Distribution function - Revision history

en>Omnipaedista: per WP:HNP

2014-12-29T08:43:27Z

← Older revision		Revision as of 10:43, 29 December 2014
Line 1:		Line 1:
	~~A '''typed [[lambda calculus]]'''~~ is ~~a typed [[formalism (mathematics)\|formalism]]~~ that ~~uses~~ the ~~lambda-symbol (~~<~~math~~>~~\lambda~~<~~/math~~>~~) to denote anonymous function abstraction~~. ~~In this context, types are usually objects of~~ a ~~syntactic nature that are assigned to lambda terms; the exact nature~~ of a ~~type depends on the calculus considered~~ (~~see kinds below~~). ~~From a certain point of view~~, ~~typed lambda calculi can~~ be ~~seen as refinements of the [[untyped lambda calculus]] but from another point of view~~, ~~they can also be considered the~~ more ~~fundamental theory and~~ '~~'untyped lambda calculus'' a special case with only one type~~.		When we were young it didnt appear to matter what you ate you always had a great deal of stamina and for most people which meant the fat would be kept down additionally. But as you get elder the bodies changes and the processes which we took for granted when you were young dont function too. Such is the case for metabolism. Unfortunately metabolism has become an convenient target over time as anything to blame fat gain on. But that certainly isnt the case; you require metabolism particularly whenever you are older. One way to grow it happens to be through strength training.<br><br>One pound = 3500 calories. Maintaining a caloric deficit of 500 calories daily for one week should signify we lose a pound (if our illustration does her circuit training, and consumes about 1700 total calories that day, then she's produced her 500 calorie deficit). For anybody who's ever dieted, though, the equations may be maddeningly inconsistent. Sure, eat less plus move more, plus you should lose weight. But that doesn't usually result. Why not?<br><br>Your basal metabolic rate (BMR)Your basal metabolic rate is important when planning a fat reduction program. It shows the rate your body burns calories really for basic metabolic functions, i.e. how several calories would we burn for a day should you simply lie inside bed. "Lying inside bed" isn't truly actual naturally, because should you are thinking or having conversation while inside bed, a body usually nevertheless burn more calories than the BMR. The BMR shows just the minimum calories required to stay alive.<br><br>But if you are able to go on with all the 500 calorie diet, a body can get used to taking in fewer calories. That means if you might go back to the aged habits eating more than 2000 calories a day, you'll have a abrupt gain inside fat, that is a lot more fat than when you didn't go on a low calorie diet to begin with.<br><br>Your daily bmr plus the calories you burn for sport plus additional escapades create the total number of calories we burn a day. This is exactly why BMR is really significant - it assists you plan a weight loss or fat gain system.<br><br>At any given moment, 25 % of all guys and 33 % of all ladies are on several type of formal diet in the United States. More than 55 % gain back all of their weight plus more than what they began with.1 Unfortunately, most diets are a one-size-fits-all approach. With any diet book you choose off the bookstore shelf, or any old diets passed down by your excellent aunt, you'll find the same diet for everyone. Some of those are completely unsound nutritionally whilst others might be backed by good nutrition principles. Yet, even those with wise nutrition principles don't personalize their approach to suit each person's body makeup. These are generally unluckily a one-size-fits-all dieting approach.<br><br>If you wish To lose weight, we have to program on eating a amount of calories that is someplace between a BMR plus the total amount of calories we burn in a day. If you like to gain weight, we need to eat more calories than we burn in a day.<br><br>If you beloved this article therefore you would like to be given more info about [http://safedietplans.com/bmr-calculator bmr calculator] generously visit our webpage.

	~~Typed lambda calculi are foundational [[programming languages]] and are the base of typed [[functional programming languages]] such as [[ML programming language\|ML]] and [[Haskell~~ (~~programming language~~)~~\|Haskell]] and, more indirectly, typed [[imperative programming\|imperative programming languages]]~~. ~~Typed lambda calculi play an important role in~~ the ~~design of [[type systems]]~~ for ~~programming languages; here typability usually captures desirable properties of the program~~, e.~~g. the program will not cause~~ a ~~memory access violation~~.

