Term

Term of Language

A finite sequence (or string) of symbols is a term of $\mathcal{L}$ if and only if it is one of the following forms:

1. A variable $v$ or constant symbol $c$;
2. $f(t_1,\dots ,t_n)$, where $f$ is an $n$-ary function symbol of $\mathcal{L}$, and each $t_i$ is a term. Alternatively, we can define terms set-theoretically: The set of terms of L is the least set S of finite sequences such that: (i) each variable and constant symbol belongs to S; (ii) if $f$ is an $n$-ary function symbol of $\mathcal{L}$ and $t_1,\dots ,t_n\in S$, then $f(t_1,\dots ,t_n)\in S$.

Term Complexity function

For terms define the term complexity function $h$ by: $h(v)=h(c)=1$, where $v$ is a variable symbol and $c$ is a constant symbol $h(f(t_1\dots t_n))=1+\max \{h(t_1)\dots h(t_n)\}$ where $f(t_1\dots t_n)$ is a term.