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

\( \mathcal{L} \)-structure

In model theory, the definition of a language involves syntax only. An \( \mathcal{L} \)-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.

\( \mathcal{L}\)-structure

Let \( \mathcal{L} \) be a language. An \( \mathcal{L} \)-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 [\( \; ? \, \to \, ? \; \)], where \( n \) is the arity of the function.

An \( \mathcal{L} \)-structure is often denoted by the Fraktur symbol \( \mathfrak{U} \).

\( c \) is a symbol often used to represent some "constant" symbol of a language, and \( c^{\mathfrak{U}} \) is often used to represent an element of the universe mapped to by the symbol represented by \( c \). For the similar purposes, the symbols \( f \), \( f^{\mathfrak{U}} \), \( R \) and \( R^{\mathfrak{U}} \) are used.

An \( \mathcal{L} \)-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 \( \mathcal{L} \) be a language, and let \( \mathfrak{U} \) be an \( \mathcal{L} \)-structure for the language. Let \( A \) be the universe of \( \mathfrak{U} \).

A variable assignment function is a mapping from the [what?] of \( \mathcal{L} \) to [what?].