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Default Reasoningand

Theory Change

G. Antoniou A.Nayak A. Ghose

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Part I

Introduction to Default Logic

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Nonmonotonic Reasoning — Motivation

How do you get to work? By bus!

Usually, I go to work by bus.

I walk to the bus stop and read:

No buses today, we’re on strike.

Now I have to take back my previous conclusion: nonmonotonicity.

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Incomplete information

Why not use the classical rule

goToWork ! ¬ strike " takeBus?

• Can I list all potential obstacles? (icy streets, being in a hurry etc.).

• I may not have the time or the resources to establish theconditions of the left hand: Incomplete information.

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Plausible conjectures

Intelligent systems need to make plausible conjectures, based on, say:

• Default rulesUsually I go to work by bus.

• IntrospectionIf the Rolling Stones were giving a concert in mycity tonight, I would have heard of that.

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Reasons for being interested in NMR

• Reasoning with incomplete information.

• Maintaining "competing" information within the same knowledge base.

• Compact representation of information.

• Pieces of information that are stable under changes.

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Default Logics — Overview

• The notion of a default

• Reiter’s default logic

- Syntax- Extensions- Properties

• Default logic variants

- Constrained default logic- Priorities among defaults

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The notion of a default

Defaults are rules of inference that can be applied if some informationis given, and some assumptions can be made.

Prototypical reasoningchild(X) : hasParents(X) / hasParents(X)

No-risk reasoningaccused(X) : innocent(X) / innocent(X)

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Defaults — Examples

Defaults in law

criminal(X) ! foreigner(X) : expel(X) / expel(X)

Exception:politicalRefugee(X) " ¬ expel(X)

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Defaults — Examples

Defaults in biology

Typically molluscs are shell-bearers.Cephalopods are molluscs.Cephalopods are not shell-bearers.

Expressed formally:

mollusc(X) : shellBearer(X) / shellBearer(X)cephalopod(X) " mollusc(X) ! ¬ shellBearer(X)

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Defaults — Examples

Closed World Assumption

true : ¬A / ¬A

for all ground facts A.

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Default logic — Types of knowledge

A default theory T consists of two kinds of knowledge:

• A set W of first order formulas called facts; certain information.

• A set D defaults; the assumptions that can be made.

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Defaults — Definition

A default d has the form

A : B1, . . . , Bn / C

with closed first order formulas A, Bi, C.

A is called the prerequisite pre(d), B1, . . ., Bn the justificationsjust(d), and C the consequent cons(d) of d.

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Defaults with variables

“Defaults” with free variables are read as schemes.

bird(X) : flies(X) / flies(X)

represents the set of defaults

bird(tweety) : flies(tweety) / flies(tweety)bird(sam) : flies(sam) / flies(sam). . .

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Extensions — Informal idea

Extensions are "world views" that are based on the given information(facts and defaults).

Extensions are obtained from the application of some defaults in D.They include always the certain information W.

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Interpretation of defaults

A : B1, . . ., Bn / C

If A is currently known,and if it is consistent to assume B1, . . ., Bn,then conclude C.

A : B1, . . ., Bn / C is applicable to a deductively closedset of formulas E iff A#E and ¬ B1$E, . . ., ¬ Bn$E.

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Extensions — Desirable properties

• They should be deductively closed under classical reasoning.

• They should be maximal: no more defaults can be applied.

Extensions represent maximal world views. In practice we may beinterested in portions of extensions (query evaluation).

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In-set and Out-set

P = (d0, d1, . . .) sequence of defaults from D without multiple occurrences.P[k] denotes the beginning part of P of length k.

• In(P) = Th(W % {cons(d) | d occurs in P}).

• Out(P) = {¬ B | B # just(d) for some d in P}.

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In-set and Out-set — Examples

aa : ¬ b / ¬ b [d1]b : c / c [d2]

Let P1 = (d1); In(P1) = Th({a, ¬ b}) and Out(P1) = {b}.

Let P2 = (d2, d1); In(P2) = Th({a, c, ¬ b}), Out(P2) = {¬ c, b}.

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Processes

Enforce that defaults can indeed be applied in the given order.P is called a process iff the defaults can be applied in the given order.

P is a process iff for every k such that P[k] is defined,dk is applicable to In(P[k]).

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Extensions — Definition

Given a process P we define the following:

• P is successful iff In(P) & Out(P) = ', otherwise it is failed.

• P is closed iff every default in D that is applicable to In(P) already occurs in P.

• E is an extension of a default theory T = (W,D) iff there is aclosed and successful process P of T such that E = In(P).

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Processes & Extensions — Example

aa : ¬ b / d [d1]true : c / b [d2]

• P1 = (d1) is successful but not closed.

• P2 = (d1, d2) is closed but failed.

