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In [[proof theory]], a '''structural rule''' is an [[inference rule]] that does not refer to any [[logical connective]], but instead operates on the [[judgment (mathematical logic)|judgment]] or [[sequent]]s directly. Structural rules often mimic intended meta-theoretic properties of the logic. Logics that deny one or more of the structural rules are classified as [[substructural logic]]s.
 
==Common structural rules==
* '''Weakening''', where the hypotheses or conclusion of a sequent may be extended with additional members. In symbolic form weakening rules can be written as <math>\frac{\Gamma \vdash \Sigma}{\Gamma, A \vdash \Sigma}</math> on the left of the [[Turnstile (symbol)|turnstile]], and <math>\frac{\Gamma \vdash \Sigma}{\Gamma \vdash A, \Sigma}</math> on the right.
* '''Contraction''', where two equal (or unifiable) members on the same side of a sequent may be replaced by a single member (or common instance). Symbolically: <math>\frac{\Gamma, A, A \vdash \Sigma}{\Gamma, A \vdash \Sigma}</math> and <math>\frac{\Gamma \vdash A, A, \Sigma}{\Gamma \vdash A, \Sigma}</math>. Also known as '''factoring''' in [[automated theorem proving]] systems using [[Resolution (logic)|resolution]].
* '''Exchange''', where two members on the same side of a sequent may be swapped. Symbolically: <math>\frac{\Gamma_1, A, \Gamma_2, B, \Gamma_3 \vdash \Sigma}{\Gamma_1, B, \Gamma_2, A, \Gamma_3 \vdash \Sigma}</math> and <math>\frac{\Gamma \vdash \Sigma_1, A, \Sigma_2, B, \Sigma_3}{\Gamma \vdash \Sigma_1, B, \Sigma_2, A, \Sigma_3}</math>. (This is also known as the ''permutation rule''.)
 
A logic without any of the above structural rules would interpret the sides of a sequent as pure [[sequence]]s; with exchange, they are [[multiset]]s; and with both contraction and exchange they are [[set (mathematics)|set]]s.
 
A famous structural rule is known as '''[[cut rule|cut]]'''. Considerable effort is spent by proof theorists in showing that cut rules are superfluous in various logics. More precisely, what is shown is that cut is only (in a sense) a tool for abbreviating proofs, and does not add to the theorems that can be proved. The successful 'removal' of cut rules, known as ''[[Cut-elimination theorem|cut elimination]]'', is directly related to the philosophy of ''[[computation]] as normalization'' (see [[Curry–Howard correspondence]]); it often gives a good indication of the [[computational complexity theory|complexity]] of [[decision problem|deciding]] a given logic.
 
==See also==
*[[Affine logic]]
*[[Linear logic]]
*[[Ordered logic]]
*[[Strict logic]]
 
{{logic}}
 
[[Category:Proof theory]]
[[Category:Rules of inference]]

Revision as of 06:52, 15 February 2014

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