Fase Game

support for either a or b in a particular context then in one admissible valuation a will be in the context while in the other valuation b would be in the context.

Example

Consider the following NATMS network with its possible admissible valuations:

D = {aou,— »SL*», bout— {A>)a, — >{(B))b ) Vj(a) = {{A }{—iB})» Vj(b) = ( {B })

V2(a) = {{A}},V2(a)-<{^A }{B }>.

Environm ent U Î A ) (B) {A B) C ontexti (using V ( ) (a) (a) {b} la b ) C ontext2 (using V2) {b } la) lb} (a b )

This is one area where it might appear that EATMS scores over the NATMS. The corresponding EATMS expansion of the non-monotonic dependencies gives the following acyclic network with just a single admissible valuation.

Figure 5.1.5 (a)

5.1.5 Multiple Interpretations (again) choose({Oa Ia }) choose([Oh Ib J) ignored Ia }) ignored lb |) V(Oa) = ({Oa )l V(Ia) = {{Ia }} V(ia) = {{Ia } (Ob }} V(Ob) = U O b )) V(Ib) = {{Ib}} V(ib) = {{Ib H Ob }} V(a) = ( (Ob }} V(b) = {{Oa }} V(_L) = {{Oa lb ) (Ob Ia } ( Ob Oa }}

Given any problem solving environment E the choose/ignore constraints will attempt to force the inclusion of one or both of Oa and Ob in the extended environment E '. As the inclusion of both is contradictory, two extensions result: E'j = |O a Ib l and E'2 = {Ob Ia ). Depending on which one of E 'j or E'2 is picked, a context is generated that contains b or a respectively. So although the problem of multiple admissible valuations is avoided, the problem of multiple contexts remains. In this situation either the user has direct control over which extension is picked or an arbitrary choice must be made by the system2.

Another problem that the EATMS solves, enabling the construction of an admissible valuation, is that of non-monotonic cycles. The expansion of non-monotonic dependencies in an odd loop will generate a stabilising assumption, allowing the cycle to be interpreted. In other words, given an odd loop a()Ut— >sLa the EATMS will assume there is independent support for a. This means a will be included in any context, and the odd cycle does not occur. The expansion of a,,ut— >sLa and its admissible valuation (see below) demonstrate this.

' This arhitnuy choice is exactly one of the reasons cited against the JTMS. However this problem will in- 'ariahly ;tnse when a system includes non-monotonic justifications. If non-monotonic cycles are allowed, mul-

5.1.5 Multiple Interpretations (again) Example choose({ Oa Ia }) ignored h ) ) V (0,) = { fO .)) V ( a ) = l |0 , ) l V(Ia) = {{Ia }} V(ia) = {{Ia }{Oa } ) V(-L) = {{Oa }}

Given Oa is contradictory, any problem solver environment is forced to include Ia, thus a will be in all contexts. Here the EATMS is forcing "a" to be in to avoid what would otherwise be a "contradiction" or unsatisfiable "cycle" (there is no actual cycle in the network itself). Because the translation scheme actually removes the dependency between nodes in any non-monotonic cycle it is possible to generate a single admissible valuation of the network, even in the case of even non-monotonic cycles where one might expect multiple possible interpretations.

So it can be seen that at least some of the problems of the NATMS are shared with existing EATMSs. One problem that is unique to the NATMS is the number and size of environments created by non-monotonic antecedents. Some further work should be done on investigating the possibility of abbreviating environments so that an environment IA] A2A3) might be replaced by {A123}, so a non-monotonic link would generate an environment {—iA 123 I rather than the environments { —iAj }, {—1A2 ) and {—1A3 ).

5.2 Default Reasoning

5.2 Default Reasoning

There is a very close connection between default reasoning and Truth Maintenance Systems. Indeed, the JTMS [Doyle 1979] can be thought of as an implementation mechanism for NML-I [McDermott and Doyle 1980]. Additionally, as mentioned in §1.2.2, there are strong associations between NML-I and autoepistemic logic (AE) [Moore 1983] and it has been demonstrated [Konolige 1986] that autoepistemic logic and Reiter’s [1980] default logic (DL) are of equivalent power and expressiveness.

In general a default rule (in DL, see 1.2.2) of the form a : MP / y is equivalent to an autoepistemic implication L a a —iL—1(3 —> y which in turn is equivalent to —iM—>a a M(3 —* y

(taking M in NML as the dual of L whereby M = —iL—i). These translations are somewhat counter-intuitive. One would expect an intuitive reading of the Reiter default to be "if a is true and it is consistent to assume p then y is true". Yet the intuitive reading of the corresponding NML-I implication is "if - i a is inconsistent and it is consistent to assume P then y is true". The truth condition on a has become somewhat distorted in form even if it is equivalent in content.

The intuitive reading of the autoepistemic implication ("if a is believed (known) and ->P is not believed (known) then y is true") is at first glance even more counter-intuitive: why should a not have to be true but be believed to be true? The answer comes from the semantics of the default. Take the standard flying bird and the default that says normally (or normal) birds fly. This can be interpreted as saying only those birds explicitly stated not to fly don’t actually fly. To be able to say a bird flies because it is not explicitly stated not to fly relies on the fact that not only is the object in question a bird, it is actually known or believed to be a bird. However this point becomes secondary when the defaults considered are of the form Mp / y.

