Belief Update, Belief Revision and Planning with the Fluent Calculus and FLUX

1. Belief Update, Belief Revision and Planning with the Fluent Calculus and FLUX Cleyton Rodrigues Marcos Aurelio Pablo Santana Jacques Robin

2. Non-MonotonicReasoning • Monotonic Reasoning • KB |= f, then g, KB  g |= f • Inference engine only performs ask and tell to the KB, never retract • Non-monotonic reasoning • Allows KB |= f, and then KB  g | f • Previously derived facts can be retracted upon arrival (for example from sensors) of new, conflicting evidence

3. Sources ofNon-monotonicity • Ontological • Non-stationary Environment • KB must reflect environment changes as time goes by • Epistemological • Partially observable Environment • Decision making requires using abduction in addition to deduction • KB must reflect changes of the agent’s beliefs as new evidences become available through sensing

4. Non-Monotonic Reasoning Case Study:The Wumpus World (from: www.javaschool.com/about/images/wumpus.world.gif)

5. The Frame Problem • Almost everything stays unchanged between states • We want thus to specify just the modifications • The wrong way: At(Agent,x,s) Ù Adjacent(x,y) Þ At(Agent,y,Result(Go(x,y),s)) • Why? • This axiom says what changes but not what stays the same!

6. The Ramification Problem • Ex: • I go from home to the store, creating a new situation S’. In S’: • I am now at the store • The number of people in the store went up by 1 • The agent moves from (x,y) to (z,w) • If he holds something, this thing moves too to (z,w) • The hunter kills the wumpus in (x,y) • The wumpus is not in (x,y) anymore • Do we want to say all that in the action definition? • Actions have: • intended effects that satisfy the agent’s goal and justify their execution • side effects related to preconditions of other actions

7. The Qualification Problem • In the real world it is difficult to define the circumstances under which a given action is guaranteed to work. • think of everything that can go wrong! • e.g. grabbing an object may not work if it is slippery,glued down, a holographic projection, or the gripper fails, or the agent is struck by lightning… • if some conditions are left outthe agent is in danger of generating false beliefs

8. Knowledge Representation Language forNon-Monotonic Reasoning • The Situation Calculus • The Event Calculus • Transaction Logic • The Fluent Calculus FAZER UM RESUMO DAS CARACTERÍSTICAS DE CADA LINGUAGEM!!!

9. FluentCalculus

10. The Fluent Calculus Layers Non-Deterministic Environment Actions with Conditional Effects Communicative Multi-Agent Environment Partially Observable Environment Environments With Infinite Fluent Domains Noisy Sensors Noisy Actuators Concurrent Asynchronous Environment Ramifying Environment Communication  Belief Revision Ramifications Concurrency Meta-reasoning   FC Planning Simple FC

11. The Simple Fluent Calculus: Key Ideas • Sorted first-order logic language • Fluent • “A fluent represents a single atomic property of the physical world which may change in the course of time, in particular through manipulation by the robot.” • Ex: player has ball, player position, ball position, player speed… • State • “…a so-called state is a snapshot of the environment at a certain moment.” • Action • Pre-conditionAxioms • StateUpdateAxioms • SensingActions

12. The Simple Fluent Calculus • Sorts • FLUENT • STATE • ACTION • SIT • Functions •  :  STATE • State : SIT  STATE • Do : ACTION x SIT  SIT • o : STATE x STATE  STATE • Predicates • Poss : ACTION x STATE

13. The Simple Fluent Calculus:Foundational Axioms

14. The Simple Fluent Calculus:Abbreviations

15. The Simple Fluent Calculus:Pure State Formulas • Ex: • Holds(InRoom(r), z), Connects(d,r,r’)

16. The Simple Fluent Calculus:Action Precondition Axioms • Ex:

17. The Simple Fluent Calculus:State Update Axioms • Ex:

18. The Simple Fluent Calculus:MOF Metamodel o * State ----------------------------------------------- + minus(f:Fluent) : State + do(a:Action) : State * poss * * Action Situation * / holds Fluent *

19. Fluent Calculus models how the current state changes given the actions performed by the agents It can be used to “simulate” the outcomes of a given action sequence Ex: Do(Open(D12), Do(Go(R1), S0)) = InRoom(R1) o Opened(D12) o ... The entire planning problem can be modeled as the problem of finding a situation in which certain goal conditions are met: Ex: s Holds(HasGold, s) Planning with the Simple Fluent Calculus

20. CONSIDERAR COMO VAMOS FAZER COM O METAMODEL The Disjunctive Fluent Calculus • Non-deterministic actions: Alternative outcomes x Vagueness • Simple disjunctive state update axioms for non-deterministic actions • θn is a first-order formula without terms of any reserved sort • Example • Alternative outcomes:Poss(Shoot, s)  (State(Do(Shoot, s)) = State(s) o DeadWumpus) (State(Do(Shoot, s)) = State(s) o LiveWumpus) • Vagueness:Poss(Shoot, s)  Holds(Life(Wumpus, x),s) y (State(Do(Shoot, s)) = State(s) o Life(Wumpus, y)  y < x) • Planning: should consider all possible outcomes!

