The Complexity and Generality of Learning Answer Set Programs (AIJ 2018) Mark Law, Alessandra Russo and Krysia Broda September 2, 2018 1/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
ILP under the Answer Set Semantics ◮ Several ILP frameworks have been proposed to learn ASP: ◮ In ILP b (resp ILP c ) at least one (resp every ) answer set of B ∪ H must cover the (atom) examples. ◮ In ILP LAS examples are partial interpretations and a combination of ILP b and ILP c can be expressed. 2/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
ILP under the Answer Set Semantics ◮ Several ILP frameworks have been proposed to learn ASP: ◮ In ILP b (resp ILP c ) at least one (resp every ) answer set of B ∪ H must cover the (atom) examples. ◮ In ILP LAS examples are partial interpretations and a combination of ILP b and ILP c can be expressed. ◮ This paper asks two fundamental questions: ◮ What class of ASP programs can each framework learn? ◮ Is there any (complexity) price paid by the more general frameworks? 2/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
ILP under the Answer Set Semantics ◮ Several ILP frameworks have been proposed to learn ASP: ◮ In ILP b (resp ILP c ) at least one (resp every ) answer set of B ∪ H must cover the (atom) examples. ◮ In ILP LAS examples are partial interpretations and a combination of ILP b and ILP c can be expressed. ◮ This paper asks two fundamental questions: ◮ What class of ASP programs can each framework learn? ◮ Is there any (complexity) price paid by the more general frameworks? ◮ In the paper we also consider ILP sm , ILP LOAS and ILP context LOAS . 2/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
One-to-one Distinguishability Definition 1 A learning framework F can distinguish H 1 from H 2 wrt B iff there is at least one task T F = � B , E F � such that H 1 ∈ F ( T F ) and H 2 �∈ F ( T F ). 3/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
One-to-one Distinguishability Definition 1 A learning framework F can distinguish H 1 from H 2 wrt B iff there is at least one task T F = � B , E F � such that H 1 ∈ F ( T F ) and H 2 �∈ F ( T F ). ◮ D 1 1 ( F ) is the set of tuples � B , H 1 , H 2 � such that F can distinguish H 1 from H 2 wrt B . 3/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
One-to-one Distinguishability Definition 1 A learning framework F can distinguish H 1 from H 2 wrt B iff there is at least one task T F = � B , E F � such that H 1 ∈ F ( T F ) and H 2 �∈ F ( T F ). ◮ D 1 1 ( F ) is the set of tuples � B , H 1 , H 2 � such that F can distinguish H 1 from H 2 wrt B . � 0 { p } 1 . � Let B = ∅ , H 1 = { p . } and H 2 = . ◮ � B , H 1 , H 2 � is not in D 1 1 ( ILP b ). 3/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
One-to-one Distinguishability Definition 1 A learning framework F can distinguish H 1 from H 2 wrt B iff there is at least one task T F = � B , E F � such that H 1 ∈ F ( T F ) and H 2 �∈ F ( T F ). ◮ D 1 1 ( F ) is the set of tuples � B , H 1 , H 2 � such that F can distinguish H 1 from H 2 wrt B . � 0 { p } 1 . � Let B = ∅ , H 1 = { p . } and H 2 = . ◮ � B , H 1 , H 2 � is not in D 1 1 ( ILP b ). E + = { p } E − = ∅ H 2 ∈ ILP b ( � B , { p } , ∅� ). 3/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
One-to-one Distinguishability Definition 1 A learning framework F can distinguish H 1 from H 2 wrt B iff there is at least one task T F = � B , E F � such that H 1 ∈ F ( T F ) and H 2 �∈ F ( T F ). ◮ D 1 1 ( F ) is the set of tuples � B , H 1 , H 2 � such that F can distinguish H 1 from H 2 wrt B . � 0 { p } 1 . � Let B = ∅ , H 1 = { p . } and H 2 = . ◮ � B , H 1 , H 2 � is not in D 1 1 ( ILP b ). ◮ � B , H 2 , H 1 � is in D 1 1 ( ILP b ). 3/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
