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Topic: Beta reduction

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	beta reduction Computer Encyclopedia Enterprise Resource Directory Complete Guide to Internet (Site not responding. Last check: 2007-10-23)
		A copy of the body of the lambda abstraction is made and occurrences of the {bound variable} being replaced by the argument.
		The opposite of beta reduction is {beta abstraction}.
		These are the two kinds of {beta conversion}.
	www.jaysir.com /computer-encyclopedia/b/beta-reduction-computer-terms.htm (86 words)

	Barendregt: Lambda Calculus (Site not responding. Last check: 2007-10-23)
		beta is a finite subset of B and b is in f(beta)}.
		Weak reduction is CR (use a reduction allowing disjoint weak reductions to be done simultaneously and show this has the diamond property and its transitive closure is weak reduction).
		Because we can standardize the reduction Z ->> N0' ->>l N' and factor this standard reduction into a leftmost part and a non-leftmost part as in Z ->>l N ->>=/=l N' (once we reduce a redex which is not leftmost, contracting a leftmost redex would violate the definition of a standard reduction).
	www.andrew.cmu.edu /user/cebrown/notes/barendregt.html (21701 words)

	Hindley. Basic Simple Type Theory. (Site not responding. Last check: 2007-10-23)
		(That is, the leftmost reduction strategy is normalizing; this follows from the Standardization Theorem.) Furthermore, an induction argument shows that every beta-normal form is of the form "lambda x1.
		We have Weak Normalization of beta and beta-eta [Turing 1942, Curry/Feys 1958] by induction on the maximum degree of beta-redex's, where the degree of [[lambda x:sigma M:tau] N:sigma] is the number of atom-occurrences in the type sigma->tau.
		Note the similarity of this proof to the one in Andrews 1971 that typed lambda terms (in the Church approach) have normal forms.
	www.andrew.cmu.edu /user/cebrown/notes/hindley97.html (4117 words)

	Lambda Calculus :: Beta Reduction (Site not responding. Last check: 2007-10-23)
		Beta (β) reduction is what is often called function application.
		Beta reduction is simply substituting a value into the lambda expression.
		The first beta reduction evaluates to another lambda expression.
	www.braznet.com /david/webmastering/LambdaCalculus/beta.html (154 words)

	Beta Reduction Constraints - Bodirsky, Erk, Koller, Niehren (ResearchIndex) (Site not responding. Last check: 2007-10-23)
		Beta Reduction Constraints - Bodirsky, Erk, Koller, Niehren (ResearchIndex)
		In this paper, we introduce beta reduction constraints to describe beta reduction steps between partially known lambda terms.
		We show that beta reduction constraints can be expressed in an extension of CLLS by group parallelism.
	citeseer.ist.psu.edu /582142.html (294 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		Thus in the case of $\beta$-reduction the effect of a parallel reduction is same as that of a ``complete development'' which is defined by using ``residuals'' of $\beta$-redexes.
		A nice feature of parallel reduction however is that it can be defined directly by induction on the structure of $\lambda$-terms (without referring to residuals or other auxiliary notions), and the inductive definition provides us exactly what we need in proving the theorem inductively.
		Moreover the notion can be easily extended to other reduction systems such as Girard's second-order system {\bf F} and G\"odel's system {\bf T}.
	theory.lcs.mit.edu /~iandc/Abstracts/takahashi95.ltx (148 words)

	Inline expansion - Wikipedia, the free encyclopedia
		The main drawback is that the expansion usually results in a larger binary code, which can actually hurt performance if it damages locality of reference or exceeds resource constraints.
		In the context of functional programming languages, inline expansion is often referred to as beta reduction, a term used in the lambda calculus, the formal language underlying these languages.
		Once the compiler has decided to inline a particular function, it is usually a simple matter to do so.
	en.wikipedia.org /wiki/In-line_expansion (1000 words)

