CHR + O Operational Semantics in Fluent Calculus (Using Knowledge & Ramifications)

1. CHR + O Operational Semantics in Fluent Calculus(Using Knowledge & Ramifications) December, 2007

2. SimpleFluent Calculus (SFC)

3. Introduction • A many-sorted first-order language with equality • Includes: • Sorts: FLUENT < STATE, ACTION, SIT • Functions: • Predicate

4. Abbreviations

5. Foundational Axioms (Fstate)

6. SFC Domain Axiomatization • State Constraints • Unique simple Action Precondition Axiom for each function symbol with range ACTION • A set of State Update Axioms • Foundational Axioms (Fstate) • Possibly further domain-specific axioms

7. Action Precondition Axiom • Ex:

8. State Update Axiom • Ex:

9. Knowledge & Sensing

10. Introduction • Extends Simple Fluent Calculus • KState : SIT x STATE • Kstate(s, st) • The state st can be the current agent state at the situation s • Models the uncertainty about current agent’s state (since the agent can’t know what state is the real one)

11. Definitions

12. Foundational Axiom

13. Knowledge Update Axiom

14. Knowledge Update Axiom Example

15. Ramifications in Fluent Calculus

16. Modeling Ramifications

17. Fluent Calculus with Ramifications • 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-

18. Abbreviations

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

20. State Update Axiomwith Ramifications

21. Causal Relations Axiomatization • Relies on the assumption that the underlying Causes relation is completely specified

22. Fluent Calculus Domain Axiomatizationwith Ramifications • State constraints • Causal Relations axiomatization • Unique action precondition axiom for each function symbol with range ACTION • Set of state update axioms (possibly with ramifications) • Foundational Axioms: Fstate and Framify • Domain Specific Axioms

23. CHR Operational Semantics in Fluent Calculus

24. Domain Sorts • RULE • Ex: gcd(0) <=> true | true. • CONSTRAINT • p(1,2,7) • UDC < CONSTRAINT • BIC < CONSTRAINT • EQUATION < BIC

25. Domain Functions • S0 :  SIT • The initial situation / the initial constraint store

26. Domain Predicates (1/3) InQuery : CONSTRAINT The constraint is in the initial goal (query) Matching : Seq(UDC) x Seq(UDC) x Set(EQUATION) x STATE Matching(h, m, e, s) The sequence of constraints m is in s and matches with h generating the matching equations in e

27. Domain Predicates (2/3) RenVars : RULE x RULE x STATE RenVars(r, rc, s) Rc is a copy of the rule r with all variables renamed (using names not already being used in s) Ex: RenVars(“p(x)  q(x)”, “p(x1)  q(x1)”, s) If x1 does not appear in s

28. Domain Predicates (3/3) entails : STATE x Set(EQUATION) x Set(BIC) entails(s, h, g) CT |= s  \exists x(h ^ g)

29. Domain Actions • Simplify: RULE x Seq(UDC)  ACTION • Propagate: RULE x Seq(UDC) ACTION • Simpagate: RULE x Seq(UDC)x Seq(UDC)  ACTION

30. Initial State

31. SimpagatePre-condition

32. SimpagateKnowledge Update Axiom

33. CHR + O Operational Semantics in Fluent Calculus(Using Knowledge & Ramifications

34. Domain Sorts • FATOM < CONSTRAINT • TAXONOMY_FATOM < FATOM < UDC • a:b • SUBCLASS_FATOM < FATOM < UDC • a::b • FIELD_FATOM < FATOM < UDC • a[bc] • OPERATION_FATOM < FATOM < BIC • a[b(c,d,e)  f]

35. Initial State

36. SimpagateKnowledge Update Axiom

37. Causal Relations Axiomatization