aa r X i v : . [ c s . A I] J u l Imparo is complete by inverse subsumption
David TothJuly 16, 2014
Abstract
In Inverse subsumption for complete explanatory induction[YII12] Yamamoto et al. investigate which inductivelogic programming systems can learn a correct hypothesis H by using the inverse subsumption instead of inverseentailment. We prove that inductive logic programming system Imparo is complete by inverse subsumption forlearning a correct definite hypothesis H wrt the definite background theory B and ground atomic examples E , byestablishing that there exists a connected theory T for B and E such that H subsumes T . Keywords.
Imparo. Inverse subsumption. Inductive Logic Programming.
A task in Inductive Logic Programming (ILP) is given logic theories background knowledge B , and examples E to find a logic theory H that explains the examples E from the background knowledge and is consistent with thebackground knowledge, i.e. B ∧ H | = E and B ∧ H = f alse . Such a logic theory H is called a correct hypothesis wrt B and E and a system that takes as an input theories B and E and returns a correct hypothesis H is calledan ILP system.ILP systems find a hypothesis H using the principle of the inverse entailment[Mug95] for theories B , E , H : B ∧ H | = E ⇐⇒ B ∧ ¬ E | = ¬ H . First they construct an intermediate theory F called a bridge theory satisfyingthe conditions B ∧ ¬ E | = F and F | = ¬ H . Then as H | = ¬ F , they generalize the negation of the bridge theory F with the anti-entailment. However, the operation of the anti-entailment since being less deterministic may becomputationally more expensive than the operation of the inverse subsumption (anti-subsumption). ThereforeYamamoto et al. [YII12] investigate how the procedure of the inverse subsumption can be realized in ILP systemsin a complete way.The negation of Imparo’s bridge theory is called a connected theory. While Kimber proves that for everyhypothesis H there exists a connected theory T such that H entails T ( H | = T ), we prove that for every hypothesis H there exists a connected theory T such that H (theory-)subsumes T ( H (cid:23) T ) and hence extend Imparo’sprocedure for finding a hypothesis from anti-entailment to anti-subsumption preserving its completeness. Definition 1. [YII12]Let S and T be two clausal theories. Then, S theory-subsumes T , denoted by S (cid:23) T , if forany clause D ∈ T , there is a clause C ∈ S such that C (cid:23) D . The inverse relation of the (theory-)subsumption iscalled anti-subsumption . Definition 2. (Definition 2.83 in [Kim12]) An open program is a triple h B, U, I i where B is a program, U is a setof predicates called undefined or abducible , and I is a set of first-order axioms. If B is a definite program and I isa set of definite goals, then P is a definite open program . Definition 3. (Correct hypothesis) Let P = h B, U, I i be a definite open program, E a logic theory theory calledexamples, H is an inductive solution for P, E iff B ∪ H | = E and B ∪ H ∪ I = f alse . Definition 4. (Definition 4.1. in [Kim12] Let C be a program clause A ← { L , ..., L n } . The atom A is denoted by C + and the set { L , ..., L n } is denoted by C − . 1 efinition 5. (Definition 4.2 in [Kim12]). Let Σ be a set of m program clauses { C , ..., C m } . The set { C +1 , ..., C + m } is denoted by Σ + and the set C − ∪ ... ∪ C − m is denoted by Σ − . Definition 6. (Definition 4.3 in [Kim12]) Let h P = B, U, I i be a definite open program, and let E be a groundatom. Let T , ..., T n be n disjoint sets of ground definite clauses defining only predicates in U . T = T ∪ ... ∪ T n isan n -layered Connected Theory for P and E if and only if the following conditions are satisfied: • B | = T − n , • B ∪ T + n ∪ ... ∪ T + i +1 | = T − i , for all i (1 ≤ i < n ), • B ∪ T + n ∪ ... ∪ T +1 | = E , and • B ∪ T ∪ I is consistent. Definition 7. (Definition 4.4 in [Kim12]). Let P = h B, U, I i be a definite open program, and let E be a groundatom. A Connected Theory for P and E is an n -layered Connected Theory for P and E , for some n ≥ Definition 8. [Kim12] Let P = h B, U, I i be an open definite program, let E be a ground atom. A set H of definiteclauses is derivable from P and E by Connected Theory Generalisation , denoted
P, E ⊢ CT G H , iff there is a T suchthat T is a Connected Theory for P and E , and H | = T , and B ∪ H ∪ I is consistent. Theorem 9. (Implication by Ground Clauses [NCDW97]). Let Σ be a non-empty set of clauses, and C be a groundclause. Then Σ | = C if and only if there is a finite set Σ g of ground instances of clauses from Σ , such that Σ g | = C . Theorem 10.
Completeness of connected theory generalization (Theorem4.6 in [Kim12]) Let h B, U, I i be a definiteopen program, let H be a definite program, and let e be an atom. If H is an inductive solution for P and E = { e } ,then H is derivable from P and E by connected theory generalisation.Proof. [Kim12] The full proof is in Kimber’s PhD thesis [Kim12]. Since H is a correct hypothesis for P and E ,then B ∪ H | = E by definition. Therefore, by 9, there is a finite set S of ground instances of clauses in B ∪ H , suchthat S | = E . Let T = S ∩ ground ( H ). Since T ⊆ S , then T is ground and finite, and since T ⊆ ground ( H ) then H | = T . Then Kimber proves that T is a connected theory for P and E . We define a derivability of the hypothesis by the inverse subsumption.
Definition 11.
Let P = h B, U, I i be an open definite program, H be a correct hypothesis wrt P and a groundexample E , then H is derivable by connected theory inverse subsumption iff there exists a connected theory T for P and E such that H (cid:23) T . We denote the statement by P, E ⊢ CT IS H .The result of this paper is: Theorem 12.
Completeness of connected theory inverse subsumption . Let h B, U, I i be a definite open program,let H be a definite program, and let e be an atom. If H is an inductive solution for P and E = { e } , then H isderivable from P and E by connected theory inverse subsumption.Proof. Construct a connected theory T = S ∩ ground ( H ) for P and E as in the proof of 10. Then H (cid:23) ground ( H ) (cid:23) S ∩ ground ( H ) = T , hence H (cid:23) T by transitivity as required. Acknowledgements
We thank Dr. Krysia Broda, Dr. Timothy Kimber, Prof. Yoshitaka Yamamoto for checking the contents of draftsof this paper. 2 eferences [Kim12] Timothy Kimber.
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New generation computing , 13(3-4):245–286, 1995.[NCDW97] Shan-Hwei Nienhuys-Cheng and Ronald De Wolf.
Foundations of inductive logic programming , volume1228. Springer, 1997.[Tot14] David Toth. Classification of inductive logic programming systems. Master’s thesis, Imperial CollegeLondon, 2014.[YII12] Yoshitaka Yamamoto, Katsumi Inoue, and Koji Iwanuma. Inverse subsumption for complete explana-tory induction.