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?].