	BRICS Mini-Course: Optimal Graph Reduction: Computation, Continuations, Complexity
		Finally, we discuss the inherent complexity of the parallel beta step, the operation that is at the heart of optimal graph reduction.
		Context semantics is a kind of rarified flow analysis, in which the traversal of a static graph is directed by a global state (the ``context'') that records the history of the path.
		While parallel beta reduction is nominally implemented by one graph step, many ``sharing reductions'' may be needed prior to that single graph step.
	www.brics.dk /MC/99/GraphReduction (1149 words)

	[No title]
		b) --------- ---- Lq is not in normal form beta reduction of Ld = (Lq.
		b) -------- - Lq is still not in normal form beta reduction of Le = (Lq.
		1) 0 ----- - beta reduction of Lb = 1 Notice we never evaluate the case not taken (0).
	www.eg.bucknell.edu /~cs208/spring06/lambda.txt (770 words)

	[No title]
		There is a development of algorithms for bracket abstraction of the various types considered by Curry, plus a development of a reasonably fast algorithm for eta-strong reduction.
		Commenting out the lines with ALLARGS convert these to single head reduction steps, which are _much_ faster.
		BETASTEP in particular is quite slow when called upon to reduce all arguments; I think that this is to be expected.
	math.boisestate.edu /~holmes/holmes/combinators.wat (473 words)

	Logic and Semantics Seminar - 5th June, 1998: Harry Mairson
		Optimal evaluation was finally realized by Lamping, who introduced a beautiful graph reduction technology for sharing evaluation contexts dual to the sharing of values.
		We prove that the cost of parallel beta-reduction -- the "shared procedure call" that is the fundamental operation of optimal reduction -- is not bounded by any Kalmar-elementary recursive function.
		The main theorem gives a lower bound on the work that must be done by any technology that implements optimal reduction.
	www.cl.cam.ac.uk /Research/LS/Talks/1997_98/98_06_05.Abstract.html (276 words)

	Lambda_calculus
		Not every lambda expression is equivalent to a normal form, but if it is, then the normal form is unique up to naming of the formal arguments.
		Furthermore, there is an algorithm for computing normal forms: keep replacing the first (left-most) formal argument with its corresponding concrete argument, until no further reduction is possible.
		Punit,Gupta, Amit and Ashutosh Agte, Untyped lambda-calculus, alpha-, beta- and eta- reductions and recursion
	www.brainyencyclopedia.com /encyclopedia/l/la/lambda_calculus.html (2218 words)

	Information and Computation Bibliography (Site not responding. Last check: 2007-10-23)
		The notion of parallel reduction is extracted from the simple proof of the Church-Rosser theorem by Tait and Martin-Löf.
		Moreover the notion can be easily extended to other reduction systems such as Girard's second-order system F and Gödel's system T.
		The abstract is also available as a LaTeX file, a DVI file, or a PostScript file.
	theory.lcs.mit.edu /~iandc/References/takahashi1995:120.html (142 words)

	The IT University of Copenhagen -- FunTechs talk (Site not responding. Last check: 2007-10-23)
		Unfortunately, heavy use of this data structure can become intractable in time and space; the typical culprit is the fundamental operation of beta reduction.
		By adding uplinks from a child to its parents, we can efficiently implement beta reduction in a bottom-up manner, thus avoiding combinatorial explosion in time required to search the term in a top-down fashion.
		I will present an algorithm for performing beta reduction on lambda terms represented as uplinked DAGs; describe its proof of correctness; discuss its relation to alternate techniques such as Lamping graphs, the suspension lambda-calculus (SLC) and director strings; and present some timings of an implementation.
	www.itu.dk /research/funtechs/coplas/talks/2005-01-19.html (306 words)

