CHORD Semantics

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CHORD Semantics

• January, 2007

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F-Atoms

• User Defined Constraint
• AB
• AB-gtC, ABgtC
• AB(V)-gtC, AB(PT0)gtT1
• Built-in Constraint
• 1 Integer, abc String
• Integer Double
• 1toString()-gt1
• ...

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Monotonic OO Semantics

• General Rules
•    XY, YZ gt XZ.
•    XY, YMgtT gt XMgtT.
•    XMgtT gt XM-gtV.
•    XMgtT, XM-gtV gt VT.
•    XM-gtV1, XM-gtV2 gt V1 V2.
• XM(V)-gtR, XM(PT0)gtT1) gt VT0, RT1.
• Simple Inheritance
• XY, XZ gt Y Z.

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Non-MonotonicOO Semantics(Overriding Multiple
Inheritance)
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CHR Semantics x Open World Assumption

• Notation
• ConstraintStore Set(Constraint)
• Program Set(Rule)
• i initial constraint store
• p current program

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CHR Semantics x Open World Assumption

• R ConstraintStore x Program x ConstraintStore
• Reachable predicate
• R(cs, p) some constraint store reachable from
  cs by iteratively applying the rules from p
• I Constraint x ConstraintStore x Program? t,
  f, u
• Interpretation function
• Computes the truth-value of a constraint given
  some initial constraint store and set of rules
• t, the constraint can be proved true
• f, the constraint can be proved false
• u, the constraint cant be proved neither true
  nor false

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CHR Semantics x Open World Assumption

• ?c?s (c ? s ? s ? R(i,p) ? I(c,i,p) t)
• Everything appearing in some reachable state is
  true
• ?c?s (I(c,i,p) u ? c ? s ? s ? R(i,p))
• Every undefined constraint does not appear in any
  reachable state
• ?c ?s (I(c,i,p) f ? c ? s ? s ? R(i,p) ? false
  ? R(i?c,p))
• Every false constraint causes a failed final
  state when added to some constraint store

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CHR Semantics x Open World Assumption

• Remarks
• We cant reason directly about negative or
  undefined facts in CHR
• The set of positive facts is partially observable
• We can prove some fact to be false by finding a
  proof for its negation (Reductio Ad Absurdum)

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OO Inheritance in OWA
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Closed World Semantics
c1 c2, c2m-gtb
true ? c1 c2, c2m-gtb, c1m-gtb false ? c1
c3, c3 c2 ... undefined ?
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Open World Semantics
c1 c2, c2m-gtb
true ??X (c1m-gtX), c1 c2, c2m-gtb false ?
c2 c1, c2m-gtc1, ... unknown ? c1m-gtb, c1
c3, c3 c2, ...
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Overriding in OWA (1st Version)

• XY, YM-gtgtV gt XM-gtgtV1
• Problems
• Too limited
• Unnatural
• Solution
• Use CWA locally for overriding missing facts
• if obtained facts do not contradict known facts
• Is this abduction?

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Locally Closed OO Semantics for CHORD
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Local Inheritance Context

• Definition
• a b, bm-gtd
• bm-gtd/a

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Local Overriding Inheritance Context

• Proposal
• Change the semantics from
• a b ? bm-gtd (1)
• To
• a b ? bm-gtd ? ??X,Y( aX ? Xb ?
  Xm-gtY) (2)
• In this case we can inherit
• am-gtd (3)
• If it comes directly from b
• If (2) is consistent with the current constraint
  store and program rules we say that (1) is a
  Consistent Overriding Local Inheritance Context

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Not all Local Overriding Inheritance Contexts are
consistent...
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c1m-gta gt c1c3. GOAL c1c2, c3c2,
c2m-gta, c3m-gtb c1c2, c2m-gta is a local
overriding inheritance context! c1m-gta/c2 is
NOT a consistent local overridinginheritance
context because ??X,Y( c1X ? Xc2 ? Xm-gtY)
is false for X c3, Y b
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Proposal
Constraint Store
Local Inheritance Contexts

• The final stores considers just the
  consistentlocal overriding inheritance contexts
• Use backtracking to find the consistentlocal
  inheritance contexts

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Important

• Theres no negation in CHR so
• Its not possible to directly prove anything
  like ??X,Y( aX ? Xb ? Xm-gtY)
• BUT we can look for a counterexample.

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Overriding in OWA (2nd Version) /1

• XY, YM-gtV gt
• XM-gtV1,
• ((VV1, XM-gtV1/Y) true)

1st option Suppose Im consistent and the value
of V can be directly inherited
2nd option Maybe Im not consistent
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Overriding in OWA (2nd Version) /2

• XM-gtV/Y, XC, CY, CM-gtVx gt false.

Is there any provable counterexample? Backtrack.
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Multiple Inheritance
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Local Source Based Multiple Inheritance Context

• Proposal
• Change the semantics from
• a b ? bm-gtd (1)
• To
• a b ? bm-gtd ? ??X,Y,T( b?X ? aX ? ?bX ?
  ?Xb ? (Xm-gtY ? XmgtT)) (2)
• In this case we can inherit
• am-gtd (3)
• If no other unrelated superclass defines m (for
  any value)
• If (2) is consistent with the current constraint
  store and program rules we say that (1) is a
  Consistent Local Source Based Multiple
  Inheritance Context

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Local Value Based Multiple Inheritance Context

• Proposal
• Change the semantics from
• a b ? bm-gtd (1)
• To
• a b ? bm-gtd ? ??X,Y,T( b?X ? aX ? ?bX ?
  ?Xb ? Xm-gtY ? Y ? d) (2)
• In this case we can inherit
• am-gtd (3)
• If no other unrelated superclass defines m to be
  d
• If (2) is consistent with the current constraint
  store and program rules we say that (1) is a
  Consistent Local Value Based Multiple Inheritance
  Context

