1. CHR + D + A + O Operational Semantics in Fluent Calculus December, 2007

2. Simple Fluent 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 (F state )

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 (F state ) Possibly further domain-specific axioms

7. Pure State Formula It may contain then: Hold(f,z), where f is a fluent and z is a state Any domain predicates referencing only the domain sorts

8. State Constraints Ex:

9. Action Precondition Axiom Ex:

10. State Update Axiom Ex: Question: Is it possible to change a previously “unknown” number of fluents at once? “Delta” is a pure state formula, so it can’t be used to generate the needed fluents. V+ and V- are just “fluent sets”... Can I use a function to generate these fluent sets?

11. CHR Operational Semantics in Fluent Calculus

12. Domain Sorts (1/2) RULE –Ex: gcd(0) true | true. RULE_ID ATOM < CONSTRAINT –p(1,2,7) UDC < ATOM BIC < ATOM EQUATION < BIC

13. Domain Sorts (2/2) FORMULA < CONSTRAINT CONJUNCTION < FORMULA EQ_CONJ < CONJUNCTION BIC_CONJ < CONJUNCTION UDC_CONJ < CONJUNCTION

14. “Domain” Sorts BICS_STATE < STATE UDCS_STATE < STATE GOAL_STATE < STATE Question: Can I specialize the reserved sorts? Are the specialized ones considered “Domain” sorts?

15. Domain Functions (1/5) S 0 :  SIT –The initial situation InGoal : CONSTRAINT  FLUENT –The constraint x is in G InUdc : UDC  FLUENT –The user defined constraint x is in U InBic : BIC  FLUENT –The built-in constraint x is in B G – Goal U – User Defined Constraints Store B – Built-in Constraint Store H – Propagation History0

16. Domain Functions (2/5) RuleId : RULE  RULE_ID –Returns an unique identifier for the passed rule InPropHistory : RULE_ID x UDC_CONJ x EQ_CONJ  FLUENT –f = InPropHistory(rule, constraints, equations) –The ‘rule’ was executed with the conjunction of matching ‘constraints’ generating the conjunction of ‘equations’

17. Domain Functions (3/5) MakeGoal : CONJUNCTION  GOAL_STATE –Makes an state containing an InGoal(c) fluent for each constraint c in the conjunction MakeBICS : BIC_CONJ  BICS_STATE –Makes an state containing an InBic(c) fluent for each constraint c in the conjunction MakeUDCS : UDC_CONJ  UDCS_STATE –Makes an state containing an InUdc(c) fluent for each constraint c in the conjunction

18. Domain Functions (4/5) “,” : CONSTRAINT x CONSTRAINT  CONJUNCTION  : CONSTRAINT x CONJUNCTION true :  BIC false :  BIC

19. Domain Functions (5/5) SimplifRule : UDC_CONJ x BIC_CONJ x CONJUNCTION  RULE PropagRule : UDC_CONJ x BIC_CONJ x CONJUNCTION  RULE SimpagRule : UDC_CONJ x UDC_CONJ x BIC_CONJ x CONJUNCTION  RULE These functions make up a rule from its parts

20. Domain Predicates (1/3) InQuery : CONSTRAINT –The constraint is in the initial goal (query) DisjointVariables : RULE x CONSTRAINT –DisjointVariables(rule, constraint) –No variable in ‘rule’ appears in ‘constraint’

21. Domain Predicates (2/3) Matching : UDC_CONJ x UDC_CONJ x EQ_CONJ –Matching(head, matching, equations) –The conjunction of udc constraints ‘head’ matches the conj. of udc constraints ‘matching’ generating the conj. of matching equations ‘equations’ Matching(“p(X), q(Y)”,“p(A), p(2)”,“X=A, Y=2”) Matching(“p(X), q(Y)”,“p(A), p(2)”, “X=2, Y=A”)

22. Domain Predicates (3/3) checkEntails : BIC_CONJ x EQ_CONJ x BIC_CONJ –checkEntails(bics, head, guard) –CT |= bics   x (head ^ guard)

23. Domain Actions Solve: BIC  ACTION Introduce: UDC  ACTION Simplify: RULE x UDC_CONJ x EQ_CONJ  ACTION Propagate: RULE x UDC_CONJ x EQ_CONJ  ACTION Simpagate: RULE x UDC_CONJ x UDC_CONJ x EQ_CONJ  ACTION

24. Abbreviations (1/2) These expressions cannot be defined as domain predicates, since a pure state formula forces all predicates to refer just to domain sorts.