	[No title]
		reduction R x y /\ reduction R y z ==> reduction R x z) /\ (!x y z.
		reduction R x y ==> reduction R (z @@ x) (z @@ y)) /\ (!x y z.
		reduction R x y ==> reduction R (x @@ z) (y @@ z)) /\ (!x y v.
	www.cs.utah.edu /~swalton/hol98/examples/lambda/chap3Script.sml (2537 words)

	reduction from FOLDOC
		The most important forms are beta reduction (application of a lambda abstraction to one or more argument expressions) and delta reduction (application of a mathematical function to the required number of arguments).
		An evaluation strategy (or reduction strategy), determines which part of an expression (which redex) to reduce first.
		See graph reduction, string reduction, normal order reduction, applicative order reduction, parallel reduction, alpha conversion, beta conversion, delta conversion, eta conversion.
	foldoc.org /foldoc/foldoc.cgi?reduction (106 words)

	BLC 1999 Abstracts (Site not responding. Last check: 2007-10-23)
		The last problem on which Curry worked before he died in 1982 was that of defining a reduction in combinatory logic to correspond closely to the usual beta-reduction in lambda-calculus.
		Several solutions to this problem have been posed since then, but despite some ingenuity in their formulation, none has been really clean and simple enough to make its development attractive.
		(It is not a "tidy" problem and it promises no beautiful solution -- but then, neither does real life!) In this talk I shall discuss criteria for acceptability of a beta reduction, and describe the known candidates, and suggest how far they succeed or fail in satisfying these.
	www.cl.cam.ac.uk /users/ad260/blc/hindley.html (136 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		It is well-known that the simply-typed lambda calculus and several standard polymorphic extensions (system F, the system of intersection types, and the system of positive recursive types) can only encode terms that strongly normalize (terminate) under beta reduction.
		We present a non-semantic proof of strong normalization of beta reduction for the system of intersection types.
		First, we introduce another notion of reduction for lambda terms and convert the problem of strong normalization of beta reduction to the problem of weak normalization of the new reduction.
	www.cs.bu.edu /colloquium/LOG-94/95-05-31.Joe_Wells (181 words)

	Unification with Expansion Variables (Site not responding. Last check: 2007-10-23)
		So beta unification is powerful, relates directly to program evaluation and a form of type inference, and by mixing substitutions with terms offers many challenges.
		In the work on unification with expansion variables we study the unification problem in isolation from beta reduction and type inference (although the link is relevent for certain properties).
		In the work on exact intersection typing inference we use this form of unification to study forms of type inference that exactly analyse the beta reduction of a term using particular reduction strategies.
	cs-people.bu.edu /bake/uwev (328 words)

	BET from FOLDOC
		A term from lambda-calculus for beta reduction or beta abstraction.
		An item "in beta test" is thus mostly working but still under test.
		Beta releases are generally made available to a small number of lucky (or unlucky), trusted customers.
	www.instantweb.com /d/dictionary/foldoc.cgi?query=BET (556 words)

	Lambda Calculus and Types: Synopsis
		Terms, free and bound variables, alpha-conversion, substitution, variable convention, contexts, the formal theory lambda beta, the n rule, fixed point combinators, brief mention of other lambda-theories.
		The general idea of compatible closure, reflexive transitive closure, diamond and Church-Rosser properties for general notions of reduction.
		beta-reduction, proof of the Church-Rosser property (via parallel reduction), connection between beta-reduction and lambda beta, consistency of lambda beta.
	web.comlab.ox.ac.uk /oucl/courses/topics04-05/lcat/synopsis.html (218 words)

	Information Bridge: DOE Scientific and Technical Information
		Beta reduction factors for protective clothing at the Oak Ridge National Laboratory
		Beta reduction factors (f{sub{beta}}) for protective clothing (PC) at the Oak Ridge National Laboratory (ORNL) have been determined for a variety of protective clothing combinations.
		Field comparison tests were then conducted to determine the validity of these beta reduction factors.
	www.osti.gov /bridge/product.biblio.jsp?osti_id=564171 (267 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		N) (if (eq x N) M N)))) ;beta-1 reduction, does all the beta reduction possible within a term without calling itself on M (defun bone (M) (cond ((abstraction?
		M) M))) ;beta reduces a term until it is in normal form, ; has grown enough times consecutively to be deemed non terminating, ; or has looped back to equal its previous smallest form.
		;these also work as tests of substitution and alpha renaming as they are used in the beta reduction process.
	www.bath.ac.uk /~cs3nb/lamb.lsp (498 words)