• P3 = (d2) is a closed and successful process. Thus E = Th({a,b}) isan extension, in fact the only extension of the default theory.

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Process tree

• Nodes: (In, Out)

• Edges: (In, Out) " (In’, Out’) iff there is a defaultd = A : B1, . . ., Bn / C such that- d is applicable to In- In’ = Th(In % {C})- Out’ = Out % {¬B1, . . ., ¬Bn}

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Process tree

• Root: (Th(W), ')

• Paths starting at the root correspond to processes.

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A procedure for determining extensions

Given a default theory T.

1. Build the process tree of T.

2. Traverse the tree, and collect all closed & successful nodes(corresponding to closed & successful processes of T).

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Extensions — Example

true : a / ¬ a

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Extensions — Example

true : p / ¬ qtrue : q / r

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Extensions — Example

green : ¬ likesCars / ¬ likesCarsaaaMember : likesCars / likesCars

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Reiter’s definition of extensions

Let E and F be deductively closed sets of formulas, and D a set of defaults.

• A default A : B1, . . ., Bn / C is applicable to F with respect to belief set E iff A # F and ¬ B1 $ E, . . ., ¬ Bn $ E.

• F is closed under D w.r.t. belief set E iff, for every default d#Dthat is applicable to F w.r.t. E, cons(d)#F.

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Reiter’s definition of extensions

For T = (W,D) let (T(E) be the smallest set of formulas that• includes W,• is deductively closed, and• is closed under D w.r.t. E.

THEOREM [Antoniou 1996] E is an extension of T iff E = (T(E).

Note: Tradeoff between guessing and search.

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Default logic — Properties

• Existence of extensions

• Joint consistency of justifications

• Cumulativity

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Existence of extensions

We saw that it is not guaranteed. Is this undesirable?

• One may say no: user is responsible, as in programming

• The opposite view expects a logic to be more "fault tolerant".Different information sources (distributed systems,information superhighway).

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Ensuring existence of extensions

• Maintain the notion of extensions, but restrict the classes of defaulttheories considered.

• Modify the concept of an extension.

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Normal default theories

If we go the first way we may restrict attention to normal defaults:

A : B / B

[Reiter 1980] shows that normal default theories have alwaysextensions, essentially because all processes are successful.

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Normal defaults — Limitations

They may be unable to express interactions among defaults.

dropout(bill)dropout(X) : adult(X) / adult(X)adult(X) : employed(X) / employed(X)

Two extensions, but we would prefer one (which?)

In general, normal default theories are strictly less expressive thangeneral default theories.

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Semi-normal defaults

One possible solution:adult(X) : employed(X) ! ¬ dropout(X) / employed(X)

Semi-normal defaults:A : B ! C / C

[Etherington 1987] gives a sufficient condition for theexistence of extensions.

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Preference among extensions

Another way to ensure extensions is to use normal defaults, but add apriority relation which can model interactions among defaults.See Priorities among defaults later on.

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Modifications of the extension concept

To ensure the existence of extensions.

• Justified Default Logic [Lukaszewicz 1988]

• Constrained Default Logic [Schaub 1992]

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Joint consistency of justifications

Given the defaults

true : p / qtrue : ¬ p / r

there is a single extension, Th({q, r}).

Jumping to conclusions versus consistent set of beliefs.

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Approaches based on joint consistency

• Constrained Default Logic [Schaub 1992]

• Rational Default Logic [Mikitiuk & Truszczynski 1993]

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Joint consistency — Discussion

Is joint consistency a desirable property? Which is the right DL approach?

It depends very much on the problem at hand!

Reiter’s approach may lead to counterintuitive results:

true : usable(l) ! ¬ broken(l) / usable(l)true : usable(r) ! ¬ broken(r) / usable(r)broken(l) ) broken(r)

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Joint consistency — Discussion

But joint consistency may also lead to counterintuitive results.

travel : goodWeather / takeSwimSuittravel : badWeather / takeRainCoattravel¬ goodWeather ) ¬ badWeather

Here a cautious traveler would wish to apply both defaults, eventhough the assumptions contradict one another.

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Inference relations

In classical logic: M |– A. In default logic:

• Skeptical reasoningW |~D A iff A is included in all extensions of (W,D).

• Credulous reasoningW |~D A iff A is included in at least one extension of (W,D).

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Cumulativity

Ensures the safe use of lemmas (in the skeptical approach):

If W |~D A, then for all formulas B,

W |~D B * W%{A} |~D B.

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Default logic violates cumulativity

Consider the default theory T = (W,D) [Makinson 1989]:

true : p / p

p ) q : ¬ p / ¬ p

The only extension is Th({p}), so ' |~D p ) q.But if we add p ) q to T, then we get two extensions

Th({p})Th({¬ p, q})

p is not included in both extensions.

