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

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

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

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

17. 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

18. 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!

19. The Ramifying Fluent Calculus • Modeling Ramifications

20. 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-

21. Abbreviations

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

23. State Update Axiomwith Ramifications • Simple • Disjunctive

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

25. 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

26. The Concurrent Fluent Calculus: Foundational Axioms

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

28. The Concurrent Fluent Calculus

29. The Concurrent Fluent Calculus

30. 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 *

31. 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

32. The Meta Fluent Calculus: The Knows Predicate

33. 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

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

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

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

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

38. 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)

39. 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

40. Flux

41. FLUX Example of Belief Updatewith the Simple Fluent Calculus

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