Definition:Substitution (Mathematical Logic)

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Definition

Mapping

Let $S$ be a set.

Let $f: S^t \to S$ be a mapping.

Let $\set {g_1: S^k \to S, g_2: S^k \to S, \ldots, g_t: S^k \to S}$ be a set of mappings.

Let the mapping $h: S^k \to S$ be defined as:

$\map h {s_1, s_2, \ldots, s_k} = \map f {\map {g_1} {s_1, s_2, \ldots, s_k}, \map {g_2} {s_1, s_2, \ldots, s_k}, \ldots, \map {g_t} {s_1, s_2, \ldots, s_k} }$

Then $h$ is said to be obtained from $f, g_1, g_2, \ldots, g_k$ by substitution.

The definition can be generalized in the following ways:

• It can apply to mappings which operate on variously different sets.
• Each of $g_1, g_2, \ldots, g_t$ may have different arities. If $g$ is a mapping of $m$ variables where $m > k$, we can always consider it a mapping of $k$ variables in which the additional variables play no part. So if $g_i$ is a mapping of $k_i$ variables, we can take $k = \max \set {k_i: i = 1, 2, \ldots, t}$ and then each $g_i$ is then a mapping of $k$ variables.

Partial Function

Let $f: \N^t \to \N$ be a partial function.

Let $\set {g_1: \N^k \to \N, g_2: \N^k \to \N, \ldots, g_t: \N^k \to \N}$ be a set of partial functions.

Let the partial function $h: \N^k \to \N$ be defined as:

$\map h {n_1, n_2, \ldots, n_k} \approx \map f {\map {g_1} {n_1, n_2, \ldots, n_k}, \map {g_2} {n_1, n_2, \ldots, n_k}, \ldots, \map {g_t} {n_1, n_2, \ldots, n_k} }$

where $\approx$ is as defined in Partial Function Equality.

Then $h$ is said to be obtained from $f, g_1, g_2, \ldots, g_k$ by substitution.

Note that $\map h {n_1, n_2, \ldots, n_k}$ is defined only when:

• All of $\map {g_1} {n_1, n_2, \ldots, n_k}, \map {g_2} {n_1, n_2, \ldots, n_k}, \ldots, \map {g_t} {n_1, n_2, \ldots, n_k}$ are defined
• $\map f {\map {g_1} {n_1, n_2, \ldots, n_k}, \map {g_2} {n_1, n_2, \ldots, n_k}, \ldots, \map {g_t} {n_1, n_2, \ldots, n_k} }$ is defined.

Sources

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