# The concept of generalized constraint-based computation

In GTU, computation/deduction is treated as an instance of question answering. With reference to Figure 6.3, the point of departure is a proposition or a system of propositions, *p,* described in a natural language *p* is referred to as the initial information set, INL. The query, q, is likewise expressed in a natural language. As was stated earlier, the first step in NL-computation involves precisiation of *p* and *q*, resulting in pre- cisiands *p** and *q**, respectively. The second step involves construction of protoforms of *p** and *q**, *p*** and *q***, respectively. In the third and last step, p** and q** are applied to the computation/deduction module, *C/D. *An additional internal source of information is world knowledge, wk. The output of *C/D* is an answer, ans(*q*|*p*).

*Examples*

The key idea in NL-computation—the meaning postulate—plays a pivotal role in computation/deduction in GTU. More specifically, *p** may be viewed as a system of generalized constraints which induces a generalized constraint on ans(q|p). In this sense, computation/deduction in GTU may be equated to generalized constraint propagation. More concretely, generalized constraint propagation is governed by what is referred to as the deduction principle. Informally, the basic idea behind this principle is the following.

*Deduction principle*

Assume that the answer to *q* can be completed if we know the values of variables *u _{i},...,u_{n}.* Thus,

Generally, what we know are not the values of the *u _{i}* but a system of generalized constraints which represent the precisiand of

*p, p**. Express the precisiand,

*p**, as a generalized constraint on the

*u*

_{i}.

At this point, what we have is GC(u_{;} ,...,u_{n}) but what we need is the generalized constraint on ans(qp), ans(qp) = *f(u _{u}...,u_{n}).* To solve this basic problem—a problem which involves constraint propagation— what is needed is the extension principle of fuzzy logic (Zadeh, 1965; 1975b). This principle will be discussed at a later point. At this juncture, a simple example will suffice.

Assume that and

*q:* What is the average height of Swedes?

Assume that we have a population of Swedes, *G = (u _{u}...,u_{n}),* with

*h*

_{i},

*i = 1,...,n,*being the height of ith Swede. Precisiends of

*p*and

*q*may be expressed as

In this instance, what we are dealing with is propagation of the constraint on p* to a constraint on ans(qp). Symbolically, the problem may be expressed as

with the understanding that the premise and the consequent are fuzzy constraints. Let m_{ave}(^{v}) be the membership function of the average height. Application of this extension principle reduces computation of the membership function of ans(qp) to the solution of the variational problem

subject to

In this simple example, computation of the answer to the query requires the use of just one rule of deduction—the extension principle. More generally, computation of the answer to a query involves application of a sequence of deduction rules drawn from the Computation/ Deduction module. The Computation/Deduction module comprises a collection of agent-controlled modules and submodules, each of which contains protoformal deduction rules drawn from various fields and various modalities of generalized constraints (see Figure. 6.4). A protoformal deduction rule has a symbolic part which is expressed in terms of protoforms, and a computational part which defines the computation that has to be carried out to arrive at a conclusion. The principal protoformal deduction rules are described in the following.