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Structure and definability in general bounded arithmetic theories

Chris Pollett
1999 Annals of Pure and Applied Logic  
Kolaitis p i (A) ({||'||}) from P p i (A) ({||'|| 2 }) for behaved ' 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] .  ... 
doi:10.1016/s0168-0072(99)00008-1 fatcat:mwyiqnncjra2ffnelkxp2qu424

Expressing versus Proving: Relating Forms of Complexity in Logic

A. Kolokolova
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).  ... 
doi:10.1093/logcom/exq008 fatcat:lpla27ghunf5fmmvdanaoasaei

Queries with arithmetical constraints

Stéphane Grumbach, Jianwen Su
1997 Theoretical Computer Science  
We prove 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).  ... 
doi:10.1016/s0304-3975(96)00194-6 fatcat:2l24uv556reahd6xv2u57wxgqu

Closure Properties of Weak Systems of Bounded Arithmetic [chapter]

Antonina Kolokolova
2005 Lecture Notes in Computer Science  
This work 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.  ... 
doi:10.1007/11538363_26 fatcat:wptivaza2nbmbkcrdmnig3eo6e

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]

Ján Kľuka, Paul J. Voda
2017 arXiv   pre-print
We then explore the space of subexponential 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 , ⊆ .  ... 
arXiv:1705.05459v1 fatcat:fsiz2psexzdrlpkpxwlxy3co5m

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

Morteza Moniri
2006 Journal of Logic and Computation  
This implies the unprovability of the scheme ¬¬PIND(Σ b+ 1 ) 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).  ... 
doi:10.1093/logcom/exi085 fatcat:gxdlsvxk6fgjbnegws5ua7ap74

Expansions of pseudofinite structures and circuit and proof complexity [article]

Jan Krajicek
2015 arXiv   pre-print
I shall describe a 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.  ... 
arXiv:1505.00118v1 fatcat:yba63k2rkbgo5kmsj73yczq5fq

On the Power of Ordering in Linear Arithmetic Theories

Dmitry Chistikov, Christoph Haase, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming  
We study the problems of deciding whether a relation 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.  ... 
doi:10.4230/lipics.icalp.2020.119 dblp:conf/icalp/ChistikovH20 fatcat:ataa2j72pzbbxjnp6acv7c4uwm

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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