First Order Bounded Arithmetic and Small Boolean Circuit Complexity Classes

In [1] S. Buss introduced systems of bounded arithmetic , , , (i = 1, 2, 3, …). and are first order systems and and are second order systems. and are closely related to and respectively in the polynomial hierarchy, and and are closely related to PSPACE and EXPTIME respectively. One of the most important problems in bounded arithmetic is whether the hierarchy of bounded arithmetic collapses, i.e. whether = or = for some i, or whether = , or whether is a conservative extension of S2 = ⋃i. These problems are relevant to the problems whether the polynomial hierarchy PH collapses or whether PSPACE = PH or whether PSPACE = EXPTIME. It was shown in [4] that = implies and consequently the collapse of the polynomial hierarchy. We believe that the separation problems of bounded arithmetic and the separation problems of computational complexities are essentially the same problem, and the solution of one of them will lead to the solution of the other.

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Circuit complexity and the expressive power of generalized first-order formulas

Automata, Languages and Programming - Lecture Notes in Computer Science ◽

10.1007/3-540-55719-9_60 ◽

1992 ◽

pp. 16-27 ◽

Cited By ~ 4

Author(s):

Howard Straubing

Keyword(s):

Circuit Complexity ◽

Expressive Power ◽

First Order

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From Bounded Arithmetic to Memory Management: Use of Type Theory to Capture Complexity Classes and Space Behaviour

Lecture Notes in Computer Science - Typed Lambda Calculi and Applications ◽

10.1007/3-540-45413-6_2 ◽

2001 ◽

pp. 2-3

Author(s):

Martin Hofmann

Keyword(s):

Type Theory ◽

Memory Management ◽

Complexity Classes ◽

Bounded Arithmetic

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Low complexity algorithms in knot theory

International Journal of Algebra and Computation ◽

10.1142/s0218196718500698 ◽

2019 ◽

Vol 29 (02) ◽

pp. 245-262

Author(s):

Olga Kharlampovich ◽

Alina Vdovina

Keyword(s):

Computational Complexity ◽

Time Complexity ◽

Linear Time ◽

Circuit Complexity ◽

Equivalence Problem ◽

Low Complexity ◽

Combinatorial Structure ◽

Complexity Classes ◽

Np Complete ◽

Almost All

Agol, Haas and Thurston showed that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. This shows that (unless P[Formula: see text]NP) the genus problem has high computational complexity even for knots in a 3-manifold. We initiate the study of classes of knots where the genus problem and even the equivalence problem have very low computational complexity. We show that the genus problem for alternating knots with n crossings has linear time complexity and is in Logspace[Formula: see text]. Alternating knots with some additional combinatorial structure will be referred to as standard. As expected, almost all alternating knots of a given genus are standard. We show that the genus problem for these knots belongs to [Formula: see text] circuit complexity class. We also show, that the equivalence problem for such knots with [Formula: see text] crossings has time complexity [Formula: see text] and is in Logspace[Formula: see text] and [Formula: see text] complexity classes.

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y = 2x VS. y = 3x

Journal of Symbolic Logic ◽

10.2307/2275554 ◽

1997 ◽

Vol 62 (2) ◽

pp. 661-672 ◽

Cited By ~ 1

Author(s):

Alexei Stolboushkin ◽

Damian Niwiński

Keyword(s):

Circuit Complexity ◽

Order Logic ◽

First Order Logic ◽

Linear Ordering ◽

Finite Model ◽

Logical Relation ◽

Open Problems ◽

First Order ◽

Standing Question

AbstractWe show that no formula of first order logic using linear ordering and the logical relation y = 2x can define the property that the size of a finite model is divisible by 3. This answers a long-standing question which may be of relevance to certain open problems in circuit complexity.

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CATEGORICAL COMPLEXITY

Forum of Mathematics Sigma ◽

10.1017/fms.2020.26 ◽

2020 ◽

Vol 8 ◽

Author(s):

SAUGATA BASU ◽

UMUT ISIK

Keyword(s):

Complexity Theory ◽

Boolean Functions ◽

Circuit Complexity ◽

Polynomial Rings ◽

Common Interest ◽

Vector Spaces ◽

Topological Spaces ◽

Complexity Classes ◽

Graded Algebras ◽

Projective Varieties

We introduce a notion of complexity of diagrams (and, in particular, of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest such as finite sets, Boolean functions, topological spaces, vector spaces, semilinear and semialgebraic sets, graded algebras, affine and projective varieties and schemes, and modules over polynomial rings. We show that on one hand categorical complexity recovers in several settings classical notions of nonuniform computational complexity (such as circuit complexity), while on the other hand it has features that make it mathematically more natural. We also postulate that studying functor complexity is the categorical analog of classical questions in complexity theory about separating different complexity classes.

