Abductive Logic Programming Agents

• The ALP agent cycle

• ALP combines backward and forward reasoning

• ALP gives a semantics to production rules

• ALP can be used for explaining observations, conditional solutions, generating actions, default reasoning

• Pre-active reasoning, combining utility and uncertainty

Deciding whether or not to carry an umbrella

The prisoner’s dilemma

Forward reasoningusing beliefs

Forward reasoningusingbeliefs

Backward Reasoningusingbeliefs

Consequences

Judgeprobabilities and utilities

Decide

Maintenance goal

Achievement goal

Observe Act

Consequences

Consequences

Abductive logic programming (ALP) agent model

The World

An agent

ALP agents combine beliefs and goals

Beliefs, represented by logic programs, describe how things are.

Goals represented by integrity constraints, prescribe how things should be. They include condition-action rules commands queries obligations & prohibitions atomic and non-atomic actions denials

The ALP agent cycle

Record current observations, Use forward reasoning to derive consequences of the

observations, triggering any integrity constraints and adding any new goals

Use backward reasoning to reduce goals to sub-goals Perform conflict-resolution to choose between

candidate sub-goals that are atomic actions. Execute the associated actions.

Conflict-resolution can be performed by using forward reasoning to derive consequences of candidate actions.

Decision theory can be used to choose actions whose consequences have maximal expected utility.

Backward reasoning can also be used to explain observations, before using forward reasoning to derive consequences.

The London underground

Goal If there is an emergency then I get help.

BeliefsA person gets help if the person alerts the driver.

A person alerts the driver if the person presses the alarm signal button.

There is an emergency if there is a fire.There is an emergency if one person attacks another.There is an emergency if someone becomes seriously ill.There is an emergency if there is an accident.

There is a fire if there are flames.There is a fire if there is smoke.

The world

If there is an emergency then get help

There is an emergency

ALP combines forward and backward reasoning

observe

act

Forwardreasoning

Backward reasoning

get help

press thealarm signal button

There is a firealert the driver

Abductive Logic Programming

Abductive Logic Programs <P, A, IC> have three components:

• P is a normal logic program. • A is a set of abducible predicates.• IC, the set of integrity constraints, is a set of first-order

sentences.

Often, ICs are expressed as conditionals:

If A1 &...& An then Bor as denials:

not (A1 &...& An & not B)

Normally, P is not allowed to contain any clauses whose conclusion contains an abducible predicate. (This restriction can be made without loss of generality.)

ALP Semantics and Proof Procedures

Semantics:

Given an abductive logic program, < P,A,IC > , an abductive explanation for a goal G is a set Δ of ground atoms in terms of the abducible predicates such that:

G holds in P ΔIC holds in P Δ or P Δ IC is consistent.

Proof procedures:

Backward reasoning to show G.Forward reasoning to show observations and

explanations satisfy IC

Different notions of “holds” are compatible with these characterisations, i.e.: truth in the “intended” minimal model, truth in all models, etc.

ALP gives a logical semantics to production rules.

• Logical rules used to reason forward can be represented by LP clauses, with forward reasoning.

• Reactive rules that implement stimulus-response associations can be represented by integrity constraints, with forward reasoning.

• Pro-active rules that simulate goal-reduction:

If goal G and conditions C then add H as a sub-goal.

can be represented by LP clauses, with backward reasoning.

ALP viewed in Active Deductive Database terms

Logic programs define data. E.g.

The bus leaves at 9:00.The bus leaves at 10:00.

The bus leaves at X:00 if X is an integer & 9 ≤ X ≤ 18.

Integrity constraints maintain integrity. E.g.

There is no bus before 9:00.

If the bus leaves at X:00, then it arrives at its destination at X:Y & 20 ≤ Y

≤ 30.

ALP can be used to explain observations

Program: Grass is wet if it rained.Grass is wet if the

sprinkler was on.The sun was shining.

Abducible predicates: it rained, the sprinkler was on

Integrity constraint: not (it rained and the sun was shining)

Observation: Grass is wet

Two potential explanations: it rained, the sprinkler was on

The only explanation that satisfies the integrity constraint is

the sprinkler was on.

ALP can be used to generate conditional solutions

Program: X citizen if X born in USA.X citizen if X born outside USA & X resident of USA & X naturalised.X citizen if X born outside USA & Y is mother of X & Y citizen & X registered.Mary is mother of John.Mary is citizen.

Abducible predicates: X born in USA, X born outside USA,

X resident of USA, X naturalised, X registered

Integrity constraint:if John resident of USA then false.

