Term structure vs. representation

Originally, logicians defined a term to be a character string adhering to certain building rules.^[2] However, since the concept of tree became popular in computer science, it turned out to be more convenient to think of a term as a tree. For example, several distinct character strings, like "(n⋅(n+1))/2", "((n⋅(n+1)))/2", and "${\displaystyle {\frac {n(n+1)}{2}}}$", denote the same term and correspond to the same tree, viz. the left tree in the above picture. Separating the tree structure of a term from its graphical representation on paper, it is also easy to account for parentheses (being only representation, not structure) and invisible multiplication operators (existing only in structure, not in representation).

Structural equality

Two terms are said to be structurally, literally, or syntactically equal if they correspond to the same tree. For example, the left and the right tree in the above picture are structurally unequal terms, although they might be considered "semantically equal" as they always evaluate to the same value in rational arithmetic. While structural equality can be checked without any knowledge about the meaning of the symbols, semantic equality cannot. If the function / is e.g. interpreted not as rational but as truncating integer division, then at n=2 the left and right term evaluates to 3 and 2, respectively. Structural equal terms need to agree in their variable names.

In contrast, a term t is called a renaming, or a variant, of a term u if the latter resulted from consistently renaming all variables of the former, i.e. if u = tσ for some renaming substitution σ. In that case, u is a renaming of t, too, since a renaming substitution σ has an inverse σ⁻¹, and t = uσ⁻¹. Both terms are then also said to be equal modulo renaming. In many contexts, the particular variable names in a term don't matter, e.g. the commutativity axiom for addition can be stated as x+y=y+x or as a+b=b+a; in such cases the whole formula may be renamed, while an arbitrary subterm usually may not, e.g. x+y=b+a is not a valid version of the commutativity axiom.^{[note 1] [note 2]}

Ground and linear terms

The set of variables of a term t is denoted by vars(t). A term that doesn't contain any variables is called a ground term; a term that doesn't contain multiple occurrences of a variable is called a linear term. For example, 2+2 is a ground term and hence also a linear term, x⋅(n+1) is a linear term, n⋅(n+1) is a non-linear term. These properties are important in, for example, term rewriting.

Given a signature for the function symbols, the set of all terms forms the free term algebra. The set of all ground terms forms the initial term algebra.

Abbreviating the number of constants as f₀, and the number of i-ary function symbols as f_i, the number θ_h of distinct ground terms of a height up to h can be computed by the following recursion formula:

• θ₀ = f₀, since a ground term of height 0 can only be a constant,
• ${\displaystyle \theta _{h+1}=\sum _{i=0}^{\infty }f_{i}\cdot \theta _{h}^{i}}$, since a ground term of height up to h+1 can be obtained by composing any i ground terms of height up to h, using an i-ary root function symbol. The sum has a finite value if there is only a finite number of constants and function symbols, which is usually the case.

Building formulas from terms

Given a set R_n of n-ary relation symbols for each natural number n ≥ 1, an (unsorted first-order) atomic formula is obtained by applying an n-ary relation symbol to n terms. As for function symbols, a relation symbol set R_n is usually non-empty only for small n. In mathematical logic, more complex formulas are built from atomic formulas using logical connectives and quantifiers. For example, letting ${\displaystyle \mathbb {R} }$ denote the set of real numbers, ∀x: x ∈ ${\displaystyle \mathbb {R} }$ ⇒ (x+1)⋅(x+1) ≥ 0 is a mathematical formula evaluating to true in the algebra of complex numbers. An atomic formula is called ground if it is built entirely from ground terms; all ground atomic formulas composable from a given set of function and predicate symbols make up the Herbrand base for these symbol sets.

Sorted terms

When the domain of discourse contains elements of basically different kinds, it is useful to split the set of all terms accordingly. To this end, a sort (sometimes also called type) is assigned to each variable and each constant symbol, and a declaration ^{[note 3]} of domain sorts and range sort to each function symbol. A sorted term f(t₁,...,t_n) may be composed from sorted subterms t₁,...,t_n only if the ith subterm's sort matches the declared ith domain sort of f. Such a term is also called well-sorted; any other term (i.e. obeying the unsorted rules only) is called ill-sorted.

For example, a vector space comes with an associated field of scalar numbers. Let W and N denote the sort of vectors and numbers, respectively, let V_W and V_N be the set of vector and number variables, respectively, and C_W and C_N the set of vector and number constants, respectively. Then e.g. ${\displaystyle {\vec {0}}\in C_{W}}$ and 0 ∈ C_N, and the vector addition, the scalar multiplication, and the inner product is declared as ${\displaystyle +:W\times W\to W,*:W\times N\to W}$, and ${\displaystyle \langle .,.\rangle :W\times W\to N}$, respectively. Assuming variable symbols ${\displaystyle {\vec {v}},{\vec {w}}\in V_{W}}$ and a,b ∈ V_N, the term ${\displaystyle \langle ({\vec {v}}+{\vec {0}})*a,{\vec {w}}*b\rangle }$ is well-sorted, while ${\displaystyle {\vec {v}}+a}$ is not (since + doesn't accept a term of sort N as 2nd argument). In order to make ${\displaystyle a*{\vec {v}}}$ a well-sorted term, an additional declaration ${\displaystyle *:N\times W\to W}$ is required. Function symbols having several declarations are called overloaded.

See many-sorted logic for more information, including extensions of the many-sorted framework described here.

Lambda terms

Terms with bound variables

Notation example	Bound variables	Free variables	Written as lambda-term
limn→∞ x/n	n	x	limit(λn. div(x,n))
${\displaystyle \sum _{i=1}^{n}i^{2}}$	i	n	sum(1,n,λi. power(i,2))
${\displaystyle \int _{a}^{b}\sin(k\cdot t)dt}$	t	a, b, k	integral(a,b,λt. sin(k⋅t))

int sum(int lwb, int upb, int fct(int)) {    // implements general sum operator
    int res = 0;
    for (int i=lwb; i<=upb; ++i)
        res += fct(i);
    return res;
}

int square(int i) { return i*i; }            // implements anonymous function (lambda i. i*i); however, C requires a name for it

#include <stdio.h>
int main(void) {
    int n;
    scanf(" %d",&n);
    printf("%d\n", sum(1, n, square));        // applies sum operator to sum up squares
    return 0;
}