Deduction modulo is an extension of first-order predicate logic where axioms are replaced by a congruence relation on propositions and where many theories, such as arithmetic, simple type theory and some variants of set theory, can be expressed. An important question in deduction modulo is to find a condition of the theories that have the strong normalization property. Dowek and Werner have giv...

In these notes, we prove some general theorems for establishing properties of untyped λ-terms, using a variant of the reducibility method. These theorems apply to (pure) λ-terms typable in the systems of conjunctive types DΩ and D. As applications, we give simple proofs of the characterizations of the terms having headnormal forms, of the normalizable terms, and of the strongly normalizing term...

Polynomial-clone reducibilities are generalizations of the truth-table reducibilities. A polynomial clone is a set of functions over a finite set X that is closed under composition and contains all the constant and projection functions. For a fixed polynomial clone C, a sequence B ∈ X is C-reducible to A ∈ X if there is an algorithm that computes B from A using only effectively selected functio...

There is little doubt that the notion of reducibility is the most useful tool that complexity theory has delivered to the rest of the computer science community. For most computational problems that arise in real-world applications, such as the Traveling Salesperson Problem, we still know little about their deterministic time or space complexity. We cannot now tell whether classes such as P and...

This paper reviews and investigates Bennett s notions of strong and weak computational depth also called logical depth for in nite binary sequences Roughly an in nite binary sequence x is de ned to be weakly useful if every element of a non negligible set of decidable sequences is reducible to x in recursively bounded time It is shown that every weakly useful sequence is strongly deep This resu...

We show that the pointlike and the idempotent pointlike problems are reducible with respect to natural signatures in the following cases: the pseudovariety of all finite semigroups in which the order of every subgroup is a product of elements of a fixed set π of primes; the pseudovariety of all finite semigroups in which every regular J-class is the product of a rectangular band by a group from...

In this paper we provide a general condition for the reducibility of the ReshetikhinTuraev quantum representations of the mapping class groups. Namely, for any modular tensor category with a special symmetric Frobenius algebra with a non-trivial genus one partition function, we prove that the quantum representations of all the mapping class groups built from the modular tensor category are redu...

The paper applies the pseudo-linear algebra to unify the results on reducibility, reduction and transfer equivalence for continuousand discrete-time nonlinear control systems. The necessary and sufficient condition for reducibility of nonlinear input-output equation is presented in terms of the greatest common left factor of two polynomials describing the behaviour of the ‘tangent linearized sy...

We investigate some aspects of proof methods for the termination of (extensions of) the second-order λ-calculus in presence of union and existential types. We prove that Girard’s reducibility candidates are stable by union iff they are exactly the non-empty sets of terminating terms which are downward-closed w.r.t. a weak observational preorder. We show that this is the case for the Curry-style...