Normal Proofs and Their Grammar

Normal Proofs and Their Grammar

We present grammatical (or equational) descriptions of the set of normal inhabitants fM j ` M : A; M in -normal form g of a given type A under a given basis , both for the standard simple type system (in partial discharge convention) and for the system in total discharge convention (or Prawitz-style natural deduction system). It is shown that in the latter system we can describe the set by a ( ...

Proof Normalization and Normal Proofs

Proofs, Disproofs, and Their Duals

Bi-intuitionistic logic, also known as Heyting-Brouwer logic or subtractive logic, is extended in various ways by a strong negation connective used to express commitments arising from denials. These logics have been introduced and investigated in [48]. In the present paper, an inferentialist semantics in terms of proofs, disproofs, and their duals is developed. Whereas the Brouwer-Heyting-Kolmo...

Expansion Tree Proofs and Their Conversion to Natural Deduction Proofs

We present a new form of Herbrand's theorem which is centered around structures called expansion trees. Such trees contains substitution formulas and selected (critical) variables at various non-terminal nodes. These trees encode a shallow formula and a deep formula the latter containing the formulas which label the terminal nodes of the expansion tree. If a certain relation among the selected ...

Normal Proofs in Intruder Theories

Given an arbitrary intruder deduction capability, modeled as an inference system S and a protocol, we show how to compute an inference system b S such that the security problem for an unbounded number of sessions is equivalent to the deducibility of some message in b S. Then, assuming that S has some subformula property, we lift such a property to b S, thanks to a proof normalisation theorem. I...