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Default Logics — Overview

• The notion of a default

• Reiter’s default logic

- Syntax- Extensions- Properties

• Default logic variants- Constrained default logic- Priorities among defaults

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Constrained default logic

JDL does not guarantee joint consistency of justifications.

true : p / qtrue : ¬ p / r

has the single modified extension Th({q, r}).

Constrained Default Logic [Schaub 1992] guarantees joint consistency.It does so by maintaining a consistent set of supporting beliefs.

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Default applicability in CDL

CDL adopts the idea of JDL not to run blindly into failure: "look ahead".

A default A : B1, . . ., Bn / C is applicable to deductively closed sets offormulas E (current knowledge) and Con (set of supporting beliefs) iff

A#E, and {B1, . . ., Bn, C} % Con is consistent.

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The supporting belief set

Let T = (W,D) be a default theory. Let P be a sequence of defaults(d0, d1, . . .) of defaults without multiple occurrences.

• Con(P) = Th(W % cons(P) % just(P)).

Exampletrue : p / q [d1]true : ¬ p / r [d2]

For the sequence P = (d1) we have Con(P) = Th({p, q}).

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Processes and extensions in CDL

• P is a constrained process iff dk is applicable to In(P[k]) and Con([k]), for all k such that P[k] is defined.

• P is a closed constrained process iff every default in D that isapplicable to In(P) and Con(P) already occurs in P.

• A pair (E, C) of deductively closed sets of formulas is aconstrained extension iff there is a closed constrained process Psuch that E = In(P) and C = Con(P).

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Constrained extensions — Example

pp : ¬ r / qp : r / r

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Constrained extensions — Example

true : usable(l) ! ¬ broken(l) / usable(l)true : usable(r) ! ¬ broken(r) / usable(r)

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Properties of CDL

THEOREM Every default theory has at least one constrained extension.

THEOREM Let E = In(P) for a closed and successful process P of T.If E % ¬ Out(P) is consistent then (E, Th(E % ¬ Out(P))) is a constrainedextension of T.

The converse is not true:true : p / ¬ p.

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Properties of CDL

THEOREM If (E,C) is a constrained extension of T then there is a modifiedextension F of T such that E + F.

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Comparison of default logics

THEOREM Let T be a normal default theory. The following statementsare equivalent:

(1) E is an extension of T.

(2) E is a modified extension of T.

(3) There is a set C such that (E,C) is a constrained extension of T.

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Priorities among defaults

• Default logics give overview of all possible extensions.

• But what if we wish to choose the most "important"(probable) possibilities?

- Medical diagnosis.

- Law.

• Technically achieved by introducing priorities among defaults.

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Default theories with priorities

penguin : ¬ flies / ¬ flies

bird : flies / flies

Give the first default higher priority than the second.

Total ordering among defaults: apply the default withthe highest priority that is applicable.

In the example above we get only one extension, Th({¬ flies}).

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Partial priority orders

Sometimes it is not reasonable to expect that the preference order be total.

conservative : taxCut ! ¬ spengingsCut / taxCut ! ¬ spengingsCut

conservative ! radical : taxCut ! spengingsCut / taxCut ! spengingsCut

socialDemocrat : ¬ taxCut ! ¬ spengingsCut / ¬ taxCut ! ¬ spengingsCut

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Partial priority orders

politicianpolitician : ¬ respected / ¬ respected [d1]politician : wellPaid / wellPaid [d2]¬ respected : ¬ wellPaid / ¬ wellPaid [d3]d2 < d3

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Consider “linearizations” of partial orders

According to all three total orderings which include <,

d1 << d2 << d3d2 << d1 << d3d2 << d3 << d1

we conclude wellPaid.

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References

G. Antoniou (1997). Nonmonotonic Reasoning. MIT Press (in press)

G. Brewka (1994). Reasoning about Priorities in Default Logic. In Proc. 12thNational Conference on Artificial Intelligence, AAAI/MIT Press, 940-945

D. Etherington (1987). Formalizing Nonmonotonic Reasoning Systems. ArtificialIntelligence 31: 41-85

W. Lukaszewicz (1988). Considerations on Default Logic.Computational Intelligence 4: 1-16

D. Makinson (1989). General theory of cumulative inference. In Proc. 2ndInternational Workshop on Non-Monotonic Reasoning, Springer LNAI 346

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References (continued)

W. Marek & M. Truszczynski (1993). Nonmonotonic Logic. Springer

R. Reiter (1980). A Logic for Default Reasoning. Artificial Intelligence 13: 81-132

T. Schaub (1992). On Constrained Default Theories. In Proc. 10th EuropeanConference on Artificial Intelligence