Yes, and not just to Boolean logic; the old linear systems theory work applies to constraint propagation as well. Start with the basic linear systems equations: X[k+1] := A*X[k] + B*I[k]; Y[k+1] := C*X[k] + D*I[k]; Look at each nonzero term in the matrices A, B, C and D, and you can express each item as a constraint, or as part of a constraint, or as an element of the entire system – kind of like SQL treats Select statements as atomic, freely substitutable into the “slots” (a la Minsky’s frames) of contexts. Given the equivalence of FOL to linear systems theory, you can inherit all the work done on optimal controls, i.e. Pontryagin’s method (Sp?) of solving differential equations works for logical equations as well, just a bit more toothfully. -Rich Sincerely, Rich Cooper EnglishLogicKernel.com Rich AT EnglishLogicKernel DOT com 9 4 9 \ 5 2 5 - 5 7 1 2 From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Philip Jackson
Sent: Saturday, October 11, 2014 3:46 AM
To: [ontolog-forum]
Subject: Re: [ontolog-forum] Constraint Solving versus Inferencing 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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