In class, we’ve been talking about the integers, which we’ve denoted Z. We started off talking about equality in Z, but then moved on to talking about congruences, which is a weaker notion. This handout explains how “congruence modulo n” is something called an equivalence relation, and we can use it to construct a set Z/nZ that’s a “quotient” of Z. The key point is that a congruence modulo n in...

Relational parametricity is a fundamental concept within theoretical computer science and the foundations of programming languages, introduced by John Reynolds [6]. His fundamental insight was that types can be interpreted not just as functors on the category of sets, but also as equality preserving functors on the category of relations. This gives rise to a model where polymorphic functions ar...

We investigate a first-order conditional probability logic with equality, which is, up to our knowledge, the first treatise of such logic. The logic, denoted LFPOIC=, allows making statements such as: CP>s(φ, θ), and CP6s(φ, θ), with the intended meaning that the conditional probability of φ given θ is at least (at most) s. The corresponding syntax, semantic, and axiomatic system are introduced...

We introduce the concepts of intuitionistic fuzzy equality and intuitionistic fuzzy mapping. And we obtain some fundamental properties of intuitionisic fuzzy mapping. Furthermore, we give the decomposition of an intuitionistic fuzzy mapping by using intuitionistic fuzzy equivalence relation. 2010 AMS Classification: 03E72, 04A72

We study isomorphisms of types in the system of simply-typed λ-calculus with inductive types and recursion. It is shown that in some cases (multiproducts, copies of types), it is possible to add new reductions in such a way that strong normalisation and confluence of the calculus is preserved, and the isomorphisms may be regarded as intensional w.r.t. a stronger equality relation.

Meta-operations on primitive recursive functions sit at the brink of what is computationally possible: the semantic equality of primitive recursive programs is undecidable, and yet this paper shows that the whole class of p.r. functions can be enumerated without semantic duplicates. More generally, the construction shows that for any equivalence relation ≈ on natural numbers, N/ ≈ is r.e. if ≈ ...

Monotonicity under coarse-graining is a crucial property of the quantum relative entropy. The aim of this paper is to investigate the condition of equality in the monotonicity theorem and in its consequences as the strong sub-additivity of von Neumann entropy, the Golden-Thompson trace inequality and the monotonicity of the Holevo quantitity. The relation to quantum Markov states is briefly ind...

We introduce a semantic encoding of partial algebras as total algebras through a Horn axiomatization of the existence equality relation interpreted as an algebraic operation. We show that this novel encoding enjoys several important properties that make it a good tool for the execution of partial algebraic specifications through means specific to ordinary algebraic reasoning, such as

This paper presents a method to translate a proof in an extensional version of the Calculus of Constructions into a proof in the Calculus of Inductive Constructions extended with a few axioms. We use a specific equality in order to translate the extensional conversion relation into an intensional system.

The usual Heisenberg uncertainty relation, ∆X∆P ≥ h̄/2, may be replaced by an exact equality for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. This exact uncertainty relation, δX∆Pnc ≡ h̄/2, can be generalised to other pairs of conjugate observables such as photon number and phase, and is sufficiently strong to provide the basis for moving fr...