Sparse array - Revision history

2.123.221.11: /* Sparse Array as Linked List */ Doubles don't have a fixed size.

2014-09-17T12:24:16Z

Sparse Array as Linked List: Doubles don't have a fixed size.

← Older revision		Revision as of 14:24, 17 September 2014
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74.71.63.252: /* Clarify that a sparse array of X bytes doesn't waste memory compared to a full array of X bytes */

2014-02-15T04:52:31Z

Clarify that a sparse array of X bytes doesn't waste memory compared to a full array of X bytes

← Older revision		Revision as of 06:52, 15 February 2014
Line 1:		Line 1:
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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]]~~

128.157.160.13: Corrected grammatical errors. Replaced first person with third person as is more standard.

2013-07-17T01:15:00Z

Corrected grammatical errors. Replaced first person with third person as is more standard.

← Older revision		Revision as of 03:15, 17 July 2013
Line 1:		Line 1:
	~~Man or woman who wrote the articles~~ is ~~called Eusebio. His~~ [~~https://www.gov.uk/search?q=friends friends~~] ~~say it's~~ not ~~good for him~~ but ~~so what~~ on ~~earth he loves doing is going to be acting and he'~~s ~~been doing it for some time now~~. ~~Filing has been his profession as news got around~~. ~~Massachusetts has always been his livelihood place and his wife and kids loves it. Go to his website to search out out~~ more~~: http://circuspartypanama~~.~~com<br><br>~~		{{Unreferenced\|date=December 2009}}
			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.

	~~Here is my web blog~~ ... [~~http~~://~~circuspartypanama~~.~~com clash~~ of ~~clans hack download~~]		==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]]

en>Timrollpickering: clean up, removed stub tag using AWB

2012-08-25T16:09:55Z

clean up, removed stub tag using AWB

New page

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