Second-order logic

second-ordersecond order logicexistential second-order logic
The field of descriptive complexity studies which computational complexity classes can be characterized by the power of the logic needed to express languages (sets of finite strings) in them. A string w = w 1 ···w n in a finite alphabet A can be represented by a finite structure with domain D = {1,...,n}, unary predicates P a for each a ∈ A, satisfied by those indices i such that w i = a, and additional predicates which serve to uniquely identify which index is which (typically, one takes the graph of the successor function on D or the order relation


Description Logic Modeling. metadata. Web Ontology Language.

Univariate analysis

Univariatesingle variable
Like other forms of statistics, it can be inferential or descriptive. The key fact is that only one variable is involved. Descriptive statistics describe a sample or population. They can be part of exploratory data analysis. The appropriate statistic depends on the level of measurement. For nominal variables, a frequency table and a listing of the mode(s) is sufficient. For ordinal variables the median can be calculated as a measure of central tendency and the range (and variations of it) as a measure of dispersion.


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.


In ontology engineering, the term semantics refers to the meaning of concepts, properties, and relationships that formally represent real-world entities, events, and scenes in a logical underpinning, such as a description logic, and typically implemented in the Web Ontology Language. The meaning of description logic concepts and roles is defined by their model-theoretic semantics, which are based on interpretations. The concepts, properties, and relationships defined in OWL ontologies can be deployed directly in the web site markup as RDFa, HTML5 Microdata, or JSON-LD, in graph databases as RDF triples or quads, and dereferenced in LOD datasets.

Semantic parameterization

All but the when questions, which require a Temporal Logic to represent such phenomena, have been aligned with the meta-model in semantic parameterization using Description Logic (DL). The semantic parameterization process is based on Description Logic, wherein the TBox is composed of words in a dictionary, including nouns, verbs, and adjectives, and the ABox is partitioned into two sets of assertions: 1) those assertions that come from words in the natural language statement, called the grounding, and 2) those assertions that are inferred by the (human) modeler, called the meta-model.


Description Logic#Modeling. metadata. Web Ontology Language.

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.

Arithmetic mean

Summary statistics.

Ian Horrocks

His research focuses on knowledge representation and reasoning, particularly ontology languages, description logic and optimised tableaux decision procedures. Horrocks completed his Bachelor of Science, Master of Science (1995) and Doctor of Philosophy (1997) degrees in the School of Computer Science at the University of Manchester. After several years as a lecturer, senior lecturer, Reader then Professor in Manchester, he moved to the University of Oxford in 2008. His work on tableau reasoning for very expressive description logics has formed the basis of most description logic reasoning systems in use today, including Racer, FaCT++, HermiT and Pellet.

Semantic reasoner

reasonerreasoning enginereasoning
Introduction to Description Logics DL course by Enrico Franconi, Faculty of Computer Science, Free University of Bolzano, Italy. Inference using OWL 2.0 Semantics by Craig Trim (IBM). Marko Luther, Thorsten Liebig, Sebastian Böhm, Olaf Noppens: Who the Heck Is the Father of Bob?. ESWC 2009: 66-80. Jurgen Bock, Peter Haase, Qiu Ji, Raphael Volz. Benchmarking OWL Reasoners. In ARea2008 - Workshop on Advancing Reasoning on the Web: Scalability and Commonsense (June 2008). Tom Gardiner, Ian Horrocks, Dmitry Tsarkov. Automated Benchmarking of Description Logic Reasoners. Description Logics Workshop 2006. OpenRuleBench Senlin Liang, Paul Fodor, Hui Wan, Michael Kifer.

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.

Two-variable logic

two-variable logic with counting
This result generalizes results about the decidability of fragments of two-variable logic, such as certain description logics; however, some fragments of two-variable logic enjoy a much lower computational complexity for their satisfiability problems. By contrast, for the three-variable fragment of first-order logic without function symbols, satisfiability is undecidable. The two-variable fragment of first-order logic with no function symbols is known to be decidable even with the addition of counting quantifiers, and thus of uniqueness quantification. This is a more powerful result, as counting quantifiers for high numerical values are not expressible in that logic.


probability weighted moments
*L-estimator 1) As summary statistics for data. 2) To derive estimators for the parameters of probability distributions, applying the method of moments to the L-moments rather than conventional moments. The L-moments page Jonathan R.M. Hosking, IBM Research. L Moments. Dataplot reference manual, vol. 1, auxiliary chapter. National Institute of Standards and Technology, 2006. Accessed 2010-05-25.

Modal logic

Description logic. Doxastic logic. Dynamic logic. Enthymeme. Hybrid logic. Interior algebra. Interpretability logic. Kripke semantics. Metaphysical necessity. Modal verb. Multi-valued logic. Possible worlds. Provability logic. Regular modal logic. Relevance logic. Research Materials: Max Planck Society Archive. Rhetoric. Strict conditional. Two dimensionalism. This article includes material from the Free On-line Dictionary of Computing, used with permission under the GFDL. Barcan-Marcus, Ruth JSL 11 (1946) and JSL 112 (1947) and "Modalities", OUP, 1993, 1995.

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.

Exploratory data analysis

explorative data analysisexploratorydata analysis
Descriptive statistics. Young, F. W. Valero-Mora, P. and Friendly M. (2006) Visual Statistics: Seeing your data with Dynamic Interactive Graphics. Wiley ISBN: 978-0-471-68160-1. Jambu M. (1991) Exploratory and Multivariate Data Analysis. Academic Press ISBN: 0123800900. S. H. C. DuToit,A. G. W. Steyn,R. H. Stumpf (1986) Graphical Exploratory Data Analysis. Springer ISBN: 978-1-4612-9371-2.


polynomial spaceAPhard
A logical characterization of PSPACE from descriptive complexity theory is that it is the set of problems expressible in second-order logic with the addition of a transitive closure operator. A full transitive closure is not needed; a commutative transitive closure and even weaker forms suffice. It is the addition of this operator that (possibly) distinguishes PSPACE from PH. A major result of complexity theory is that PSPACE can be characterized as all the languages recognizable by a particular interactive proof system, the one defining the class IP. In this system, there is an all-powerful prover trying to convince a randomized polynomial-time verifier that a string is in the language.

Formal concept analysis

concept latticeconcept analysisconcept learning
Description logic. Factor analysis. Graphical model. Grounded theory. Inductive logic programming. Pattern theory. Statistical relational learning. Schema (genetic algorithms). A Formal Concept Analysis Homepage. Demo. 11th International Conference on Formal Concept Analysis. ICFCA 2013 – Dresden, Germany – May 21–24, 2013.


elementary recursiveelementary recursive functionbounded sums and products
The class of lower elementary functions has a description in terms of composition of simple functions analogous to that we have for elementary functions. Namely, a polynomial-bounded function is lower elementary if and only if it can be expressed using a composition of the following functions: projections, n+1, nm, n\wedge m,, one exponential function (2^n or n^m) with the following restriction on the structure of formulas: the formula can have no more than two floors with respect to an exponent (for example, xy(z+1) has 1 floor, has 2 floors, 2^{2^x} has 3 floors). Here n\wedge m is a bitwise AND of and. In descriptive complexity, ELEMENTARY is equal to the class of high order queries.


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.

Knowledge representation and reasoning

knowledge representationrepresentationknowledge
Introduction to Description Logics course by Enrico Franconi, Faculty of Computer Science, Free University of Bolzano, Italy. DATR Lexical knowledge representation language. Loom Project Home Page. Description Logic in Practice: A CLASSIC Application. The Rule Markup Initiative. Nelements KOS - a non-free 3d knowledge representation system.