Pruebas

Formally, a labelling of the nodes in N, or more specifically a valuation V over a network <N,D> is an assignment of values to nodes V: N —> T'. Given a fixed ordering on N we can write V as an n-tuple (V(nodei) ,..., V(nodeiNi)) and the set of all possible valuations is VINI.

We are now in a position to write formal conditions to capture the idea that each node’s value, with respect to a particular valuation, must correspond to the value supported by the dependencies attached to it.

The value of a given node n under a valuation V, written VN(n, V), is defined to be the sum (as defined by the summation functions as outlined in the preceding section) of the values of the dependencies, V®(di, V), for each dependency attached to n:

Vs : N x V INI — > V,

(n,V) l— » SN(KD(di ,V )...KD(dm, V)) where dj e £)(n)

The value of each dependency d, ^ ’(d, V), is similarly defined with the value being equal to the sum of the values, ^ ( A j j, V), of the antecedent classes:

TD: I) x Vini — * "V,

(d.V) l > 5»d)(FA(A ,, V ) FA(Ap, V)) for d = (A ,... Ap, c)

Finally, the value for each class of antecedents is defined to be the sum of the values assigned to each node in the class.

3.2.3 Formal Semantics of Support

VA: Pw{ N )xVini — (A ,V)(—>S^A)(V (a ,)...Via,)) for A = { ai aq }

The valuation functions provide a framework in which to place the user defined summation functions. Strictly speaking each summation function is a one place function taking an arbitrary n-tuple of values X = (vj ,..., vn) £ l / and returns a single value v e V, i.e.

S:Pw(‘S )—>V\

(v, ,...,vn)l— » S ( ( v , .... vn) )

For convenience sake, to prevent a proliferation of brackets, I have treated them as variable arity functions thus

(vi vn)l— > 5 ( v , vn )

There are no restrictions placed on the summation functions save the following: a node with no attached dependencies shall have a value nil, and a node with a single dependency shall have the value of that dependency. I.e.

Defn (n, V) = SN( 0 ) = nil where D(n) = 0 , and

TN (n, V) = SN(v) = v where I)(n) = {d } and l^’fd, V) = v

The first restriction forces a correspondence between the meaning of nil as the value corresponding to "no supported value exists", and the fact that only dependencies are capable of supporting values. The second restriction corresponds to the intuition that the summation functions are a way of combining information about evidence or belief. If there is only a single value, i.e. there is no other support to be considered, then that value should remain unchanged.

We are now in a position to define those valuations in which the values assigned to the nodes correspond to the support given to them by their attached dependencies. We shall call such valuations admissible.

3.2.3 Formal Semantics of Support

Defn A valuation V of <N,D> is admissible iff V n e N, V(n) = ^ ( n , V) I shall write VA(<N,D>) for the set of admissible valuations of <N,D>.

We can also formally define the notion of a dependency supporting the value of a node, given a particular valuation. Intuitively we want d to support n if the value of n is a function of the value of d so that if the value of d is changed then the value of n is changed. Thus Defn dj e D(rt) = ]d[ dm } supports VN(n, V) iff 3 v e T's.t.

SN(1/D(d,, V ) ^ ( d j - t . V ) , v, V/D(dj+,,V ).... VD(dm,V)) * SN(KD( d , , V ) V D(dm,V))

and v is a value that could actually assigned to some dependency4.

3.3 Semantics of II)Ns

So far, this chapter has covered the need for formal semantics; sketched the informal semantics of support; defined the network and interpretation structures for IDNs; and provided a formal definition of a well supported interpretation, i.e. an admissible valuation. This in itself is sufficient for the work that follows in the rest of the thesis.

However, if we wish to relate the structure of the network to the admissible valuations in such a way as to be able to prove relationships between the values of different nodes we

need more structure. I.e. for IDNs to be a formal system with a well defined semantics, equivalent to a logic of some sort, we must have the following (from |Patel-Schneider 1987]) • a syntactic language of sentences £;

• a class of semantic structures C; • a set of truth values V,

• and an interpretation function i that maps sentences in the language to elements of the set of truth values, given a particular semantic structure. I.e.

V a e C ,S e C.i(a . S) e V

1 Given the presence of assertionul dependencies, this should always he the case.

3.3 Semantics of IDNs

For example, first-order logic consists of a language containing constant and variable symbols, function and predicate symbols, and quantifiers and connectives e.g. a, v, V. In Tarskian semantics of FOL [Boolos et al 1974] the semantic structures consist of:

• a domain or universe of discourse from which objects can be assigned to variables; • a mapping from constants in the language to objects in the domain (denotation); • a mapping from functions in the language to functions in the domain; and

• a characteristic function for each predicate, mapping sets of objects in the domain to truth values.

The set of truth values is { "true", "false" }, and the interpretation function is just the set of rules for determining the truth value of a compound wff from its constituent elements.

The preceding section defined a variety of structures that could potentially fill one or more of the roles outlined above. The rest of this section is devoted to exploring one possible way of mapping IDN structures to the elements outlined above.

In attempting to relate IDNs to the constituents of a formal system, the first problem is deciding on the language: what are the things we actually want to talk about? Obviously we need to talk about networks and we have defined above what constitutes such a network. We defined a class of semantic structures or valuations. The semantics of support through valuation and summation functions give us an interpretation function for assigning values of "satisfied" or "unsatisfied" to the network, given a particular valuation:

Defn /(V, < N ,I)» = "satisfied" if f V e FA(<N,1)>) = "unsatisfied" otherwise

Additionally, we want to talk about individual nodes or dependencies within the network and the relationships between them. For example, if we have an SL-dependency linking a and b, a —>sl b, that represents an inference of b is believed from a, we want to be able to write theorems like:

Ihm "a — >SLb g <N,D> and a being believed (to be true) entails b being believed

3.3 Semantics of IDNs

and

Thm "a being believed (to be true) entails a — »sl b supports b (to be true)”

Before formally defining entailment, we must consider how to formally write such theorems. In FOL, propositions are either true or false and given the two-valued nature of the semantics one can write " a is false" as —>a and write theorems like:

Thm ->b , a -> b |=fOI -.a

The semantics of (FOL) entailment allow a reading of "not b and a implies b entails not a” or "if ‘not b’ is true and ‘a implies b’ is true then ‘not a’ is true” or "if b is false and a implies b then a is false".

Given we may be dealing with three or more values in V, we cannot get away with this (i.e. have prefix modifiers to indicate intended truth values or restrictions on interpretations), but must include the restrictions we want to make in the sentences (this is like introducing meta level predicates). So instead of writing |= a I will write |= (a, true). Similarly, instead of |= -iCt I shall write |= (a, false). So the basic formulae in our logic of IDNs are pairs consisting of a node or dependency and a value, or extra-logical statements for example about the structure of networks. The nodes/dependencies value pairs are interpreted as follows:

I)efn /(V, (n, v)) = "true" iff V(n) = v = "false" otherwise Defn /(V, (d, v)) = "true" iff FD(d) = v

= "false" otherwise

We can now formalise the theorems above and write I'hm a — >sl b e <N,I)>, (a, in) |=,DN (b, in)

and

Thm (a, in) |=,I)N (a — >SL b, in)