	~~Typed lambda calculi~~ are closely related to [[mathematical logic]] and [[proof theory]] via the [[Curry–Howard isomorphism]] and they can be considered as the [[internal language]] of classes of [[category theory\|categories]], ~~e.g.~~ the ~~[[simply typed lambda calculus]] is the language of [[cartesian closed category\|Cartesian closed categories]] (CCCs).~~

	~~== Kinds of typed lambda calculi ==~~
	~~Various typed lambda calculi have been studied~~. The ~~[[simply typed lambda calculus]] has only one [[type constructor]],~~ the ~~arrow~~ <~~math~~>~~\to~~<~~/math~~>~~, and its only types~~ are ~~[[basic type]]s and [[function type]]s <math>\sigma\~~to~~\tau</math>. [[Dialectica interpretation\|System T]] extends the simply typed lambda calculus~~ with ~~a type of natural numbers and higher order primitive recursion; in this system~~ all ~~functions provably recursive in [[Peano arithmetic]] are definable. [[System F]] allows polymorphism by using universal quantification over all types; from~~ a ~~logical perspective it~~ can ~~describe all functions that are provably total~~ in ~~[[second-order logic]]~~. ~~Lambda calculi with [[dependent types]] are~~ the ~~base of [[intuitionistic type theory]]~~, ~~the [[calculus of constructions]] and the [[LF (logical framework)\|logical framework]] (LF)~~, a ~~pure lambda calculus~~ with ~~dependent types~~. ~~Based on work by Berardi on [[pure type system]]s, [[Barendregt]] proposed~~ the ~~[[Lambda cube]] to systematize~~ the ~~relations~~ of ~~pure typed lambda calculi (including simply typed lambda calculus, System F, LF and the calculus of constructions).~~

	~~Some typed lambda calculi introduce~~ a ~~notion of ''[[subtype\|subtyping]]'', i~~.~~e. if <math>A</math>~~ is a ~~subtype of~~ <~~math~~>B<~~/math~~>, ~~then~~ all ~~terms~~ of type ~~<math>A</math> also have type <math>B</math>~~. ~~Typed lambda calculi~~ with ~~subtyping~~ are ~~the simply typed lambda calculus with conjunctive types and [[System F~~-~~sub\|System F<sub><:</sub>]]~~.

	~~All~~ the ~~systems mentioned so far~~, ~~with the exception of the untyped lambda calculus~~, ~~are~~ '~~'[[strongly normalizing]]'': all computations terminate~~. ~~As a consequence they~~ are ~~consistent as a logic, i~~.~~e. there are uninhabited types. There exist~~, ~~however, typed lambda calculi that are not strongly normalizing. For example the dependently typed lambda calculus~~ with ~~a type of all types (Type : Type) is not normalizing due~~ to ~~[[Girard~~'s ~~paradox]]~~. ~~This system is also the simplest pure type system,~~ a ~~formalism which generalizes the [[Lambda cube]]~~. ~~Systems with explicit recursion combinators~~, ~~such as [[Gordon Plotkin\|Plotkin's]] PCF, are not normalizing, but they are not intended~~ to ~~be interpreted as~~ a ~~logic. Indeed, [[Programming language for Computable Functions\|PCF]] (for Programming language for Computable Functions)~~ is a ~~prototypical, typed functional programming language, where types are used to ensure that programs are well-behaved but not necessarily terminating.~~

	~~== Applications to programming languages ==~~
	~~In [[computer programming]],~~ the ~~routines (functions, procedures, methods)~~ of ~~[[strongly typed programming language]]s closely correspond to typed lambda expressions~~. ~~[[Eiffel (programming language)\|Eiffel]] has a notion of "inline agent" that makes it possible~~ to ~~define and manipulate typed lambda expressions directly~~, ~~through such expressions as '''agent''' (p: PERSON): STRING '''do Result''' := p.spouse.name '''end''', denoting an object that represents a function which returns~~ a ~~person's spouse's name~~.

	~~== See also ==~~
	* [[Kappa calculus]]—an analogue of typed lambda calculus which excludes higher-order functions

	~~== References ==~~
	<~~references/~~>
	* Henk Barendregt, [~~ftp~~://~~ftp~~.~~cs.ru.nl~~/~~pub/CompMath.Found/HBK.ps Lambda Calculi with Types~~]~~, Handbook of Logic in Computer Science, Volume II, Oxford University Press~~.

