Proof theory is a flexible tool that naturally leads to a precise, mathematical specification of type systems that can be very expressive.

Arithmetic,

Proof Theory, and Computational Complexity.

Gallier introduces mathematical logic, emphasizing

proof theory and procedures for constructing formal proofs of formulae algorithmically.

Although his methods, rules and structures have lead to verification programs essential to computer science, and his work on natural deduction, the sequent calculus and ordinal

proof theory are still considered advanced, his life eventually ran contrary to the passions of his nation.

For

proof theory without contraction, excellent references, for example, are [5], [6], [12].

When we talk of Hilbert's finitism, we ordinarily mean his finitist

proof theory or metamathematics rather than what he takes to be the finitist portion of a formalized mathematical theory.

The remaining essays concern Prior's formal legacy, with articles on applications of temporal logic in computer science, particularly database theory, and articles on modal

proof theory and modal semantics.

In his

proof theory he stated counterparts of Gentzen's cut rule and Hauptsatz.

2 [Logics and Meanings of Programs): Semantics of Programming Languages -- denotational semantics General Terms: Languages, Theory, Verification Additional Key Words and Phrases: Constraint programming, dynamic scheduling,

proof theory, strongest postcondition

Shin provides two theories of diagrams, Venn-I and Venn-II, in which (1) the main syntactical notion is that of (well-formed) diagrams; (2) the

proof theory consists of a set of rules that tell us how to operate with diagrams; and (3) the semantics is a generalization of the usual semantics for Venn diagrams.

Her argument thus implies that actual self-defense never constitutes a failure of

proof theory.

The fifth chapter,

Proof Theory and Decidability, might better be labelled

Proof Theory and Undecidability, although there are a few decidability results mentioned.