Something else to note, which doesn't seem to have been mentioned in the thread, is that a wide class of constraint satisfaction problems are theoretically equivalent to determining the satisfiability of expressions in Boolean logic. This is the complexity class of NP-complete problems, which includes many decision and optimization problems.
http://en.wikipedia.org/wiki/Boolean_satisfiability_problem Phil
> Date: Sat, 11 Oct 2014 02:14:17 -0400
> From: sowa@xxxxxxxxxxx
> To: ontolog-forum@xxxxxxxxxxxxxxxx
> Subject: Re: [ontolog-forum] Constraint Solving versus Inferencing
>
> I just wanted to clarify some points in this thread.
>
> > For a constraint to be expressed in Boolean logic, aren't you then
> > moving out of the realm of propositional logic and into the realm
> > of first order predicate logic?
>
> No. Boolean logic is usually used as a synonym for propositional logic.
>
> But it's important to note that Boolean algebra is actually more general
> than propositional logic, since George Boole himself applied the algebra
> to several different domains:
>
> 1. If the letters represent propositions, you get propositional logic:
> 1 represents truth, 0 falsity, AxB is AND, A+B is OR, -A is NOT.
> Peirce observed that if A implies B, the truth value of A must be
> less than or equal to the truth value of B. Therefore, A≤B can
> be used to represent "If A, then B."
>
> 2. But GB also used the letters to represent a Boolean algebra of
> monadic predicates: 1 is the predicate that is true of everything,
> 0 is true of nothing, AxB is the predicate that is true of
> everything for which A and B are both true, etc.
>
> 3. GB also used the letters to represent sets: 1 is the universe
> of discourse, 0 is the empty set, A+B is the union of A and B,
> AxB is the intersection, and -A is the complement of A. But
> Boole's version of set theory was actually a version of mereology:
> he did not distinguish a single value x from the set {x}.
>
> Description logics take advantage of points #2 and #3 by treating
> a class a pair: a set and a predicate that is true of everything
> in the set. The same Boolean operators apply to both the sets and
> the predicates that determine them.
>
> > Even if you only use equality (=), and non-equality ('=)
> > to express your constraints, don't those two act as predicates?
>
> Well, yes. But you don't get predicate calculus without quantifiers.
> If you only have constants as the arguments of the predicates, the
> expressive power is limited to propositional logic.
>
> You can have constraint logic programming (CLP) with just equality.
> If you extend the notation to support inequalities, the solution
> to a CLP problem may consist of a family of possible options.
>
> John
>
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