Related papers: Limits of real numbers in the binary signed digit …
We encode arrays as functions which, in turn, are encoded as sets of ordered pairs. The set cardinality of each of these functions coincides with the length of the array it is representing. Then we define a fragment of set theory that is…
This paper is concerned with the exact solution of mixed-integer programs (MIPs) over the rational numbers, i.e., without any roundoff errors and error tolerances. Here, one computational bottleneck that should be avoided whenever possible…
Milner (1984) defined an operational semantics for regular expressions as finite-state processes. In order to axiomatize bisimilarity of regular expressions under this process semantics, he adapted Salomaa's proof system that is complete…
We present necessary and sufficient conditions for the termination of linear homogeneous programs. We also develop a complete method to check termination for this class of programs. Our complete characterization of termination for such…
We introduce a full binary directed tree structure to represent the set of natural numbers, further categorizing them into three distinct subsets: pure odd numbers, pure even numbers, and mixed numbers. We adopt a binary string…
$\{log\}$ is a programming language at the intersection of Constraint Logic Programming, set programming and declarative programming. But $\{log\}$ is also a satisfiability solver for a theory of finite sets and finite binary relations.…
Applying machine learning to mathematical terms and formulas requires a suitable representation of formulas that is adequate for AI methods. In this paper, we develop an encoding that allows for logical properties to be preserved and is…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
Number fields and their rings of integers, which generalize the rational numbers and the integers, are foundational objects in number theory. There are several computer algebra systems and databases concerned with the computational aspects…
Arithmetic automata recognize infinite words of digits denoting decompositions of real and integer vectors. These automata are known expressive and efficient enough to represent the whole set of solutions of complex linear constraints…
We present a formal logic for quantitative reasoning about security properties of network protocols. The system allows us to derive concrete security bounds that can be used to choose key lengths and other security parameters. We provide…
We present {\lambda}ert, a type theory supporting refinement types with explicit proofs. Instead of solving refinement constraints with an SMT solver like DML and Liquid Haskell, our system requires and permits programmers to embed proofs…
In this paper we examine a number of term rewriting system for integer number representations, building further upon the datatype defining systems described in [2]. In particular, we look at automated methods for proving confluence and…
In this work, we consider the properties of the two-term Machin-like formula and develop an algorithm for computing digits of $\pi$ by using its rational approximation. In this approximation, both terms are constructed by using a…
We present a comprehensive programme analysing the decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof…
We advocate a declarative approach to proving properties of logic programs. Total correctness can be separated into correctness, completeness and clean termination; the latter includes non-floundering. Only clean termination depends on the…
Coding theory is very useful for real world applications. A notable example is digital television. Basically, coding theory is to study a way of detecting and/or correcting data that may be true or false. Moreover coding theory is an area…
Logic programming (LP) is a programming language based on first-order Horn clause logic that uses SLD-resolution as a semi-decision procedure. Finite SLD-computations are inductively sound and complete with respect to least Herbrand models…
We show that the number of short binary signed-digit representations of an integer $n$ is equal to the $n$-th term in the Stern sequence. Various proofs are provided, including direct, bijective, and generating function proofs. We also show…
We present a method for verifying partial correctness properties of imperative programs that manipulate integers and arrays by using techniques based on the transformation of constraint logic programs (CLP). We use CLP as a metalanguage for…