Complexity class

complexity classescomputational complexityclasses
Many complexity classes can be characterized in terms of the mathematical logic needed to express them; see descriptive complexity. The most commonly used problems are decision problems. However, complexity classes can be defined based on function problems (an example is FP), counting problems (e.g. #P), optimization problems, promise problems, etc. The most common model of computation is the deterministic Turing machine, but many complexity classes are based on nondeterministic Turing machines, boolean circuits, quantum Turing machines, monotone circuits, etc.

Seven-number summary

Bowley's seven-figure summaryseven-figure summary
In descriptive statistics, the seven-number summary is a collection of seven summary statistics, and is an extension of the five-number summary. There are two similar, common forms. As with the five-number summary, it can be represented by a modified box plot, adding hatch-marks on the "whiskers" for two of the additional numbers. The following percentiles are evenly spaced under a normally distributed variable: The middle three values – the lower quartile, median, and upper quartile – are the usual statistics from the five-number summary and are the standard values for the box in a box plot.

Finite model theory

finite modelsfinite model
Thus the main application areas of FMT are: descriptive complexity theory, database theory and formal language theory. FMT is mainly about discrimination of structures. The usual motivating question is whether a given class of structures can be described (up to isomorphism) in a given language. For instance, can all cyclic graphs be discriminated (from the non-cyclic ones) by a sentence of the first-order logic of graphs? This can also be phrased as: is the property "cyclic" FO expressible?.

Query (complexity)

In descriptive complexity, a query is a mapping from structures of one signature to structures of another vocabulary. Neil Immerman, in his book Descriptive Complexity, "use[s] the concept of query as the fundamental paradigm of computation" (p. 17). Given signatures \sigma and \tau, we define the set of structures on each language, and. A query is then any mapping Computational complexity theory can then be phrased in terms of the power of the mathematical logic necessary to express a given query. A query is order-independent if the ordering of objects in the structure does not affect the results of the query.

PH (complexity)

PHpolynomial hierarchy PHpolynomial hierarchy
In computational complexity theory, the complexity class PH is the union of all complexity classes in the polynomial hierarchy:

Neil Immerman

He is one of the key developers of descriptive complexity, an approach he is currently applying to research in model checking, database theory, and computational complexity theory. Professor Immerman is an editor of the SIAM Journal on Computing and of Logical Methods in Computer Science. He received B.S. and M.S. degrees from Yale University in 1974 and his Ph.D. from Cornell University in 1980 under the supervision of Juris Hartmanis, a Turing award winner at Cornell. His book "Descriptive Complexity" appeared in 1999.

Computational complexity theory

computational complexitycomplexity theorycomplexity
Descriptive complexity theory. Game complexity. List of complexity classes. List of computability and complexity topics. List of important publications in theoretical computer science. List of unsolved problems in computer science. Parameterized complexity. Proof complexity. Quantum complexity theory. Structural complexity theory. Transcomputational problem.


AC 0 ACAC''' 0
From a descriptive complexity viewpoint, DLOGTIME-uniform AC 0 is equal to the descriptive class FO+BIT of all languages describable in first-order logic with the addition of the BIT predicate, or alternatively by FO(+, ×), or by Turing machine in the logarithmic hierarchy. In 1984 Furst, Saxe, and Sipser showed that calculating the parity of an input cannot be decided by any AC 0 circuits, even with non-uniformity. It follows that AC 0 is not equal to NC 1, because a family of circuits in the latter class can compute parity. More precise bounds follow from switching lemma. Using them, it has been shown that there is an oracle separation between the polynomial hierarchy and PSPACE.

NP (complexity)

NPnondeterministic polynomial timeclass NP
In terms of descriptive complexity theory, NP corresponds precisely to the set of languages definable by existential second-order logic (Fagin's theorem). NP can be seen as a very simple type of interactive proof system, where the prover comes up with the proof certificate and the verifier is a deterministic polynomial-time machine that checks it. It is complete because the right proof string will make it accept if there is one, and it is sound because the verifier cannot accept if there is no acceptable proof string.

SO (complexity)

Second-ordersecond-order logicsecond-order logic over finite structures
In descriptive complexity we can see that the languages recognised by SO formulae are exactly equal to the languages decided by Turing machines in the polynomial hierarchy. Extensions of SO with some operators also give us the same expressivity given by some well known complexity class, so it is a way to do proofs about the complexity of some problems without having to go to the algorithmic level. We define second-order variable, a SO variable has got an arity k and represent any proposition of arity k, i.e. a subset of the k-tuples of the universe. They are usually written in upper-case.

Statistical dispersion

Summary statistics. Qualitative variation. Robust measures of scale. Measurement uncertainty.

HO (complexity)

High orderhigh order queries
In descriptive complexity we can see that it is equal to the ELEMENTARY functions. There is a relation between the ith order and non determinist algorithm the time of which is with i-1 level of exponentials. We define high-order variable, a variable of order i>1 has got an arity k and represent any set of k-tuples of elements of order i-1. They are usually written in upper-case and with a natural number as exponent to indicate the order. High order logic is the set of FO formulae where we add quantification over higher-order variables, hence we will use the terms defined in the FO article without defining them again. HO^i is the set of formulae where variable's order are at most i.

Least fixed point

greatest fixed pointleast fixpointfixed point logic
Descriptive Complexity, 1999, Springer-Verlag. Libkin, Leonid. Elements of Finite Model Theory, 2004, Springer.

Mode (statistics)

Descriptive statistics. Moment (mathematics). Summary statistics. Unimodal function.

Spectrum of a sentence

Fagin's theorem is a result in descriptive complexity theory that states that the set of all properties expressible in existential second-order logic is precisely the complexity class NP. It is remarkable since it is a characterization of the class NP that does not invoke a model of computation such as a Turing machine. The theorem was proven by Ronald Fagin in 1974 (strictly, in 1973 in his doctoral thesis). As a corollary, Jones and Selman showed that a set is a spectrum if and only if it is in the complexity class NEXP.

P (complexity)

Ppolynomial timepolynomial-time
In descriptive complexity, P can be described as the problems expressible in FO(LFP), the first-order logic with a least fixed point operator added to it, on ordered structures. In Immerman's 1999 textbook on descriptive complexity, Immerman ascribes this result to Vardi and to Immerman. It was published in 2001 that PTIME corresponds to (positive) range concatenation grammars. Kozen states that Cobham and Edmonds are "generally credited with the invention of the notion of polynomial time." Cobham invented the class as a robust way of characterizing efficient algorithms, leading to Cobham's thesis. However, H. C.


elementary recursiveelementary recursive functionbounded sums and products
In descriptive complexity, ELEMENTARY is equal to the class of high order queries. This means that every language in the ELEMENTARY complexity class can be written as a high order formula that is true only for the elements on the language. More precisely, where ⋯ indicates a tower of exponentiations and is the class of queries that begin with existential quantifiers of th order and then a formula of (i − 1) th order. * Rose, H.E., Subrecursion: Functions and hierarchies, Oxford University Press, 1984. ISBN: 0-19-853189-3 1) Zero function. Returns zero: f(x) = 0. 2) Successor function: f(x) = x + 1. Often this is denoted by S, as in S(x).


percentiles50th percentile85th percentile speed
Summary statistics. Percentile rank. Free Online Software (Calculator) computes Percentiles for any dataset according to 8 different percentile definitions. Percentiles: Measures of Relative Standing of an observation in data set. Percentiles for grouped and ungrouped data.