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Math and science::Theory of Computation::Modal theory

L-structure

In model theory, the definition of a language involves syntax only. An L-structure moves beyond syntax and gives a first-order language a set theoretic interpretation.

The back side has a recap for concepts referenced in the below definition.

L-structure

Let L be a language. An L-structure consists of the following:

  1. A [something], which is called the universe.
  2. A map, from constant symbols → [to what?].
  3. A map, from relation symbols → relations over [what sets?], where n is the arity of the relation.
  4. A map, from function symbols → functions with signature [??], where n is the arity of the function.

An L-structure is often denoted by the Fraktur symbol U.

c is a symbol often used to represent some "constant" symbol of a language, and cU is often used to represent an element of the universe mapped to by the symbol represented by c. For the similar purposes, the symbols f, fU, R and RU are used.

An L-structure doesn't give an interpretation to the variable symbols of a language. Variable symbols are mapped to elements of the universe by a variable assignment function.

Variable assignment function

Let L be a language, and let U be an L-structure for the language. Let A be the universe of U.

A variable assignment function is a mapping from the [what?] of L to [what?].