Polynomial time ultrapowers and the consistency of circuit lower bounds

A polynomial time ultrapower is a structure given by the set of polynomial time computable functions modulo some ultrafilter. They model the universal theory ∀ PV of all polynomial time functions. Generalizing a theorem of Hirschfeld (Israel J Math 20(2):111–126, 1975 ), we show that every countable...

Celý popis

Uloženo v:
Podrobná bibliografie
Vydáno v:Archive for mathematical logic Ročník 59; číslo 1-2; s. 127 - 147
Hlavní autoři: Bydžovský, Jan, Müller, Moritz
Médium: Journal Article Publikace
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2020
Springer Nature B.V
Témata:
Algebra
Circuits
Informàtica
Informàtica teòrica
Lower bounds
Mathematical analysis
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Original Study
Polinomis
Polynomials
Production scheduling
Substructures
Time functions
Àrees temàtiques de la UPC
03C20
03C98
03B70
Bounded arithmetic
68Q17
Circuit lower bounds
Restricted ultrapowers
ISSN:0933-5846, 1432-0665
On-line přístup:Získat plný text
Tagy: Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
Popis
Shrnutí:A polynomial time ultrapower is a structure given by the set of polynomial time computable functions modulo some ultrafilter. They model the universal theory ∀ PV of all polynomial time functions. Generalizing a theorem of Hirschfeld (Israel J Math 20(2):111–126, 1975 ), we show that every countable model of ∀ PV is isomorphic to an existentially closed substructure of a polynomial time ultrapower. Moreover, one can take a substructure of a special form, namely a limit polynomial time ultrapower in the classical sense of Keisler (in: Bergelson, V., Blass, A., Di Nasso, M., Jin, R. (eds.) Ultrafilters across mathematics, contemporary mathematics vol 530, pp 163–179. AMS, New York, 1963 ). Using a polynomial time ultrapower over a nonstandard Herbrand saturated model of  ∀ PV we show that ∀ PV is consistent with a formal statement of a polynomial size circuit lower bound for a polynomial time computable function. This improves upon a recent result of Krajíček and Oliveira (Logical methods in computer science 13 (1:4), 2017 ).
Bibliografie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0933-5846
1432-0665
DOI:10.1007/s00153-019-00681-y