395 research outputs found
An Intuitionistic Formula Hierarchy Based on High-School Identities
We revisit the notion of intuitionistic equivalence and formal proof
representations by adopting the view of formulas as exponential polynomials.
After observing that most of the invertible proof rules of intuitionistic
(minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms
corresponding to the high-school identities, we show that one can obtain a more
compact variant of a proof system, consisting of non-invertible proof rules
only, and where the invertible proof rules have been replaced by a formula
normalisation procedure.
Moreover, for certain proof systems such as the G4ip sequent calculus of
Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the
non-invertible proof rules as strict inequalities between exponential
polynomials; a careful combinatorial treatment is given in order to establish
this fact.
Finally, we extend the exponential polynomial analogy to the first-order
quantifiers, showing that it gives rise to an intuitionistic hierarchy of
formulas, resembling the classical arithmetical hierarchy, and the first one
that classifies formulas while preserving isomorphism
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 + âred, 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 + âred and that G*1 is strictly stronger under standard complexity-theoretic hardness assumptions.
Secondly, we show a correspondence of EF + âred to bounded arithmetic: EF + âred 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 + âred proofs, and EF + âred is in a sense the weakest system with this property. Finally, we show that unconditional lower bounds for EF + âred 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 + âred naturally unites the central problems from circuit and proof complexity.
Technically, our results rest on a formalised strategy extraction theorem for EF + âred akin to witnessing in intuitionistic S12 and a normal form for EF + âred proofs
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Mathematical Logic: Proof Theory, Constructive Mathematics
[no abstract available
Frege systems for quantified Boolean logic
We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C. Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC0[p] circuits. In contrast, any non-trivial lower bound for propositional AC0[p]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.This research was supported by grant nos. 48138 and 60842 from the John Templeton Foundation, EPSRC grant
EP/L024233/1, and a Doctoral Prize Fellowship from EPSRC (third author). The second author was funded by the European
Research Council under the European Unionâs Seventh Framework Programme (FP7/2007â2013)/ERC grant agreement
no. 279611 and under the European Unionâs Horizon 2020 Research and Innovation Programme/ERC grant agreement no.
648276 AUTAR. The fourth author was supported by the Austrian Science Fund (FWF) under project number P28699 and by
the European Research Council under the European Unionâs Seventh Framework Programme (FP7/2007-2014)/ERC Grant
Agreement no. 61507.
Part of this work was done when Beyersdorff and Pich were at the University of Leeds and Bonacina at Sapienza University
Rome.Peer ReviewedPostprint (published version
Mathematical Logic: Proof theory, Constructive Mathematics
The workshop âMathematical Logic: Proof Theory, Constructive Mathematicsâ was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit
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