Term Language

logic

Definition

Term Language

A term langauge is built from variables and function
symbols. They are a general representation of hierarchical structures.

The set $\mathcal{T}(\mathcal{F},\mathcal{V})$, short $\mathcal{T}$ is the
smallest set of terms over $\mathcal{F}$ and $\mathcal{V}$,
such that:

$$\begin{array}{rl}\mathcal{V}\subseteq \mathcal{T}(\mathcal{F},\mathcal{V})& \\ f\in \mathcal{T}(\mathcal{F},\mathcal{V})& ⟺f/0\in \mathcal{F}\\ f(t_1,\dots ,t_n)\in \mathcal{T}(\mathcal{F},\mathcal{V})& ⟺f/n\in \mathcal{F}\wedge t_1,\dots ,t_n\in \mathcal{T}(\mathcal{F},\mathcal{V})\end{array}$$

Note that $f/0$ denotes the arity of $f$.

Semantics

Interpretation $I$ of a Term Language

Semantics are introduced using an interpretation $I$.

The interpretation $I$ over a domain $\mathcal{U}$, called the universe, maps
each function symbol of $f\in \mathcal{F}$ to a function $I(f)\in \mathcal{U}$:

$$\begin{array}{rl}I(f)\in \mathcal{U}& ⟺f/0\in \mathcal{F}\\ I(f):{\mathcal{U}}^n\to \mathcal{U}& ⟺f/n\in \mathcal{F},n>0\end{array}$$

Note that $f/n$ denotes the arity of $f$.

The values of variable symbols depend on the current value assignments.

Notations

Prefix Notation

Definition

Prefix Notation

Prefix notation is a notation for expressions of a
term languages where the operators
are placed before the terms.

Example:

$$\wedge ab$$

where $a,b$ are terms and $\wedge /2$ is an operator.

The prefix notation is primarily used for functions
and predicates.

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Postfix Notation

Definition

Postfix Notation

Postfix notation is a notation for expressions of a
term languages where the operators
are placed after the terms.

Example:

$$ab\wedge$$

where $a,b$ are terms and $\wedge /2$ is an operator.

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Infix Notation

Definition

Infix Notation

Infix notation is a notation for expressions of a
term languages where the operators
are placed between the terms.

Example:

$$a\wedge b$$

where $a,b$ are terms and $\wedge /2$ is an operator.

The infix notation is primarily used for operators.

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