21. The Ramifying Fluent Calculus • Modeling Ramifications

22. The Ramifying Fluent Calculus • Sorted second-order logic language • Reserved Predicates: • Causes : STATE x STATE x STATE x STATE x STATE x STATE • Causes(z1, e1+, e1-, z2, e2+, e2-) If z1 is the result of positive effects e1+ and negative effects e1-, then an additional effect is caused which leads to z2 (now the result of positive and negative effects e2+ and e2-, resp.) • Ramify : STATE x STATE x STATE x STATE • Ramify(z, e+, e-, z’) z’ can be reached by iterated application of the underlying casual relation, starting in state z with momentum e+ and e-

23. Abbreviations

24. Foundational Axioms (Reflexive and Transitive Closure of Causes)

25. State Update Axiomwith Ramifications • Simple • Disjunctive

26. CONSIDERAR COMO VAMOS FAZER COM O METAMODEL Causal Relations Axiomatization • Relies on the assumption that the underlying Causes relation is completely specified

27. The Concurrent Fluent Calculus • Allows agents to execute more than one action at a time concurrently • Motivation • Parallel actions lead to shorter plans and less execution time • Certain effects may only be achievable by simultaneous actions • Sorted second-order logic language • Extensions to Simple Fluent Calculus • Sorts • ACTION < CONCURRENT • Functions • :  CONCURRENT • . : CONCURRENT x CONCURRENT  CONCURRENT • Do : CONCURRENT x SIT  SIT • DirState, DirEffect+, DirEffect-: CONCURRENT x SIT  STATE • Predicates • Poss : CONCURRENT x STATE • Affects : CONCURRENT x CONCURRENT

28. The Concurrent Fluent Calculus: Foundational Axioms

29. The Concurrent Fluent Calculus • Action Preconditions • Recursive state update axiom • Disjunctive recursive state update axiom

30. The Concurrent Fluent Calculus

31. The Concurrent Fluent Calculus

32. The Concurrent Fluent Calculus:MOF Metamodel affects . * * o poss Concurrent * * State ----------------------------------------------- + minus(f:Fluent) : State + do(a:Action) : State Action * * / holds * Fluent Situation -------------------------------------------------------- + do(c:Concurrent) : Situatiuon + dirState(c:Concurrent) : State + dirEffectsPlus(c:Concurrent) : State) + dirEffectsMinus(c:Concurrent) : State *

33. The Meta Fluent Calculus • Extends the Simple Fluent Calculus in order to allow handling partial knowledge about the environment • Extensions: • Predicates • KState : SIT x STATE • Knows : FluentFormula x SIT • Kwhether : FluentFormula x SIT • Unknown : FluentFormula x SIT • Functions •  :  ACTION • ; : ACTION x ACTION  ACTION • If : FLUENT x ACTION x ACTION  ACTION

34. The Meta Fluent Calculus: The Knows Predicate

35. The Meta Fluent Calculus: The Unknown Predicate ?

36. The Meta Fluent Calculus: Knowledge Update Axioms • Define how states satisfying KState(s,z) relate to the states satisfying KState(Do(a,s),z) • Sensing Actions • Reduce the set of possible states • Physical Actions • Reflect what the agent knows about the effects of the respective action

37. The Meta Fluent Calculus: Knowledge Update by Sensing Actions • Knowledge Update Axioms for accurate sensing • Ex:

38. The Meta Fluent Calculus: Knowledge Update by Sensing Actions • Knowledge Update Axioms for sensing • Ex:

39. The Meta Fluent Calculus: Knowledge Update by Physical Actions • Knowledge Update Axioms • Ex:

40. The Meta Fluent Calculus: Conditional Actions • Extending the foundational axioms: • Ex: Ex: Do(Sense(Closed(DA3)) ; If(Closed(DA3), SendId, ); Enter(R403), S0)

41. The Meta Fluent Calculus: Planning • Simple fluent calculus does not allows planning in platially known states (with unknown fluents) • Ex: • Conditional Actions allow agents to condition their actions on the outcome of sensing:Ex: Do(Sense(Closed(DA3)) ; If(Closed(DA3), SendId, ); Enter(R403), S0)

42. The Meta Fluent Calculus: MOF metamodel o KState * * ; * State ----------------------------------------------- + minus(f:Fluent) : State + do(a:Action) : State * poss * * Action Situation * * * * false true \ Unknown If \ Kwhether / holds \ Knows * * Fluent condition

43. Flux

44. FLUX Predicates for Belief Updatewith the Simple Fluent Calculus

45. FLUX Example of Belief Updatewith the Simple Fluent Calculus

46. FLUX Predicates for Planning and Actingwith the Simple Fluent Calculus

47. FLUX Example of Planning and Acting with the Simple Fluent Calculus

48. FLUX Predicates for Belief Updatewith the Ramifying Fluent Calculus

49. FLUX Example of Belief Updatewith the Ramifying Fluent Calculus

50. FLUX Predicates for Belief Updatethe Disjunctive Fluent Calculus