Related papers: A Note on OTM-Realizability and Constructive Set T…
This paper addresses the longstanding problem of determining the structure of the $\leq_{\mathrm{LT}}$-order in the Effective Topos, known to effectively embed the Turing degrees. In a surprising discovery, we show that the…
The usual reading of logical implication "A implies B" as "if A then B" fails in intuitionistic logic: there are formulas A and B such that "A implies B" is not provable, even though B is provable whenever A is provable. Intuitionistic…
The provability logic of a theory $T$ captures the structural behavior of formalized provability in $T$ as provable in $T$ itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability…
Realizability, introduced by Kleene, can be understood as a concretization of the Brouwer-Heyting-Kolmogorov (BHK) interpretation of proofs, providing a framework to interpret mathematical statements and proofs in terms of their…
Church's Higher Order Logic is a basis for influential proof assistants -- HOL and PVS. Church's logic has a simple set-theoretic semantics, making it trustworthy and extensible. We factor HOL into a constructive core plus axioms of…
Fairly deep results of Zermelo-Frenkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is K*K = K,…
For any first order theory T we construct a Boolean valued model M, in which precisely the T--provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a first order…
The ordered structures of natural, integer, rational and real numbers are studied in this thesis. The theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order…
The ordered structures of natural, integer, rational and real numbers are studied here. It is known that the theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language…
The complexity class $NP$ can be logically characterized both through existential second order logic $SO\exists$, as proven by Fagin, and through simulating a Turing machine via the satisfiability problem of propositional logic SAT, as…
In this paper we consider first-order logic theorem proving and model building via approximation and instantiation. Given a clause set we propose its approximation into a simplified clause set where satisfiability is decidable. The…
We introduce infinite time computable model theory, the computable model theory arising with infinite time Turing machines, which provide infinitary notions of computability for structures built on the reals R. Much of the finite time…
We show that descriptive complexity's result extends in High Order Logic to capture the expressivity of Turing Machine which have a finite number of alternation and whose time or space is bounded by a finite tower of exponential. Hence we…
This work introduces a novel framework of uniform realizability that unifies and generalizes various realizability interpretations of logic, particularly focussing on the treatment of atomic formulas and quantifiers. Traditional…
We introduce the logic $\sf ITL^e$, an intuitionistic temporal logic based on structures $(W,\preccurlyeq,S)$, where $\preccurlyeq$ is used to interpret intuitionistic implication and $S$ is a $\preccurlyeq$-monotone function used to…
In previous papers on this project a general static logical framework for formalizing and mechanizing set theories of different strength was suggested, and the power of some predicatively acceptable theories in that framework was explored.…
Probabilistic team semantics is a framework for logical analysis of probabilistic dependencies. Our focus is on the axiomatizability, complexity, and expressivity of probabilistic inclusion logic and its extensions. We identify a natural…
We generalize Fitting's work on Intuitionistic Kripke models of Set Theory using Ono and Komori's Residuated Kripke models. Based on these models, we provide a generalization of the von Neumann hierarchy in the context of Modal Residuated…
We construct a countable simple theory which, in Keisler's order, is strictly above the random graph (but "barely so") and also in some sense orthogonal to the building blocks of the recently discovered infinite descending chain. As a…
We argue that in some KR applications, we want to quantify over sets of concepts formally represented by symbols in the vocabulary. We show that this quantification should be distinguished from second-order quantification and…