This book introduces the semantic aspects of natural language processing and its applications. Topics covered include: measuring word meaning similarity, multi-lingual querying, and parametric theory, named entity recognition, semantics, query language, and the nature of language. The book also emphasizes the portions of mathematics needed to understand the discussed algorithms.

is equivalent to A ¢ P(M).(φ ° φ)(A) = φ(A). (2) A subset A ¢ P(A) is closed if and only if A ¢ φ(P(M)). Proof. Suppose A is closed, then A = φ(A), hence A ¢ φ(P(M)). Conversely, if A ¢ φ(P(M)), then A = φ(X), with X ¡ A. Hence φ(A) = (φ ° φ)(X) = φ(X) = A, i.e., A is closed. Relations 39 (3) If we take instead of P(M) an arbitrary ordered set, we can define a closure operator over an ordered set as well. Example 2.6.2. Let (P, w) be an ordered set. A closure operator is given by ↓ [–]: P(P)

definitions of closure system and closure operators can be easily generalized on ordered sets. Definition 2.6.3. A closure operator on an ordered set (P, w) is a map φ: P P, satisfying for every a, b ¢ P the following conditions: (1) Extensivity: a w φ(a); (2) Monotony: If a w b, then φ(a) w φ(b); (3) Idempotency: φ(a) = φ(φ(a)). If a ¢ P, then φ(a) is called the closure of a. An element a ¢ P is called closed if a = φ(a). Remark 13 The subset of P consisting of all closed elements with respect

operators can all be defined and built in a vector space such as WORDSPACE and these are the natural logical operations in this space, just as the Boolean logic gates are the natural logical operations for set theory. Classical set theory is binary and discrete, whereas vector spaces are smooth and continuous. You find this difference between continuous and discrete interpretations of logic in very simple sentences. If somebody says “you should take bus 60 or bus 70,” and you later discover that

Formally, a Turing machine is denoted M = (S, Σ, Γ, δ, s0 , B, F), where • • • • • S is a finite set of states, Γ is the finite set of allowable tape symbols, B ¢ Γ is the blank symbol, Σ Γ not including B is the set of input symbols, δ, is the next move function, a mapping S × Γ → S × Γ × {L, R}, with δ maybe undefined for some arguments and L is the move of the head to the left, R is the move of the head to the right, • s0 is a particular state of S called the initial state, • F ¡ S is the

of constructing a query when the meaning of the entry term has already been clarified completely. For instance, starting with the term purpose, i.e., let’s talk about “purpose”, the system suggests Mines clearing action and Explosive devices as potential clarifications of ”purpose”, i.e., to give an answer to the question are we talking about the purpose of Mines clearing action or that of Explosive devices? before proceeding with the further refinement of the query/question. BEGIN Algorithm 2.0