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Delineating classes of computational complexity via second order theories with weak set existence principles. I

Journal of Symbolic Logic ◽

10.2307/2275511 ◽

1995 ◽

Vol 60 (1) ◽

pp. 103-121 ◽

Cited By ~ 1

Author(s):

Aleksandar Ignjatović

Keyword(s):

Computational Complexity ◽

Polynomial Time ◽

Second Order ◽

Complexity Classes ◽

Bounded Arithmetic ◽

Recursive Functions ◽

Speed Up ◽

Bounded Complexity ◽

Polynomial Time Hierarchy ◽

Comprehension Axiom

AbstractIn this paper we characterize the well-known computational complexity classes of the polynomial time hierarchy as classes of provably recursive functions (with graphs of suitable bounded complexity) of some second order theories with weak comprehension axiom schemas but without any induction schemas (Theorem 6). We also find a natural relationship between our theories and the theories of bounded arithmetic (Lemmas 4 and 5). Our proofs use a technique which enables us to “speed up” induction without increasing the bounded complexity of the induction formulas. This technique is also used to obtain an interpretability result for the theories of bounded arithmetic (Theorem 4).

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Finite Variable Logics in Descriptive Complexity Theory

Bulletin of Symbolic Logic ◽

10.2307/420954 ◽

1998 ◽

Vol 4 (4) ◽

pp. 345-398 ◽

Cited By ~ 17

Author(s):

Martin Grohe

Keyword(s):

Model Theory ◽

Order Logic ◽

First Order Logic ◽

Complexity Classes ◽

Finite Model ◽

Polynomial Space ◽

Finite Model Theory ◽

First Order ◽

First Order Theory ◽

Finite Structures

Throughout the development of finite model theory, the fragments of first-order logic with only finitely many variables have played a central role. This survey gives an introduction to the theory of finite variable logics and reports on recent progress in the area.For each k ≥ 1 we let Lk be the fragment of first-order logic consisting of all formulas with at most k (free or bound) variables. The logics Lk are the simplest finite-variable logics. Later, we are going to consider infinitary variants and extensions by so-called counting quantifiers.Finite variable logics have mostly been studied on finite structures. Like the whole area of finite model theory, they have interesting model theoretic, complexity theoretic, and combinatorial aspects. For finite structures, first-order logic is often too expressive, since each finite structure can be characterized up to isomorphism by a single first-order sentence, and each class of finite structures that is closed under isomorphism can be characterized by a first-order theory. The finite variable fragments seem to be promising candidates with the right balance between expressive power and weakness for a model theory of finite structures. This may have motivated Poizat [67] to collect some basic model theoretic properties of the Lk. Around the same time Immerman [45] showed that important complexity classes such as polynomial time (PTIME) or polynomial space (PSPACE) can be characterized as collections of all classes of (ordered) finite structures definable by uniform sequences of first-order formulas with a fixed number of variables and varying quantifier-depth.

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Information Flows, Graphs and their Guessing Numbers

The Electronic Journal of Combinatorics ◽

10.37236/962 ◽

2007 ◽

Vol 14 (1) ◽

Cited By ~ 20

Author(s):

Søren Riis

Keyword(s):

Network Coding ◽

Circuit Complexity ◽

Information Flows ◽

Boolean Circuit ◽

Graph Theoretic ◽

Common Information ◽

Shift Problem ◽

Cyclic Shifts ◽

The Common ◽

Multiuser Information Theory

Valiant's shift problem asks whether all $n$ cyclic shifts on $n$ bits can be realized if $n^{(1+\epsilon)}$ input output pairs ($\epsilon < 1$) are directly connected and there are additionally $m$ common bits available that can be arbitrary functions of all the inputs. If it could be shown that this is not realizable with $m = O({n\over \log \log n})$ common bits then a significant breakthrough in Boolean circuit complexity would follow. In this paper it is shown that in certain cases all cyclic shifts are realizable with $m=(n- n^{\epsilon})/2$ common bits. Previously, no solution with $m < n - o(n)$ was known, and Valiant had conjectured that $m < n/2$ was not achievable. The construction therefore establishes a novel way of realizing communication in the manner of Network Coding for the shift problem, but leaves the viability of the common information approach to proving lower bounds in Circuit Complexity open. The construction uses the graph-theoretic notion of guessing number. As a by-product the paper also establish an interesting link between Circuit Complexity and Network Coding, a new direction of research in multiuser information theory.

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