Complexity of Normalization in the Pure Typed Lambda – Calculus

Publisher Summary This chapter describes estimates for the number of reduction steps necessary to reach the normal form and sets up a specific normalization procedure for which an ɛ 4 upper bound on the number of reduction steps can be obtained easily. The pure typed λ-calculus means the system of terms built up from typed variables x τ , y τ , … and maybe typed constants a τ , b τ , … by means of application (t σ→τ s σ ) and λ-abstraction (λx σ t τ ) σ→τ . To get an estimate for the number of reduction steps needed, a number is associated with any given term, and it is shown that this number decreases with any reduction step. The chapter reveals that there is an ɛ 4 function f such that for all closed type-0-terms t , the above normalization procedure terminates in m , there is an elementary recursive function g m such that for all closed type-0-terms t with L(t) m (Ih(t)) steps.