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'''Affine logic''' is a [[substructural logic]] whose proof theory rejects the [[structural rule]] of [[Idempotency of entailment|contraction]]. It can also be characterized as [[linear logic]] with [[weakening]].  
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The name "affine logic" is associated with [[linear logic]], to which it differs by allowing the weakening rule. [[Jean-Yves Girard]] introduced the name as part of the [[geometry of interaction]] semantics of linear logic, which characterises linear logic in terms of linear algebra; here he alludes to [[affine transformation]]s on vector spaces.<ref>[[Jean-Yves Girard]], 1997. '[http://www.seas.upenn.edu/~sweirich/types/archive/1997-98/msg00134.html Affine]'.  Message to the TYPES mailing list.</ref>
 
The logic predated linear logic.  V. N. Grishin used this logic in 1974,<ref>Grishin, 1974, and later, Grishin, 1981.</ref> after observing that [[Russell's paradox]] cannot be derived in a set theory without contraction, even with an [[unrestricted comprehension|unbounded comprehension axiom]].<ref>Cf.  [[Frederic Fitch]]'s [[demonstrably consistent set theory]]</ref>  Likewise, the logic formed the basis of a decidable subtheory of [[predicate logic]], called 'Direct logic' (Ketonen & Wehrauch, 1984; Ketonen & Bellin, 1989).
 
Affine logic can be embedded into linear logic by rewriting the affine arrow <math>A \rightarrow B</math> as the linear arrow <math>A {-\!\circ} B \otimes \top</math>.
 
Whereas full linear logic (i.e. propositional linear logic with multiplicatives, additives and exponentials) is undecidable, full affine logic is decidable.
 
Affine logic forms the foundation of [[ludics]].
 
== Notes ==
<references />
 
==References==
* V.N. Grishin, 1974. “A nonstandard logic and its application to set theory,” (Russian). Studies in Formalized Languages and Nonclassical Logics (Russian), 135-171. Izdat, “Nauka,” Moskow. .
* V.N. Grishin, 1981. “Predicate and set-theoretic calculi based on logic without contraction rules,” (Russian). Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 45(1):47-68. 239.  Math. USSR Izv., 18, no.1, Moscow.
* Ketonen and Weyhrauch, 1984, A decidable fragment of predicate calculus. Theoretical Computer Science 32:297-307.
* Ketonen and Bellin, 1989. A decision procedure revisited: notes on Direct Logic.  In ''Linear Logic and its Implementation''.
 
==See also==
 
* [[Strict logic]] and [[relevant logic]]
* [[Affine type system]], a [[substructural type system]]
 
[[Category:Substructural logic]]
 
 
{{logic-stub}}

Revision as of 01:15, 9 February 2014

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