Goal: John citizen

Two abductive solutions: John born in USA,

John born outside USA & John registered

ALP can be used to generate actions

Program:

there is an emergency if there is a fireyou get help if you alert the driveryou alert the driver if you press the alarm signal button

Abducible predicates there is a fire, you press the alarm signal button

Integrity constraint functioning as a maintenance goal:

If there is an emergency, then you get help

Abductive solutionyou press the alarm signal button

ALP can be used for default reasoning

Program:

X can fly if X is a bird and normal XX is a bird if X is a penguin

Abducible predicate: normal X

Integrity constraint functioning as a denial:

If normal X and penguin X then false

Observation: tweety is a bird

Abductive consequence:

tweety can fly, assuming normal tweety

New observation: Consequence withdrawn

ALP agents can reason pre-actively, taking into account utility and uncertainty

A common form of belief has the form:

Different effects have different utilities

an effect takes place if an agent does something and some conditions hold in the environment

The same belief can be used:• to reason forwards from observations• to reason backwards from desired effects• to reason forwards from candidate actions• To reason backwards from observed effects

The state of the environment is uncertain

Combining utility and uncertaintywith pre-active thinking

To get rich, I am thinking about robbing a bank

But before constructing a plan in all its detail, I mentally infer the possible consequences. Apart from any moral considerations, if I rob a bank, get caught, and am convicted, then I will end up in jail. But I don’t want to go to jail.

I can control whether or not I try to rob a bank.But I can not control whether I will be caught or be convicted. I can only judge their likelihood. If I judge that the likelihood of getting caught and being convicted is high, then I will decide not to rob a bank, because I don’t want to go to jail.

I will not even think about how I might rob a bank, because all of the alternatives lead to the same undesirable

consequence.

Forward reasoning

Forward reasoning

Backward reasoning

Consequences

Judgeprobabilities and utilities

Decide

Maintenance goal

Achievement goal

ObserveAct

Consequences

Consequences

Preactive thinking can be applied at different levels of detail

Pre-active thinking

Goal I carry an umbrella or I do not carry an umbrella.

Beliefs I stay dry if I carry an umbrella.I get wet if I do not carry an umbrella and it

rains.I stay dry if it doesn’t rain.

Assume I carry an umbrella .

Infer I stay dry (whether or not it rains).

Assume I do not carry an umbrella .

Infer I get wet if it rains.

I stay dry if it doesn’t rain.(whether or not I carry an umbrella).

action1

action2

p11

p12

p13

p14

p21

p22

p23

p24

u11

u12

u23

u13

u14

u21

u22

u24

Expected utility of action1

p11·u11+p12·u12+p13·u13+p14·u14

Expected utility of action2

p21·u21+p22·u22+p23·u23+p24·u24

Decision Theory:to find the expected utility of a proposed action,

find all the alternative resulting states of affairs,weigh the utility of each such state by its probability, andadd them all up.

Choose the action of highest expected utility

Deciding whether or not to carry an umbrellaAssume Probability it rains = .1

Probability it doesn’t rain = .9

Utility of getting wet = – 10 Utility of staying dry = 1

Utility of carrying an umbrella = – 2 Utility of not carrying an umbrella = 0

Assume I carry an umbrella .Infer I stay dry with probability 1.Expected utility -2 + 1 = -1

Assume I do not carry an umbrella .Infer I get wet with probability .1.

I stay dry with probability .9Expected utility 0 -10·.1 + 1·.9 = -1 + .9 = -.1

Decide I do not carry an umbrella!

A more practical alternative might be to use maintenance goals or condition-action rules instead:

If I leave home and it is raining then I take an umbrella.

If I leave home and there are dark clouds in the sky then I take an umbrella.

If I leave home and the weather forecast predicts rain then I take an umbrella.

The maintenance goals compile decision-making into the thinking component of the agent cycle. The compilation might be an exact implementation of the Decision Theoretic specification. Or it might be only an approximation.

The Prisoner’s Dilemma

Goal I turn state witness or I do not turn state witness

Beliefs A prisoner gets 0 years in jail if the prisoner turns state witnessand the other prisoner does not.

A prisoner gets 4 years in jail if the prisoner does not turn state witness and the other prisoner does.

A prisoner gets 3 years in jail if the prisoner turns state witnessand the other prisoner does too.

 A prisoner gets 1 year in jail if the prisoner does not turn state witnessand the other prisoner does not turn state witness

too. 

Preactive thinking

Assume I turn state witness

Infer I get 0 years in jail if the other prisoner does not turn state witness.

I get 3 years in jail if the other prisoner turns state witness .

Assume I do not turn state witness

Infer I get 4 years in jail if the other prisoner turns state witness.

I get 1 year in jail if the other prisoner does not turn state witness.

In Classical Logic

Given the additional belief

the other prisoner turns state witnessor the other prisoner does not turn state

witness.

Infer

If I turn state witnessthen I get 0 years in jail or I get 3 years in

jail.

If I do not turn state witness then I get 4 years in jail or I get 1 year in jail.

In Decision Theory

Assume Probability the other prisoner turns state witness= .5Probability the other prisoner does not turn state witness =.5Utility of getting N years in jail = N

Assume I turn state witness

Infer Probability I get 0 years in jail = .5Probability I get 3 years in jail = .5

Expected utility .5·0 + .5·3 = 1.5 years in jail.

Assume I do not turn state witness

Infer Probability I get 4 years in jail = .5Probability I get 1 years in jail = .5

Expected utility .5·4 + .5·1 = 2.5 years in jail

Decide I turn state witness

Forward reasoning

Forward reasoning

Backward reasoning

Consequences

Judgeprobabilities and utilities

Decide

Maintenance goal

Achievement goal

ObserveAct

Consequences

Consequences

Conclusion: Logic can be used to combine proactive, reactive and proactive thinking together with Decision Theory (and other ways of making decisions)