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 $${\displaystyle \operatorname {append} (\operatorname {cons} (x,y),z)\to \operatorname {cons} (x,\operatorname {append} (y,z))}$$ and the term $${\displaystyle \operatorname {append} (\operatorname {cons} (x,\operatorname {cons} (y,\mathrm {nil} )),\operatorname {cons} (3,\mathrm {nil} )),}$$ attempting to find a matching substitution of the rule's left-hand side, ${\displaystyle \operatorname {append} (\operatorname {cons} (x,y),z)}$, within ${\displaystyle \operatorname {append} (\operatorname {cons} (x,\operatorname {cons} (y,\mathrm {nil} )),\operatorname {cons} (3,\mathrm {nil} ))}$ will fail, since ${\displaystyle y}$ cannot match ${\displaystyle \operatorname {cons} (y,\mathrm {nil} )}$. However, if the rule is replaced by a fresh copy ^[a] $${\displaystyle \operatorname {append} (\operatorname {cons} (v_{1},v_{2}),v_{3})\to \operatorname {cons} (v_{1},\operatorname {append} (v_{2},v_{3}))}$$ before, matching will succeed with the answer substitution $${\displaystyle \{v_{1}\mapsto x,\;v_{2}\mapsto \operatorname {cons} (y,\mathrm {nil} ),\;v_{3}\mapsto \operatorname {cons} (3,\mathrm {nil} )\}.}$$

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.