Re: Apparent inconsistency regarding equality in RIF-PRD

Hi Jesse,

Thanx for the feedback (which could not be more appropriately timed: 
indeed, we are preparing a revised edition to correct the errors found in 
the first edition :-)

I will examine the three problems that you have found re equality, 
subclasses and action variables and let you know.

Cheers,

Christian

IBM
9 rue de Verdun
94253 - Gentilly cedex - FRANCE
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From:   Jesse Weaver <weavej3@rpi.edu>
To:     team-rif-chairs@w3.org
Cc:     cawelty@gmail.com, Christian De Sainte Marie/France/IBM@IBMFR, 
sandro@w3.org, Ankesh <ankesh@cs.rpi.edu>, public-rif-wg@w3.org, 
Adrian.Paschke@gmx.de
Date:   05/07/2012 21:22
Subject:        Apparent inconsistency regarding equality in RIF-PRD



RIF Chairs,

In short, if a=b and b=c, then intuitively, it should hold that a=c, but 
the operational semantics do not seem to ensure such.

By definition of atomic formula (section 2.1.2) and informal definition of 
fact (section 2.2.2), a set of facts may include ground, equality, atomic 
formulas.  Consider then the following set of facts where t_1, t_2, and 
t_3 are different ground terms: \Phi = {t_1=t_2, t_2=t_3}.  By 
(operational) definition of state of the fact base (section 2.2.2), \Phi 
represents a state of the fact base.  According to the interpretation of 
condition formulas (section 12.2), I_{truth}(I(x=y)) iff I(x)=I(y). 
Intuitively, then, it should hold that since t_1=t_2 \in \Phi implies 
I(t_1)=I(t_2), and since t_2=t_3 \in \Phi implies I(t_2)=I(t_3), by 
transitivity of mathematical equality, I(t_1)=I(t_3).  However, there 
exists no ground substitution that matches t_1=t_3 to \Phi, and since 
ground, equality, atomic formulas cannot be inferred by rules (they are 
not syntactically allowed as the target of an assert action; see 
definition of atomic action, section 3.1.1), then there is also no 
subsequent state in which there exists a ground substitution that matches 
t_1=t_3 to the state's associated set of facts.  This seems like an 
inconsistency.

A similar argument can be made for the simpler (but perhaps less 
interesting and less obvious) case of when \Phi = {t_1=t_2}, where t_1 and 
t_2 are different ground terms.  Then no ground substitution exists that 
matches t_2=t_1 to \Phi or to any subsequently inferred sets of facts.

I can think of a number of ways to resolve this apparent inconsistency, 
but I wish to know: have I correctly determined an inconsistency, and if 
so, what were the intended (consistent) semantics (or definitions)?

Jesse Weaver
Ph.D. Student, Patroon Fellow
Tetherless World Constellation
Rensselaer Polytechnic Institute
http://www.cs.rpi.edu/~weavej3/index.xhtml



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