal Logic - A Sketch

First-order modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like Frege’s morning star/evening star puzzle arise. The introduction of predicate abstraction machinery provides a natural extension in which such difficulties can be addressed. But this machinery can also be thought of as part of a move to a full higher-order modal logic. In this ...

Applying Gödel's Dialectica Interpretation to Obtain a Constructive Proof of Higman's Lemma

We use Gödel's Dialectica interpretation to analyse Nash-Williams' elegant but non-constructive 'minimal bad sequence' proof of Higman's Lemma. The result is a concise constructive proof of the lemma (for arbitrary decidable well-quasi-orders) in which Nash-Williams' combinatorial idea is clearly present, along with an explicit program for finding an embedded pair in sequences of words.

Reflections on Gödel’s Ontological Argument

The Complexity of the Disjunction and ExistentialProperties in

This paper considers the computational complexity of the disjunc-tion and existential properties of intuitionistic logic. We prove that the disjunction property holds feasibly for intuitionistic propositional logic; i.e., from a proof of A _ B, a proof either of A or of B can be found in polynomial time. For intuitionistic predicate logic, we prove superexponential lower bounds for the disjunct...