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Multiple Inheritance in OWA

• We cant do this change only by the means of
  extra rules
• We cant prove negative constraints
  like??X,Y,T( b?X ? aX ? ?bX ? ?Xb ?
  (Xm-gtY ? XmgtT)))
• We cant find a counterexample for it
• We would need to prove ?Xb
• We need to change the default semantics of CHR

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Possible Solutions

• Adopt a less restrictive multiple inheritance
  semantics
• Extend CHR semantics to handle negation

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Less Restrictive Multiple Inheritance Semantics
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Proposal

• We do not need extra rules to handle multiple
  inheritance. Ex
• ab, ac, bm-gt2, bm-gt3
• This constraint store is false, because we are
  going to inherit am-gt2 and am-gt3 for a
• This is a similar approach to current programming
  languages supporting multiple inhertance (e. g.
  C)
• Multiple inheritance conflicts cause compilation
  or runtime errors.

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Changing default CHR semantics
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Source Based Multiple inheritance in OWA

• XM-gtV/Y, XC, ?CY, ?YC, CM-gtVx gt
  Y?C false.
• XM-gtV/Y, XC, ?CY, ?YC, CMgtT gt
  Y?C false.

Is there any provable counterexample? Backtrack.
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Value Based Multiple inheritance in OWA

• XM-gtV/Y, XC, ?CY, ?YC, CM-gtV gt
  Y?C false.

Is there any provable counterexample? Backtrack.
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CHRD
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Negation as Absense

• Extending CHR with negation as Absence
  Schrijvers et al 2006
• p \\ q gt r
• If p is present and q is absent, then add r
• Conclusion
• Lost declarativity
• Lost theoretical properties
• Non-logical negation
• We need logical negation!

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Negation in Integrity Constraints Abduction

• An Experimental CLP Platform for Integrity
  Constraints and Abduction Abdennadher, 2000
• For each predicate p, generate an abducible
  predicate ?p characterized by the integrity
  constraint
• p, ?p gt false

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Negation in Integrity Constraints Abduction

• Conclusion
• Doesnt properly handle negation in rule head
• ?p may be true even if theres no ?p in the
  constraint store
• a gt false
• p gt a
• b, ?p gt c
• d, c gt t
• Initial store b, d
• t is true, however Abdennadher cant prove it

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My Proposal

• For each user defined constraint p
• allow the constraint ?p having the same arity
• add the following integrity constraint
• p(X), ?p(Y) gt X Y false
• Change rule head matching semantics

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New rule semantics

• p0, ..., pn gt g b
• If
• p0,...,pk match some constraint set in the
  current constraint store
• At least one constraint in the rule head must be
  on the constraint store (avoids trivial non
  termination)
• Adding (?pk1 ... ?pn) to current
  constraint store doesnt lead to a failed state
• Guard holds
• Then
• Add body to the current constraint store

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Remarks

• This approach
• adds logical negation to CHR
• Generalizes the semantics of CHR rule matching
• CHRD rule fires if theres a set of matching
  constraints on the constraint store
• CHRD ? rule fires if there is a proof for the
  existence of matching constraints for the rule
  head
• Looking for matching constraints is still proving
  them
• Adding (r ? s) to current constraint store
  means
• T ? (r ? s ) ?
• T (r ? s ) ? ?
• T ?(r ? s )
• T ?r ? ?s

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Example

• R1 _at_ p gt false
• R2 _at_ b, ?p gt c
• R3 _at_ d, c gt t
• Initial store b, d

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Example Extended Program

• R1 _at_ p gt false
• R2 _at_ b, ?p gt c
• R3 _at_ d, c gt t
• E1 _at_ b, ?b gt false
• E2 _at_ c, ?c gt false
• E3 _at_ d, ?d gt false
• E4 _at_ p, ?p gt false

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Example Execution

• R1 _at_ p gt false
• R2 _at_ b, ?p gt c
• R3 _at_ d, c gt t
• E1 _at_ b, ?b gt false
• E2 _at_ c, ?c gt false
• E3 _at_ d, ?d gt false
• E4 _at_ p, ?p gt false

Store b, d Rule try R3, (trying ?c)
Store b, d, ?c Rule try R2 (trying p)
Store b, d, ?c, p Rule R1 failed state,
backtrack Store b, d, ?c, c Rule E2
failed state, backtrack Store b, d, t
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Example with Variables

• R1 _at_ p(2) gt false.
• R2 _at_ q(X), ?p(X) gt s(X).
• Initial store q(2)

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Example with Variables Extended Program

• R1 _at_ p(2) gt false.
• R2 _at_ q(X), ?p(X) gt s(X).
• E1 _at_ p(X), ?p(Y) gt X Y false.
• E2 _at_ q(X), ?q(Y) gt X Y false.
• E3 _at_ s(X), ?s(Y) gt X Y false.

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Example with Variables Execution

• R1 _at_ p(2) gt false.
• R2 _at_ q(X), ?p(X) gt s(X).
• E1 _at_ p(X), ?p(Y) gt X Y false
• E2 _at_ q(X), ?q(Y) gt X Y false
• E3 _at_ s(X), ?s(Y) gt X Y false

Store q(2) Rule try R2 (trying p(2))
Store q(2), p(2) Rule R1, failed state,
backtrack Store q(2), s(2)
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Future Work on CHRD

• Investigate termination of CHRD programs
• Rules may be appliable even with no matching
  constraint at current constraint store
• Investigate variables in rule head
• How to deal with not found constraints containing
  uninstantiated variables?