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From: srlm@ukc.UUCP (S.R.L.Meira)
Newsgroups: net.math,net.ai
Subject: combinatory logic
Message-ID: <4280@ukc.UUCP>
Date: Thu, 28-Jun-84 09:56:08 EDT
Article-I.D.: ukc.4280
Posted: Thu Jun 28 09:56:08 1984
Date-Received: Fri, 22-Jun-84 08:01:34 EDT
Organization: Computing Lab. Kent University, England
Lines: 64

[-: kipple :-]

in the hope that many of you are also interested in combinatory logic...
please have a look at this and mail me any suggestions, references, etc.

------------------

[by a. pettorossi in notre dame j. form. logic 22 (4) 81]

define:
     marking

	is a function that assigns, for each combinator in a term (tree)
	the number of left choices (of path) that one has to make to go
	from the root to the combinator.
	ex.:
	marking SII = 

     the set right applied subterms of a combinator X is defined as:

	1) if X is a basic combinator or a variable ras(X) = {X)
	2) if X is (YZ) then ras(YZ) = union (ras(X)) Z

     a combinator X with reduction axiom X x1 x2 x3 ... xk -> Y
     has non-ascending property iff
	
	for all i, 1<=i<=k, if  occurs in marking (X x1...xk)
	and  occurs in marking Y, then p >= q.

     a combinator (X x1 x2 ... xk -> Y) has compositive effect iff

	a right applied subterm of Y is not a variable.

------------------
Theorem:
	given a subbase B={X1,...Xk} such that all Xi in B have non-ascending
	property and no compositive effect, every reduction strategy applied
	to any Y in B+ leads to normal form.

------------------
Open Problem:
	does the theorem hold if non-ascending property is the only condition?

------------------
My personal questions:
	
	if one specifies leftmost-outermost reduction only, would the Open
	Problem be any easier?

	how much of combinatory logic can we do with B?

	and with non-ascending property only?



	silvio lemos meira

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