Related papers: A new proof of Euclid's algorithm
Inductive theorem proving is an important long-standing challenge in computer science. In this extended abstract, we first summarize the recent developments of proof by induction for Isabelle/HOL. Then, we propose united reasoning, a novel…
Calculi with control operators have been studied to reason about control in programming languages and to interpret the computational content of classical proofs. To make these calculi into a real programming language, one should also…
Recent work by Faizal et al. (2025) claims that G\"odelian undecidability of non-algorithmic truths in our universe imply the impossibility of a formal, algorithmic simulation of the universe. This paper clarifies the distinction between…
In the first of this pair of papers, it was proven that that no physical computer can correctly carry out all computational tasks that can be posed to it. The generality of this result follows from its use of a novel definition of…
We define the regular Euclidean algorithm and the general form which leads to the method of least absolute remainders and also the method of negative remainders. We are going to show that if looked from the perspective of subtraction, the…
Higher-order modal fixpoint logic (HFL) is a higher-order extension of the modal mu-calculus, and strictly more expressive than the modal mu-calculus. It has recently been shown that various program verification problems can naturally be…
A coordinate system is a foundation for every quantitative science, engineering, and medicine. Classical physics and statistics are based on the Cartesian coordinate system. The classical probability and hypothesis testing theory can only…
Higher-Order Fixpoint Logic (HFL) is a hybrid of the simply typed \lambda-calculus and the modal \lambda-calculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of…
The set of pure terms which are typable in the $\lambda$$\Pi$-calculus in a given context is not recursive. So there is no general type inference algorithm for the programming language Elf and, in some cases, some type information has to be…
We present a hypersequent calculus $\text{G}^3\text{\L}\forall$ for first-order infinite-valued {\L}ukasiewicz logic and for an extension of it, first-order rational Pavelka logic; the calculus is intended for bottom-up proof search. In…
It is well known that a rigid motion of the Euclidean plane can be written as the composition of at most three reflections. It is perhaps not so widely known that a similar result holds for Euclidean space in any number of dimensions. The…
Recent results by Harrow et. al. and by Ta-Shma, suggest that quantum computers may have an exponential advantage in solving a wealth of linear algebraic problems, over classical algorithms. Building on the quantum intuition of these…
Using the programming language Haskell, we introduce an implementation of propositional calculus, number theory, and a simple imperative language that can evaluate arithmetic and boolean expressions. Finally, we provide an implementation of…
Currently, there is a lack of rigorous theoretical system for systematically generating non-trivial and logically valid theorems. Addressing this critical gap, this paper conducts research to propose a novel automated theorem generation…
We show how to extract a monotonic learning algorithm from a classical proof of a geometric statement by interpreting the proof by means of interactive realizability, a realizability sematics for classical logic. The statement is about the…
In this research, an optimal algorithm for the Collatz conjecture is presented. Properties such as the convergence of the algorithm and an equation that relates the algorithm to the classical Collatz conjecture are obtained. It is validated…
We investigate the problem of safety verification of infinite-state parameterized programs that are formed based on a rich class of topologies. We introduce a new proof system, called parametric proof spaces, which exploits the underlying…
We give a new simple and short ("one-line") analysis for the runtime of the well-known Euclidean Algorithm. While very short simple, the obtained upper bound in near-optimal.
Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly…
Formal, automated theorem proving has long been viewed as a challenge to artificial intelligence. We introduce here a new approach to computer theorem proving, one that employs specialized language models for Lean4 proof generation combined…