One-to-one Distinguishability Definition 1 A learning framework F can distinguish H 1 from H 2 wrt B iff there is at least one task T F = � B , E F � such that H 1 ∈ F ( T F ) and H 2 �∈ F ( T F ). ◮ D 1 1 ( F ) is the set of tuples � B , H 1 , H 2 � such that F can distinguish H 1 from H 2 wrt B . � 0 { p } 1 . � Let B = ∅ , H 1 = { p . } and H 2 = . ◮ � B , H 1 , H 2 � is not in D 1 1 ( ILP b ). ◮ � B , H 2 , H 1 � is in D 1 1 ( ILP b ). E + = ∅ E − = { p } H 2 ∈ ILP b ( � B , ∅ , { p }� ) but H 1 �∈ ILP b ( � B , ∅ , { p }� ). 3/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
One-to-one Distinguishability Definition 1 A learning framework F can distinguish H 1 from H 2 wrt B iff there is at least one task T F = � B , E F � such that H 1 ∈ F ( T F ) and H 2 �∈ F ( T F ). ◮ D 1 1 ( F ) is the set of tuples � B , H 1 , H 2 � such that F can distinguish H 1 from H 2 wrt B . � 0 { p } 1 . � Let B = ∅ , H 1 = { p . } and H 2 = . ◮ � B , H 1 , H 2 � is not in D 1 1 ( ILP b ). ◮ � B , H 2 , H 1 � is in D 1 1 ( ILP b ). ◮ � B , H 1 , H 2 � is in D 1 1 ( ILP c ). 3/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
One-to-one Distinguishability Definition 1 A learning framework F can distinguish H 1 from H 2 wrt B iff there is at least one task T F = � B , E F � such that H 1 ∈ F ( T F ) and H 2 �∈ F ( T F ). ◮ D 1 1 ( F ) is the set of tuples � B , H 1 , H 2 � such that F can distinguish H 1 from H 2 wrt B . � 0 { p } 1 . � Let B = ∅ , H 1 = { p . } and H 2 = . ◮ � B , H 1 , H 2 � is not in D 1 1 ( ILP b ). ◮ � B , H 2 , H 1 � is in D 1 1 ( ILP b ). ◮ � B , H 1 , H 2 � is in D 1 1 ( ILP c ). E + = { p } E − = ∅ H 1 ∈ ILP c ( � B , ∅ , { p }� ) but H 2 �∈ ILP c ( � B , ∅ , { p }� ). 3/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
One-to-one Distinguishability Conditions Sufficient/necessary condition for � B , H 1 , H 2 � to be in D 1 Framework F 1 ( F ) ILP b AS ( B ∪ H 1 ) �⊆ AS ( B ∪ H 2 ) AS ( B ∪ H 1 ) �⊆ AS ( B ∪ H 2 ) ILP sm ILP c AS ( B ∪ H 1 ) � = ∅ ∧ ( AS ( B ∪ H 2 ) = ∅ ∨ ( E c ( B ∪ H 1 ) �⊆ E c ( B ∪ H 2 ))) ILP LAS AS ( B ∪ H 1 ) � = AS ( B ∪ H 2 ) ILP LOAS ( AS ( B ∪ H 1 ) � = AS ( B ∪ H 2 )) ∨ ( ord ( B ∪ H 1 ) � = ord ( B ∪ H 2 )) ( B ∪ H 1 �≡ s B ∪ H 2 ) ∨ ILP context LOAS ( ∃ C ∈ ASP ch st ord ( B ∪ H 1 ∪ C ) � = ord ( B ∪ H 2 ∪ C )) 4/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
One-to-one Distinguishability Conditions Sufficient/necessary condition for � B , H 1 , H 2 � to be in D 1 Framework F 1 ( F ) ILP b AS ( B ∪ H 1 ) �⊆ AS ( B ∪ H 2 ) AS ( B ∪ H 1 ) �⊆ AS ( B ∪ H 2 ) ILP sm ILP c AS ( B ∪ H 1 ) � = ∅ ∧ ( AS ( B ∪ H 2 ) = ∅ ∨ ( E c ( B ∪ H 1 ) �⊆ E c ( B ∪ H 2 ))) ILP LAS AS ( B ∪ H 1 ) � = AS ( B ∪ H 2 ) ILP LOAS ( AS ( B ∪ H 1 ) � = AS ( B ∪ H 2 )) ∨ ( ord ( B ∪ H 1 ) � = ord ( B ∪ H 2 )) ( B ∪ H 1 �≡ s B ∪ H 2 ) ∨ ILP context LOAS ( ∃ C ∈ ASP ch st ord ( B ∪ H 1 ∪ C ) � = ord ( B ∪ H 2 ∪ C )) ◮ Neither ILP b of ILP sm can distinguish H ∪ C from H for any constraint C and any H – in practice, neither ILP b nor ILP sm can learn constraints. 4/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
One-to-one Distinguishability Conditions Sufficient/necessary condition for � B , H 1 , H 2 � to be in D 1 Framework F 1 ( F ) ILP b AS ( B ∪ H 1 ) �⊆ AS ( B ∪ H 2 ) AS ( B ∪ H 1 ) �⊆ AS ( B ∪ H 2 ) ILP sm ILP c AS ( B ∪ H 1 ) � = ∅ ∧ ( AS ( B ∪ H 2 ) = ∅ ∨ ( E c ( B ∪ H 1 ) �⊆ E c ( B ∪ H 2 ))) ILP LAS AS ( B ∪ H 1 ) � = AS ( B ∪ H 2 ) ILP LOAS ( AS ( B ∪ H 1 ) � = AS ( B ∪ H 2 )) ∨ ( ord ( B ∪ H 1 ) � = ord ( B ∪ H 2 )) ( B ∪ H 1 �≡ s B ∪ H 2 ) ∨ ILP context LOAS ( ∃ C ∈ ASP ch st ord ( B ∪ H 1 ∪ C ) � = ord ( B ∪ H 2 ∪ C )) ◮ ILP LAS can distinguish any two hypotheses, so long as they have different answer sets (when combined with B ). 4/8 Mark Law, Alessandra Russo and Krysia Broda The Complexity and Generality of Learning Answer Set Programs (AIJ 2018)
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