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Structure and definability in general bounded arithmetic theories

1999
*
Annals of Pure and Applied Logic
*

Kolaitis p i (A) ({||'||}) from P p i (A) ({||'|| 2 }) for behaved '

doi:10.1016/s0168-0072(99)00008-1
fatcat:mwyiqnncjra2ffnelkxp2qu424
*and*deduce*theory*separations. We lastly introduce a notion of a model separating two*theories**and*derive some consequences. (C. ... The author would like to thank Steve Fenner, Lance Fortnow,*and*Steve Homer for the references Buhrman*and*Torenvliet [4]*and*Mocas [20]*and*would also like to thank Lane Hemaspaandra*and*Harold Hempel ... Acknowledgements This paper consists of both a revision*and*extension of some of the results from the author's dissertation [23] . ...##
###
Expressing versus Proving: Relating Forms of Complexity in Logic

2010
*
Journal of Logic and Computation
*

*In*particular, we show how the complexity of logics

*in*the setting of finite model

*theory*is used to obtain results

*in*

*bounded*

*arithmetic*, stating which functions are provably total

*in*certain weak systems ... of

*arithmetic*. ... The results on connections between finite model

*theory*

*and*

*bounded*

*arithmetic*mentioned here come from my joint work with Stephen Cook (which resulted

*in*my PhD thesis). ...

##
###
Queries with arithmetical constraints

1997
*
Theoretical Computer Science
*

We prove

doi:10.1016/s0304-3975(96)00194-6
fatcat:2l24uv556reahd6xv2u57wxgqu
*in*particular that linear queries can be evaluated*in*AC0 over finite integer databases,*and**in*NC' over linear constraint databases. This improves previously known*bounds*. ... We prove that these two queries are first-order*definable**in*the presence of (enough)*arithmetic*. ... to y*in*G. dy 'dz Vk Vk' Vk" Theorem 4. 8 . 8 Every*generic*query*in*the*arithmetical*hierarchy is*definable**in*the context*structure*(N, <, +, x). ...##
###
Closure Properties of Weak Systems of Bounded Arithmetic
[chapter]

2005
*
Lecture Notes in Computer Science
*

This work

doi:10.1007/11538363_26
fatcat:wptivaza2nbmbkcrdmnig3eo6e
*generalizes*the results of [8]*and*[9] 1 . ...*In*this paper we study the properties of systems of*bounded**arithmetic*capturing small complexity classes*and*state conditions sufficient for such systems to capture the corresponding complexity class ...*In*particular, the main ideas used for the SL*theory*were suggested to me by him. ...##
###
Page 2855 of Mathematical Reviews Vol. , Issue 87f
[page]

1987
*
Mathematical Reviews
*

The

*structures*are similar*in*that every degree d that is the Turing jump of an upper*bound*for ar*bounds*a minimal upper*bound*for ar*and*every minimal upper 87f:03121 03D Recursion*theory*87£:03123*bound*...*In*contrast, not every*generalized*high(ar) degree*bounds*a minimal upper*bound*. ...##
###
First- and Second-Order Models of Recursive Arithmetics
[article]

2017
*
arXiv
*
pre-print

We then explore the space of subexponential

arXiv:1705.05459v1
fatcat:fsiz2psexzdrlpkpxwlxy3co5m
*arithmetic**theories*between IΔ_0*and*IΔ_0(exp). ... We introduce*and*study first-*and*second-order*theories*of recursive*arithmetic*ARA_1*and*ARA_2 capable of characterizing various computational complexity classes*and*based on function algebras A, studied ... We use the last*structure**and*the infinite tree ∈ ⊆ ∸1*in*Lemma 5.7 to obtain a*generic*branch Branch , ⊆ . ...##
###
Page 3220 of Mathematical Reviews Vol. , Issue 95f
[page]

1995
*
Mathematical Reviews
*

The

*theory*of canonical height for abelian varieties has recently been*generalized*to any pair (V,g), where V is a smooth projec- tive variety*defined*over a global field K*and*g an endomorphism of V/K ... Let D be a*bounded*symmetric domain, I an*arithmetic*subgroup of the group AutD of all complex automorphisms of D*and*X(T) be the quotient space D/I’. ...##
###
Page 7680 of Mathematical Reviews Vol. , Issue 2001K
[page]