25. Abbreviations (2/2)

26. Initial State (State Constraint)

27. CHR Transitions

28. Introduce g – goal being introduced (UDC) z – state (STATE) s – situation (SIT)

29. Solve g – goal being solved (BIC) z – state (STATE) s – situation (SIT)

30. Simplify hR – Rule Head (UDC_CONJ) g – Rule guard (BIC_CONJ) b – Rule body (UDC_CONJ) z – state (STATE) m – matching constraints (UDC_CONJ) e – matching equations (EQ_CONJ) bics – the BIC store as a conjunction of BICS (BIC_CONJ)

31. hR – Rule Head (UDC_CONJ) g – Rule guard (BIC_CONJ) b – Rule body (UDC_CONJ) s – situation (SIT) m – matching constraints (UDC_CONJ) e – matching equations (EQ_CONJ) Simplify

32. Propagate hk – Rule Head (UDC_CONJ) g – Rule guard (BIC_CONJ) b – Rule body (UDC_CONJ) z – state (STATE) m – matching constraints (UDC_CONJ) e – matching equations (EQ_CONJ)

33. Propagate hk – Rule Head (UDC_CONJ) g – Rule guard (BIC_CONJ) b – Rule body (UDC_CONJ) s – situation (SIT) m – matching constraints (UDC_CONJ) e – matching equations (EQ_CONJ)

34. Simpagate hk – Rule Head Keep (UDC_CONJ) hR – Rule Head Remove (UDC_CONJ) g – Rule guard (BIC_CONJ) b – Rule body (UDC_CONJ) z – state (STATE) mk – matching head keep constraints (UDC_CONJ) mR – matching head remove constraints (UDC_CONJ) e – matching equations (EQ_CONJ)

35. Simpagate hk – Rule Head Keep (UDC_CONJ) hR – Rule Head Remove (UDC_CONJ) g – Rule guard (BIC_CONJ) b – Rule body (UDC_CONJ) s – situation (SIT) mk – matching head keep constraints (UDC_CONJ) mR – matching head remove constraints (UDC_CONJ) e – matching equations (EQ_CONJ)

36. CHR + D Operational Semantics in Fluent Calculus

37. Nondeterminism in Fluent Calculus Simple disjunctive state update axiom –θ n is a first-order formula without terms of any reserved sort

38. Domain Sorts DISJUNCTION < FORMULA CHOICE < DISJUNCTION –((a11,..., a1n) ;... ; (am1,...,amn)), where all aij are ATOM’s

39. Domain Functions “;” : CONSTRAINT x CONSTRAINT  DISJUNCTION  : CONSTRAINT x FORMULA

40. Domain Actions Split : CHOICE  ACTION

41. Actions

42. Split c – choice being splitted (CHOICE) bi – a component of the choice (CONJUNCTION) z – state (STATE) s – situation (SIT)

43. CHR + D + A Operational Semantics in Fluent Calculus

44. Domain Sorts JUSTIFICATION < CONJUNCTION –“a1,..., an”, where all ai are atoms

45. Domain Functions (1/3) Just : CONSTRAINT x JUSTIFICATION  FLUENT –Just(c,j) –The constraint c is justified by j MakeJust : CONJUNCTION x JUSTIFICATION  STATE –Makes an state containing an Just(c,j) fluent for each constraint c in the conjunction J - Justifications

46. Domain Functions (2/3) SelectJust : JUSTIFICATION x STATE  STATE –SelectJust(j,z) –Selects the fluents of the form “Just(c,jc)” in z such that JustifiedBy(c, j, z) SelectConstraint : JUSTIFICATION x STATE  STATE –SelectConstraint(j,z) –Selects the fluents of the form “InUdc(c)”, “InBic(c)” or “InGoal(c)” in z such that JustifiedBy(c, j, z)

47. Domain Functions (3/3) SelectPropHistory : JUSTIFICATION x STATE  STATE –SelectPropHistory(j,z) –Selects the fluents of the form “PropHistory(rule,matched,equations)” in z such that for some constraint c in ‘matched’, JustifiedBy(c, j, z)

48. Abbreviations

49. Domain Actions AddConstraint : CONSTRAINT x JUSTIFICATION  ACTION RemoveConstraint : JUSTIFICATION  ACTION

50. Initial State

51. State Constraints

52. Actions

53. AddConstraint

54. RemoveConstraint

55. Simpagate j – the justification for the entire rule head

56. Split / Choice