Related papers: Formative processes with applications to the decis…
We formalize the theory of forcing in the set theory framework of Isabelle/ZF. Under the assumption of the existence of a countable transitive model of ZFC, we construct a proper generic extension and show that the latter also satisfies…
Fix a set-theoretic universe $V$. We look at small extensions of $V$ as generalised degrees of computability over $V$. We also formalise and investigate the complexity of certain methods one can use to define, in $V$, subclasses of degrees…
Argumentation frameworks, consisting of arguments and an attack relation representing conflicts, are fundamental for formally studying reasoning under conflicting information. We use methods from mathematical logic, specifically…
We define a certain finite set in set theory $\{x\mid\varphi(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any…
We show that the finite satisfiability problem for the unary negation fragment with arbitrary number of transitive relations is decidable and 2-ExpTime-complete. Our result actually holds for a more general setting in which one can require…
We present an extension to the $\mathtt{mathlib}$ library of the Lean theorem prover formalizing the foundations of computability theory. We use primitive recursive functions and partial recursive functions as the main objects of study, and…
We use fast-growing finite and infinite sequences of natural numbers and more complicated constructs to define models of hypercomputation and interpret non-arithmetic predicates, with the strongest extensions reaching full second order…
We study the satisfiability problem of symbolic finite automata and decompose it into the satisfiability problem of the theory of the input characters and the monadic second-order theory of the indices of accepted words. We use our…
In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing.…
Instruction sequence is a key concept in practice, but it has as yet not come prominently into the picture in theoretical circles. This paper concerns instruction sequences, the behaviours produced by them under execution, the interaction…
We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the small finitely proper posets is compatible with a…
We show that including degrees of a particular kind of provability in the search target for any theorem-prover in sufficiently powerful formal systems over finite-sized statements preserves well-definition and a sufficient consistency while…
Over the past two decades several fragments of first-order logic have been identified and shown to have good computational and algorithmic properties, to a great extent as a result of appropriately describing the image of the standard…
In this chapter we survey two topics that have recently been investigated in frame theory. First, we give an overview of the class of scalable frames. These are (finite) frames with the property that each frame vector can be rescaled in…
The first part is expository: it explains how finite fields may be used to prove theorems on infinite fields by a reduction mod p process. The second part gives a variant of P.Smith's fixed point theorem which applies in any characteristic.
We study the structure of the category of representations of $\mathbf{FA}$, the category of finite sets and all maps, mostly working over a field of characteristic zero. This category is not semi-simple and exhibits interesting features. We…
Transformers pretrained via next token prediction learn to factor their world into parts, representing these factors in orthogonal subspaces of the residual stream. We formalize two representational hypotheses: (1) a representation in the…
We isolate a combinatorial property of capacities leading to a construction of proper forcings. Then we show that many classical capacities such as the Newtonian capacity satisfy the property.
Mathematicians still use Naive Set Theory when generating sets without danger of producing any contradiction. Therefore their working method can be considered as a consistent inference system with an experience of over 100 years. My…
In this article, we study connections between representation theory and efficient solutions to the conjugacy problem on finitely generated groups. The main focus is on the conjugacy problem in conjugacy separable groups, where we measure…