Abductive Logic Programming Agents

1. 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

2. Abductive logic programming (ALP) agent model An agent Maintenance goal Achievement goal Judge probabilities and utilities Forward reasoning using beliefs Backward Reasoning using beliefs Consequences Consequences Consequences Forward reasoning using beliefs Decide Observe Act The World

3. 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

4. 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.

5. The London underground GoalIf there is an emergency then I get help. Beliefs A 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.

6. ALP combines forward and backward reasoning The world If there is an emergency then get help There is an emergency get help Forward reasoning alert the driver There is a fire Backward reasoning press the alarm signal button observe act

7. 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 B or 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.)

8. 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  Δ • IC 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.

9. ALP gives a logical semantics to production rules. • Logical rules used to reason forward canbe • represented by LP clauses, with forward reasoning. • Reactive rulesthat implement stimulus-response associations can be represented by integrity constraints, with forward reasoning. • Pro-active rulesthat 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.

10. 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.

11. 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 onlyexplanation that satisfies the integrity constraint is • the sprinkler was on.

12. 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

13. ALP can be used to generate actions Program: there is an emergency if there is a fire you get help if you alert the driver you 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 solution you press the alarm signal button

14. ALP can be used for default reasoning Program: X can fly if X is a bird and normal X X 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

15. 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

16. 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.

17. Preactive thinking can be applied at different levels of detail Maintenance goal Achievement goal Judge probabilities and utilities Consequences Backward reasoning Forward reasoning Consequences Consequences Forward reasoning Decide Act Observe

18. 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).

19. 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, and add them all up. u11 p11 u12 p12 Expected utility of action1 p11·u11+p12·u12+p13·u13+p14·u14 u13 action1 p13 u14 p14 u21 action2 p21 Expected utility of action2 p21·u21+p22·u22+p23·u23+p24·u24 u22 p22 u23 p23 p24 u24 Choose the action of highest expected utility

20. Deciding whether or not to carry an umbrella Assume 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 .9 Expected utility 0 -10·.1 + 1·.9 = -1 + .9 = -.1 Decide I do not carry an umbrella!

21. 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.

22. The Prisoner’s Dilemma GoalI turn state witness or I do not turn state witness BeliefsA prisoner gets 0 years in jail if the prisoner turns state witness and 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 witness and the other prisoner does too. A prisoner gets 1 year in jail if the prisoner does not turn state witness and the other prisoner does not turn state witness too. 

23. 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.

24. In Classical Logic Given the additional belief the other prisoner turns state witness or the other prisoner does not turn state witness. Infer If I turn state witness then 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.

25. In Decision Theory • Assume Probability the other prisoner turns state witness = .5 • Probability the other prisoner does not turn state witness =.5 • Utility of getting N years in jail = N • Assume I turn state witness • Infer Probability I get 0 years in jail = .5 • Probability 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 = .5 • Probability I get 1 years in jail = .5 • Expected utility .5·4 + .5·1 = 2.5 years in jail • DecideI turn state witness

26. Conclusion: Logic can be used to combine proactive, reactive and proactive thinking together with Decision Theory (and other ways of making decisions) Maintenance goal Achievement goal Judge probabilities and utilities Consequences Backward reasoning Forward reasoning Consequences Consequences Forward reasoning Decide Act Observe