339 research outputs found
Provably total recursive functions and MRDP theorem in Basic Arithmetic and its extensions
We study Basic Arithmetic, BA introduced by W. Ruitenburg. BA is an
arithmetical theory based on basic logic which is weaker than intuitionistic
logic. We show that the class of the provably recursive functions of BA is a
proper sub-class of primitive recursive functions. Three extensions of BA,
called BA+U, BA_c and EBA are investigated with relation to their provably
recursive functions. It is shown that the provably recursive functions of these
three extensions of BA are exactly primitive recursive functions. Moreover,
among other things, it is shown that the well-known MRDP theorem doesn't hold
in BA, BA+U, BA_c, but holds in EBA
Unifying Functional Interpretations: Past and Future
This article surveys work done in the last six years on the unification of
various functional interpretations including G\"odel's dialectica
interpretation, its Diller-Nahm variant, Kreisel modified realizability,
Stein's family of functional interpretations, functional interpretations "with
truth", and bounded functional interpretations. Our goal in the present paper
is twofold: (1) to look back and single out the main lessons learnt so far, and
(2) to look forward and list several open questions and possible directions for
further research.Comment: 18 page
The scope of Feferman’s semi-intuitionistic set theories and his second conjecture
The paper is concerned with the scope of semi-intuitionistic set theories that relate to various foundational stances. It also provides a proof for a second conjecture of Feferman’s that relates the concepts for which the law of excluded middle obtains to those that are absolute with regard to the relevant test structures, or more precisely of ∆1 complexity. The latter is then used to show that a plethora of statements is indeterminate with respect to various semi-intuitionistic set theories
Understanding Gentzen and Frege Systems for QBF
Recently Beyersdorff, Bonacina, and Chew [10] introduced a natural class of Frege systems for quantified Boolean formulas (QBF) and showed strong lower bounds for restricted versions of these systems. Here we provide a comprehensive analysis of the new extended Frege system from [10], denoted EF + ∀red, which is a natural extension of classical extended Frege EF. Our main results are the following: Firstly, we prove that the standard Gentzen-style system G*1 p-simulates EF + ∀red and that G*1 is strictly stronger under standard complexity-theoretic hardness assumptions.
Secondly, we show a correspondence of EF + ∀red to bounded arithmetic: EF + ∀red can be seen as the non-uniform propositional version of intuitionistic S12. Specifically, intuitionistic S12 proofs of arbitrary statements in prenex form translate to polynomial-size EF + ∀red proofs, and EF + ∀red is in a sense the weakest system with this property. Finally, we show that unconditional lower bounds for EF + ∀red would imply either a major breakthrough in circuit complexity or in classical proof complexity, and in fact the converse implications hold as well. Therefore, the system EF + ∀red naturally unites the central problems from circuit and proof complexity.
Technically, our results rest on a formalised strategy extraction theorem for EF + ∀red akin to witnessing in intuitionistic S12 and a normal form for EF + ∀red proofs
On Tao's "finitary" infinite pigeonhole principle
In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard
analysis and the finitization of soft analysis statements into hard analysis
statements. One of his main examples was a quasi-finitization of the infinite
pigeonhole principle IPP, arriving at the "finitary" infinite pigeonhole
principle FIPP1. That turned out to not be the proper formulation and so we
proposed an alternative version FIPP2. Tao himself formulated yet another
version FIPP3 in a revised version of his essay.
We give a counterexample to FIPP1 and discuss for both of the versions FIPP2
and FIPP3 the faithfulness of their respective finitization of IPP by studying
the equivalences IPP FIPP2 and IPP FIPP3 in the context of reverse
mathematics. In the process of doing this we also introduce a continuous
uniform boundedness principle CUB as a formalization of Tao's notion of a
correspondence principle and study the strength of this principle and various
restrictions thereof in terms of reverse mathematics, i.e., in terms of the
"big five" subsystems of second order arithmetic
A predicative variant of a realizability tripos for the Minimalist Foundation.
open2noHere we present a predicative variant of a realizability tripos validating
the intensional level of the Minimalist Foundation extended with Formal Church
thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel
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