	Type-checking PTS (sources) (Site not responding. Last check: 2007-10-23)
		Since the development is parametric, the parameters are defined in the main file, which glues together all the others, using Load.
		First example of subtyping: when the subtyping relation is the reflexive-symmetric-transitive closure of a reduction rule:
		Second example: the subtyping relation is cumulativity, parameterized by the inclusion between sorts and a reduction rule:
	pauillac.inria.fr /~barras/pts_proofs/PTS/main.html (90 words)

	Bibliography
		Abstract: We present a simple proof of Hindley's Theorem: that it is decidable whether a term of the untyped lambda calculus is the image under type-erasing of a term of the simply typed lambda calculus.
		The proof proceeds by a direct reduction to the unification problem for simple terms.
		A program from a source language is translated (via a semantic definition) to trees of combinators; the tree is simplified via associative and distributive laws) to a linear, assembly-language-like format; the ``compiler writer's virtual machine'' operates by simulating a reduction sequence of the simplified tree.
	www.ccs.neu.edu /home/wand/pubs.html (7027 words)

	The Church Project: A linearization of the lambda-calculus
		The standard notion of beta-reduction for Lambda corresponds to two new notions of reduction, beta
		We establish various connections between the three notions of reduction, beta, beta
		As a consequence, we provide an alternative framework to study the relationship between beta-weak normalization and beta-strong normalization, and give a new proof of the oft-mentioned equivalence between beta-strong normalization of standard lambda-terms and typability in a system of “intersection types''.
	www.church-project.org /reports/Kfoury:JLC-2000-v10n3.html (281 words)

	BETA from FOLDOC (Site not responding. Last check: 2007-10-23)
		Some subterm of the original expression becomes the argument of the abstraction and the rest becomes its body.
		A term from lambda-calculus for beta reduction or beta abstraction.
		Nearby terms: Best Fit « best of all possible worlds « beta abstraction « beta conversion » beta reduction » bewusstsein » Bezier curve
	lgxserver.uniba.it /lei/foldop/foldoc.cgi?BETA (197 words)

	SciDok - Underspecified beta reduction
		For ambiguous sentences, traditional semantics construction produces large numbers of higher-order formulas,which must then be beta-reduced individually.
		Underspecified versions can produce compact descriptions of all readings, but it is not known how to perform beta reduction on these descriptions.
		We show how to do this using beta reduction constraints in the constraint language for lambda-structures (CLLS).
	scidok.sulb.uni-saarland.de /volltexte/2004/301 (103 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		EVALUATION ---------- So far we have assumed "normal order beta reduction".
		Normal Order Beta Reduction --------------------------- Normal order beta reduction requires the evaluation of the leftmost redex in an expression.
		Applicative Order Beta Reduction -------------------------------- Applicative order beta reduction requires the evaluation of both the function and the argument expression.
	www.dcs.gla.ac.uk /~pat/ml/lambda/sec-7 (215 words)

	Bezier Surface
		A surface defined by mathematical formulae, used terms: beta reduction " bewusstsein " Bezier curve " Bezier surface " biconditional " bijection " binaries
		BÃ©zier surfaces were first described in 1972 by the French engineer Pierre BÃ©zier who used them to design automobile bodies.
		The knot vectors have exactly two knots, normally 0 and 1, which are multiplied by the surface order in both parameter directions.
	www.number1-freeware.com /bezier_surface.asp (264 words)

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