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

References

1. Carmen Bruni (2018). Predicate Logic: Natural Deduction (PDF) (Lecture Slides). Univ. of Waterloo. Here: slide 13/26.
2. Michael Färber (Feb 2023). Denotational Semantics and a Fast Interpreter for jq (Technical Report). Univ. of Innsbruck. arXiv:2302.10576. Here: p.4.
3. Gordon, Andrew D.; Melham, Thomas F. (1996). "Five axioms of alpha-conversion". In von Wright, Joakim; Grundy, Jim; Harrison, John (eds.). Theorem Proving in Higher Order Logics, 9th International Conference, TPHOLs'96, Turku, Finland, August 26-30, 1996, Proceedings. Lecture Notes in Computer Science. Vol. 1125. Springer. pp. 173–190. doi:10.1007/BFB0105404. ISBN 978-3-540-61587-3.
4. Cohen, Edward (1990). "Loops B — On replacing constants by fresh variables". Programming in the 1990s. Monographs in Computer Science. New York: Springer. pp. 149–194. doi:10.1007/978-1-4613-9706-9. ISBN 9781461397069. S2CID 1509875.