2001
*
Mathematical Reviews
*

With these motiva- tions

*in*mind the author develops a*theory*of Euler*structures**and*proves a solvability criterion for uniform families of linear polyno- mials,*generalizing*a result of Ajtai from finite ... By interpreting these*theories**in*the systems T,*and*7,* of constructive functionals*in*finite types, the author gives a characterization of the*definable*set functions of CZF~*and*CZF*in*terms of constructive ...##
###
Page 5456 of Mathematical Reviews Vol. , Issue 98I
[page]

1998
*
Mathematical Reviews
*

It is shown

*in*the paper under review that if a is any uniform upper*bound*of the*arithmetic*degrees*and*b is any upper*bound*of the*arithmetic*degrees with b <a, then there is a minimal upper*bound*c ... Then we explain some basic concepts used*in*this field: propositional proof systems*and**bounded**arithmetic*. ...##
###
Page 6026 of Mathematical Reviews Vol. , Issue 2000i
[page]

2000
*
Mathematical Reviews
*

*In*particular a

*definable*set

*in*an o-minimal

*structure*is

*definably*compact (with respect to the subspace topology) if

*and*only if it is closed

*and*

*bounded*. ... of models of

*arithmetic*, as well as its refinements by Gaifman

*and*Phillips, can be

*generalized*to models of set

*theory*, often with the help of large cardinal axioms.” ...

##
###
Page 739 of Mathematical Reviews Vol. , Issue 83b
[page]

1983
*
Mathematical Reviews
*

Therefore

*generalizations*of the numerical spaces mentioned above can be studied as special cases of the*theory**and*all operations*in*the subsets can be*defined*by semimorphism. ...*In*the elaboration of the*theory*the authors were motivated by principles of geometry where the*structures*of the spaces occurring are*defined*as invariants with respect to a group of transformations. ...##
###
An Independence Result for Intuitionistic Bounded Arithmetic

2006
*
Journal of Logic and Computation
*

This implies the unprovability of the scheme ¬¬PIND(Σ b+ 1 )

doi:10.1093/logcom/exi085
fatcat:gxdlsvxk6fgjbnegws5ua7ap74
*in*the mentioned*theory*. However, this*theory*contains the sentence ∀x, y¬¬∃z ≤ y(x ≤ |y| → x = |z|). ... The above independence result is proved by constructing an ω-chain of submodels of a countable model of S 2 + Ω 3 + ¬exp such that none of the worlds*in*the chain satisfies the sentence,*and*interpreting ... Acknowledgement This research was*in*part supported by a grant from IPM (No. CS1383-4-07). ...##
###
Expansions of pseudofinite structures and circuit and proof complexity
[article]

2015
*
arXiv
*
pre-print

I shall describe a

arXiv:1505.00118v1
fatcat:yba63k2rkbgo5kmsj73yczq5fq
*general*model-theoretic task to construct expansions of pseudofinite*structures**and*discuss several examples of particular relevance to computational complexity. ... Paris*and*Dimitracopoulos [19] studied the problem of for how large m > n does the*theory*of the*arithmetic**structure*on [n] determine the*theory*of the*arithmetic**structure*on [m]*and*proved that it ...*bound*was by model*theory*of*arithmetic*although he eventually chose to present the result combinatorially. ...##
###
On the Power of Ordering in Linear Arithmetic Theories

2020
*
International Colloquium on Automata, Languages and Programming
*

We study the problems of deciding whether a relation

doi:10.4230/lipics.icalp.2020.119
dblp:conf/icalp/ChistikovH20
fatcat:ataa2j72pzbbxjnp6acv7c4uwm
*definable*by a first-order formula*in*linear rational or linear integer*arithmetic*with an order relation is*definable**in*absence of the order relation ... Our contribution is to establish a full geometric characterisation of those sets*definable*without order which*in*turn enables us to prove coNP-completeness of this problem over the rationals*and*to establish ... The complexity upper*bounds*we obtain*in*this paper rely on*bounds*on the constants*in*the*generator*representation of sets*definable**in*linear*arithmetic**theories*. ...##
###
Page 2998 of Mathematical Reviews Vol. , Issue 91F
[page]

1991
*
Mathematical Reviews
*

Buss [

*Bounded**arithmetic*, Bibliopolis, Naples, 1986; MR 89h:03104] introduced the hierarchy £° of*bounded*formulas of the language of*arithmetic**and*studied the*theories*S,, T; axiomatized by BASIC+Z°- ...*structures*that are*in*some sense infinitely close to standard*in*- finitary*structures*of classical mathematics. ...
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