Related papers: Back to Classics: Teaching Limits Through Infinite…
The problem of giving a computational meaning to classical reasoning lies at the heart of logic. This article surveys three famous solutions to this problem - the epsilon calculus, modified realizability and the dialectica interpretation -…
An alternative organization for Differential and Integral Calculus, based on an extension of real numbers that include infinitesimal and infinite quantities, is presented. Only Elementary Set Theory is used, without reference to methods or…
We propose a definition of quantum computable functions as mappings between superpositions of natural numbers to probability distributions of natural numbers. Each function is obtained as a limit of an infinite computation of a quantum…
Existing refinement calculi provide frameworks for the stepwise development of imperative programs from specifications. This paper presents a refinement calculus for deriving logic programs. The calculus contains a wide-spectrum logic…
Cut-elimination is the bedrock of proof theory. It is the algorithm that eliminates cuts from a sequent calculus proof that leads to cut-free calculi and applications. Cut-elimination applies to many logics irrespective of their semantics.…
The main object of this paper is to give the generalized von mangoldt function using the L-additive function which can help us to make it possible to calculate The Dirichlet series of the arithmetic derivative $\delta$ and Dirichlet series…
We consider the problem of sensitivity of threshold risk, defined as the probability of a function of a random variable falling below a specified threshold level $\delta >0.$ We demonstrate that for polynomial and rational functions of that…
Possibilistic logic, an extension of first-order logic, deals with uncertainty that can be estimated in terms of possibility and necessity measures. Syntactically, this means that a first-order formula is equipped with a possibility degree…
We define $\overline{\psi}$ to be the multiplicative arithemtic function that satisfies \[\overline{\psi}(p^{\alpha})=\begin{cases} p^{\alpha-1}(p+1), & \mbox{if } p\neq 2; \\ p^{\alpha-1}, & \mbox{if } p=2 \end{cases}\] for all primes $p$…
Predicting the winner of an election is a favorite problem both for news media pundits and computational social choice theorists. Since it is often infeasible to elicit the preferences of all the voters in a typical prediction scenario, a…
Non-Archimedean mathematics is an approach based on fields which contain infinitesimal and infinite elements. Within this approach, we construct a space of a particular class of generalized functions, ultrafunctions. The space of…
We extend the {\lambda}-calculus with constructs suitable for relational and functional-logic programming: non-deterministic choice, fresh variable introduction, and unification of expressions. In order to be able to unify…
The notion of a real-valued function is central to mathematics, computer science, and many other scientific fields. Despite this importance, there are hardly any positive results on decision procedures for predicate logical theories that…
We highlight the fact that in undergraduate calculus, the number pi is defined via the length of the circle, the length of the circle is defined as a certain value of an inverse trigonometric function, and this value is defined via pi, thus…
We develop a new semantics for defeasible inference based on extended probability measures allowed to take infinitesimal values, on the interpretation of defaults as generalized conditional probability constraints and on a preferred-model…
Classical mathematical models used in the semantics of programming languages and computation rely on idealized abstractions such as infinite-precision real numbers, unbounded sets, and unrestricted computation. In contrast, concrete…
We provide a general and modular criterion for the termination of simply-typed $\lambda$ -calculus extended with function symbols defined by user-defined rewrite rules. Following a work of Hughes, Pareto and Sabry for functions defined with…
In this paper we extend a decision procedure for the Boolean algebra of finite sets with cardinality constraints ($\mathcal{L}_{\lvert\cdot\rvert}$) to a decision procedure for $\mathcal{L}_{\lvert\cdot\rvert}$ extended with set terms…
We propose a formalism to derive the maximal bound of generalized Bell type inequalities and shows that the formalism can be applied to various form of Bell functions. The generic Bell function is defined to generate the combinations of all…
In polymeric quantum theories, a natural question pertains to the so called continuum limit, corresponding to the limit where the `discreteness parameter' $\lambda$ approaches zero. In particular one might ask whether the limit exists and,…