Morris |1987] explicitly links defaults and Truth Maintenance in proposing the use of the JTMS as a default reasoning device. This is a departure from the normal problem solver/JTMS interaction whereby the TMS does actually instantiate "default" rules but only in the creation of assumptions to guide the search process and control dependency directed backtracking. By coding defaults as dependencies Morris claims that the problem of

5.2 Default Reasoning

anomalous extensions can be cured. Consider the following problem (a non-temporal version of the Yale Shooting Stick problem devised by Morris): normal animals (nA) can’t fly, winged animals (W) are abnormal animals (abA) (where abnormal = not normal) with respect to the property of not flying (i.e. winged animals can normally fly which does not fit the animal norm of not flying), all birds (B) are animals and normal birds (nB) have wings. Does Tweety the bird fly? Translating this into (Reiter’s) default logic produces the following implications3:

A a nA —» —iF W —> abA B —> A B a nB —» W

The default rules or assumptions capture the intuition that things are "normally" normal! I.e. if it is consistent to assume normality (there is no evidence of abnormality) then assume normality:

:M nA /nA (RA) :M nB /nB (RB)

In default logic (be it Reiter’s as illustrated above, or Moore’s or McDermott and Doyle’s) adding the fact that Tweety is a bird produces two extensions. In the first RB (or the equivalent) is used to assume the normality of Tweety as a bird which in turn implies Tweety has wings and is therefore an abnormal animal w.r.t. flying and the system is agnostic as to whether Tweety flies or not - as Tweety is an abnormal animal RA cannot be applied. In the second RA is used to assume normality of Tweety as an animal implying that Tweety cannot fly. As a side effect, Tweety’s normality implies (contra-positively) that Tweety doesn’t have wings and is therefore an abnormal bird with respect to having (or not having) wings.

The default theory can be converted into dependencies (in the spirit of Morris) by converting each implication into a monotonic dependency with each proposition represented

This axiomatisation omits some obvious implications, e.g. W -» F. However, this does not affect Morris' example which centres around which defaults hike precedence, and whether or not multiple interpretations :ire Possible.

5.2 Default Reasoning

by a node4; the default rules are represented as non-monotonic justifications so that if abA is out (abA «-» -inA) then n is in.

{A , nA )in— >sl- ,F { W U— >SLabA ( B U — »sl A {B , nB)in— »slW {abA}OUI— >sLnA {abB}out— >sLnB

This network, being acyclic, gives rise to a single admissible valuation and lo and behold the intuitive answer (that Tweety the bird does not necessarily not fly) is supported in this valuation.

4 The scheme represented here is a propositional case, but as shown in §5.2.4 there are a variety of tech­ niques for converting propositional representations into predicate representations.

5.2 Default Reasoning

This example works for the following reason: the semantics of the TMS puts an implicit ordering on the abnormality assumptions and the one way nature of support makes it impossible for the justifications to be used in a contra-positive way. A more realistic set of dependencies to model the set of assumptions would include the contra-positives of the implications, in particular:

{B , -iW}jn— >SLabB

( n A ) i n —

This is not motivated by the desire to follow the default logic reasoning but by intuitive notions of "normal" birds and animals and their lack, or otherwise, of wings.

The addition of these two dependencies creates a cycle (a necessary condition for multiple interpretations in deterministic IDNs - see §4.1.4) and two admissible valuations exist for this network.

nodes A w B —iF aA aB nA

Vi in in in out in out out

V2 in out in in out in in

Figure 5.2 (b)

5.2 Default Reasoning

The problem of unwanted extra interpretations has reappeared.

Given the position of this thesis, i.e. that dependencies are not just records of inferences but can actually be used to make inferences, it is hardly surprising that by being selective about the set of inferences used to capture certain knowledge one can achieve the desired results.

In standard first order or propositional logic the inferences that can be made from a given set of propositions are given by the inference rules which in turn (certainly in the case of natural deduction systems [Lemmon 1965]) are derived from the (intuitive/truth functional) semantics of the connectives. Proof theoretic default or non-monotonic logics are also trying to capture the intuitive notion of defaults by adding logical machinery that allows or generates the intuitive inferences from standard constructs. One cannot be arbitrary in the choice of logical construct, changing the forms on an ad hoc basis to achieve the desired results.

In the following section I shall present a systematic representation of defaults that captures a particular set of intuitions about defaults that gives plausible results (in terms of the inferences made - even when considering sets of defaults) and an easy way of specifying interactions between defaults.

5.2.1 The Structure o f Defaults

Consider some of the forms used in the literature on default or non-monotonic logics: birds fly by default; normally birds fly; normal (prototypical) birds fly; typically birds fly. There are different possible interpretations of very similar forms - is the default expressing the fact that the consequent holds for the majority of objects satisfying the antecedent, or for the normative examples satisfying the antecedent, or for some specified set of "normal" individuals? I take the approach that the latter should be the normal reading of a default - any bird that is believed to be normal (with respect to flying) will fly.

This "normality" check is not a simple (!) appeal to the ability to consistently assert the desired conclusion - which in this case would correspond to the fact "Tweety flying" being consistent with him being a bird and an animal and anything that can be derived from these