Related papers: Unravelling Cyclic First-Order Arithmetic

We show that Hertling-Manin F-manifolds provide the appropriate theoretical framework for studying the integrability of quasilinear systems of first-order evolutionary partial differential equations of the form ${\bf u}_t=X\circ {\bf u}_x$…

Harvey Friedman shows that, over Peano Arithmetic, the consistency statement for a finitely axiomatised theory $A$ can be characterised as the weakest statement $C$ over Peano Arithmetic such that ${\sf PA}+C$ interprets $A$. We study which…

G\"odel's Dialectica interpretation was designed to obtain a relative consistency proof for Heyting arithmetic, to be used in conjunction with the double negation interpretation to obtain the consistency of Peano arithmetic. In recent…

Cut-elimination is one of the most famous problems in proof theory, and it was defined and solved for first-order sequent calculus by Gentzen in his celebrated Hauptsatz. Ceres is a different cut-elimination algorithm for first- and…

We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is…

Cycles are fundamental elements in graph-structured data and have demonstrated their effectiveness in enhancing graph learning models. To encode such information into a graph learning framework, prior works often extract a summary quantity,…

We present a memetic algorithm (\maa) approach for finding a Hamiltonian cycle in a Hamiltonian graph. The \ma is based on a proven approach to the Asymmetric Travelling Salesman Problem (\atspp) that, in this contribution, is boosted by…

We report on a recent conjecture by Gisin on a restriction of physical processes in sets of finite information numbers (FIN) and further analyze the entropic constraint associated with the proposed algorithm. In the course, we provide a…

We derive several existence results concerning cycle types and, more generally, the "mapping behavior" of complete mappings. Our focus is on so-called first-order cyclotomic mappings, which are functions on a finite field $\mathbb{F}_q$…

Basic proof-search tactics in logic and type theory can be seen as the root-first applications of rules in an appropriate sequent calculus, preferably without the redundancies generated by permutation of rules. This paper addresses the…

Separation logic is successful for software verification of heap-manipulating programs. Numbers are necessary to be added to separation logic for verification of practical software where numbers are important. However, properties of the…

Uniform proofs are sequent calculus proofs with the following characteristic: the last step in the derivation of a complex formula at any stage in the proof is always the introduction of the top-level logical symbol of that formula. We…

We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment $x \in \{0,1\}^n$ to a CNF formula $\varphi$ is shared between two parties, where Alice knows $x_1, \dots, x_{n/2}$, Bob knows…

In this paper, we study Cyclic Henkin Logic CHL, a logic that can be described as provability logic without the third L\"ob condition, to wit, that provable implies provably provable (aka principle 4). The logic CHL does have full modalised…

For a graph with edge ordering, a linear order on the edge set, we obtain a permutation of vertices by considering the edges as transpositions of endvertices. It is known from D\'enes' results that the permutation of a tree is a full cyclic…

We present a generalization of first-order unification to a term algebra where variable indexing is part of the object language. We exploit variable indexing by associating some sequences of variables ($X_0,\ X_1,\ X_2,\dots$) with a…

There are two well known systems formalizing total recursion beyond primitive recursion (\textbf{PR}), system \textbf{T} by G\"odel and system \textbf{F} by Girard and Reynolds. system \textbf{T} defines recursion on typed objects and can…

We show that no total functional can uniformly transform $\Pi_1$ primality into explicit $\Sigma_1$ witnesses without violating normalization in $\mathsf{HA}$. The argument proceeds through three complementary translations: a geometric…

Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles…

We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. En route, we prove some sharp estimates on power…