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On Measure Quantifiers in First-Order Arithmetic

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12813)

Abstract

We study the logic obtained by endowing the language of first-order arithmetic with second-order measure quantifiers. This new kind of quantification allows us to express that the argument formula is true in a certain portion of all possible interpretations of the quantified variable. We show that first-order arithmetic with measure quantifiers is capable of formalizing simple results from probability theory and, most importantly, of representing every recursive random function. Moreover, we introduce a realizability interpretation of this logic in which programs have access to an oracle from the Cantor space.

Keywords

  • Probabilistic computation
  • Peano Arithmetic
  • Realizability

Supported by ERC CoG “DIAPASoN”, GA 818616.

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Notes

  1. 1.

    Here, we will focus on this structure as a “standard model” of \(\mathsf {MQPA}\), leaving the study of alternative models for future work.

  2. 2.

    For the sake of readability, \(F\) has been written with a little abuse of notation the actual \(\mathsf {MQPA}\) formula being \(\forall x.\mathbf {C}^{1/z}( \mathrm {EXP}(z,x) \wedge \forall {y}.( \exists w.(y+w=x)\rightarrow \mathsf {FLIP}(y))\), where \(\mathrm {EXP}(z,x)\) is an arithmetical formula expressing \(z=2^{x}\) and \(\exists w.y+w=x\) expresses \(y\le x\).

  3. 3.

    For further details, see [2], where the model theory and proof theory of an extension of propositional logic with counting quantifiers is studied (in particular, the logic \(\mathsf {CPL}_{0}\) can be seen as a “finitary” fragment of \(\mathsf {MQPA}\)).

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Antonelli, M., Dal Lago, U., Pistone, P. (2021). On Measure Quantifiers in First-Order Arithmetic. In: De Mol, L., Weiermann, A., Manea, F., Fernández-Duque, D. (eds) Connecting with Computability. CiE 2021. Lecture Notes in Computer Science(), vol 12813. Springer, Cham. https://doi.org/10.1007/978-3-030-80049-9_2

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