Fresh variable

In formal reasoning, in particular in mathematical logic, computer algebra, and automated theorem proving, a fresh variable is a variable that did not occur in the context considered so far.^{[1][citation needed]} The concept is often used without explanation.^{[2][citation needed]}

Fresh variables may be used to replace other variables, to eliminate variable shadowing or capture. For instance, in alpha-conversion, the processing of terms in the lambda calculus into equivalent terms with renamed variables, replacing variables with fresh variables can be helpful as a way to avoid accidentally capturing variables that should be free.^[3] Another use for fresh variables involves the development of loop invariants in formal program verification, where it is sometimes useful to replace constants by newly introduced fresh variables.^[4]

Example

For example, in term rewriting, before applying a rule ${\displaystyle l\to r}$ to a given term ${\displaystyle t}$, each variable in ${\displaystyle l\to r}$ should be replaced by a fresh one to avoid clashes with variables occurring in ${\displaystyle t}$.^{[citation needed]} Given the rule

${\displaystyle append(cons(x,y),z)\to cons(x,append(y,z))}$

and the term

${\displaystyle append(cons(x,cons(y,nil)),cons(3,nil))}$,

attempting to find a matching substitution of the rule's left-hand side, ${\displaystyle append(cons(x,y),z)}$, within ${\displaystyle append(cons(x,cons(y,nil)),cons(3,nil))}$ will fail, since ${\displaystyle y}$ cannot match ${\displaystyle cons(y,nil)}$. However, if the rule is replaced by a fresh copy^[a]

${\displaystyle append(cons(v_{1},v_{2}),v_{3})\to cons(v_{1},append(v_{2},v_{3}))}$

before, matching will succeed with the answer substitution ${\displaystyle \{v_{1}\mapsto x,\;v_{2}\mapsto cons(y,nil),\;v_{3}\mapsto cons(3,nil)\}}$.

Notes

1. that is, a copy with each variable consistently replaced by a fresh variable