	~~{{DEFAULTSORT:Typed Lambda Calculus}}~~
	~~[[Category:Lambda calculus]]~~
	~~[[Category:Logic in computer science]]~~
	~~[[Category:Theory of computation]]~~
	~~[[Category:Type theory]]~~

en>Nanite: clarify the phase space is single-particle, not the ordinary (whole system) phase space

2014-01-23T19:15:45Z

clarify the phase space is single-particle, not the ordinary (whole system) phase space

← Older revision		Revision as of 21:15, 23 January 2014
Line 1:		Line 1:
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			Typed lambda calculi are foundational [[programming languages]] and are the base of typed [[functional programming languages]] such as [[ML programming language\|ML]] and [[Haskell (programming language)\|Haskell]] and, more indirectly, typed [[imperative programming\|imperative programming languages]]. Typed lambda calculi play an important role in the design of [[type systems]] for programming languages; here typability usually captures desirable properties of the program, e.g. the program will not cause a memory access violation.

			Typed lambda calculi are closely related to [[mathematical logic]] and [[proof theory]] via the [[Curry–Howard isomorphism]] and they can be considered as the [[internal language]] of classes of [[category theory\|categories]], e.g. the [[simply typed lambda calculus]] is the language of [[cartesian closed category\|Cartesian closed categories]] (CCCs).

			== Kinds of typed lambda calculi ==
			Various typed lambda calculi have been studied. The [[simply typed lambda calculus]] has only one [[type constructor]], the arrow <math>\to</math>, and its only types are [[basic type]]s and [[function type]]s <math>\sigma\to\tau</math>. [[Dialectica interpretation\|System T]] extends the simply typed lambda calculus with a type of natural numbers and higher order primitive recursion; in this system all functions provably recursive in [[Peano arithmetic]] are definable. [[System F]] allows polymorphism by using universal quantification over all types; from a logical perspective it can describe all functions that are provably total in [[second-order logic]]. Lambda calculi with [[dependent types]] are the base of [[intuitionistic type theory]], the [[calculus of constructions]] and the [[LF (logical framework)\|logical framework]] (LF), a pure lambda calculus with dependent types. Based on work by Berardi on [[pure type system]]s, [[Barendregt]] proposed the [[Lambda cube]] to systematize the relations of pure typed lambda calculi (including simply typed lambda calculus, System F, LF and the calculus of constructions).

			Some typed lambda calculi introduce a notion of ''[[subtype\|subtyping]]'', i.e. if <math>A</math> is a subtype of <math>B</math>, then all terms of type <math>A</math> also have type <math>B</math>. Typed lambda calculi with subtyping are the simply typed lambda calculus with conjunctive types and [[System F-sub\|System F<sub><:</sub>]].

			All the systems mentioned so far, with the exception of the untyped lambda calculus, are ''[[strongly normalizing]]'': all computations terminate. As a consequence they are consistent as a logic, i.e. there are uninhabited types. There exist, however, typed lambda calculi that are not strongly normalizing. For example the dependently typed lambda calculus with a type of all types (Type : Type) is not normalizing due to [[Girard's paradox]]. This system is also the simplest pure type system, a formalism which generalizes the [[Lambda cube]]. Systems with explicit recursion combinators, such as [[Gordon Plotkin\|Plotkin's]] PCF, are not normalizing, but they are not intended to be interpreted as a logic. Indeed, [[Programming language for Computable Functions\|PCF]] (for Programming language for Computable Functions) is a prototypical, typed functional programming language, where types are used to ensure that programs are well-behaved but not necessarily terminating.

			== Applications to programming languages ==
			In [[computer programming]], the routines (functions, procedures, methods) of [[strongly typed programming language]]s closely correspond to typed lambda expressions. [[Eiffel (programming language)\|Eiffel]] has a notion of "inline agent" that makes it possible to define and manipulate typed lambda expressions directly, through such expressions as '''agent''' (p: PERSON): STRING '''do Result''' := p.spouse.name '''end''', denoting an object that represents a function which returns a person's spouse's name.

			== See also ==
			* [[Kappa calculus]]—an analogue of typed lambda calculus which excludes higher-order functions

			== References ==
			<references/>
			* Henk Barendregt, [ftp://ftp.cs.ru.nl/pub/CompMath.Found/HBK.ps Lambda Calculi with Types], Handbook of Logic in Computer Science, Volume II, Oxford University Press.

			{{DEFAULTSORT:Typed Lambda Calculus}}
			[[Category:Lambda calculus]]
			[[Category:Logic in computer science]]
			[[Category:Theory of computation]]
			[[Category:Type theory]]

en>PKT: Help needed: Maxwellian

2012-07-24T12:59:34Z

Help needed: Maxwellian

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