We provide here a brief introduction and proof of soundness and completeness of the ILP system ASPAL. This document is in support of our ICLP 2011 submission, for the reviewers' bene ts

Corapi, D

Russo, A

English

We provide here a brief introduction and proof of soundness
and completeness of the ILP system ASPAL. This document is in support
of our ICLP 2011 submission, for the reviewers' bene ts

Corapi, Domerico

Russo, Alessandra

Spiral - Imperial College Digital Repository

ASPAL. Proof of soundness and completenessDomenico Corapi and Alessandra RussoDepartment of ComputingImperial College London180 Queen’s Gate, SW7 2AZLondon, UKE-mail: {d.corapi,a.russo}@ic.ac.ukAbstract. We provide here a brief introduction and proof of soundnessand completeness of the ILP system ASPAL. This document is in supportof our ICLP 2011 submission, for the reviewers’ benefits.Inductive Logic Programming (ILP) [4] is a machine learning technique con-cerned with the induction of logic theories, called target theories, that generalise(positive and negative) examples with respect to a prior background knowledge.For example, from the observations fly(a), f ly(b),¬fly(c) and a backgroundknowledge containing the two facts bird(a) and bird(b), we can generalise theconcept fly(X) ← bird(X). In non-trivial problems it is crucial to define thespace of possible solutions accurately. Target theories are within a space definedby a language bias, which can be expressed using mode declarations [4].We refer to [3] for notations and preliminary definitions on logic program-ming. For a given theory T we denote with BT the Herbrand base of T .Definition 1. A mode declaration is either a head declaration, written modeh(s),or a body declaration, written modeb(s), where s is a schema. A schema is aground literal containing special terms called placemarkers. A placemarker is ei-ther ‘+type’, ‘−type’ or ‘#type’ where type denotes the type of the placemarkerand the three symbols ‘+’, ‘−’ and ‘#’ indicate that the placemarker is an input,an output, a constant placemarker, respectively.In the previous example a possible language bias would be given by thethree mode declarations modeh(fly(+animal)), modeb(bird(+animal)) andmodeb(fish(+animal)).A rule h ← b1, ..., bn is compatible with a set M of mode declarations iff (a)h is the schema of a head declaration in M and bi are the schemas of bodydeclarations in M where every input and output placemarkers are replaced byvariables and constant placemarkers are replaced by constants; (b) every inputvariable in any atom bi is either an input variable in h or an output variablein some bj , j < i; and (c) all variables and constants are of the correspondingtype. From a user perspective, mode declarations establish how rules in the finalhypotheses are structured, defining literals that can be used in the head and inthe body of a well-formed hypothesis. s(M) is the set of all the rules compatible2 Domenico Corapi, Alessandra Russowith M . For a set M of mode declarations we denote with Mb the subset ofbody declaration and with Mh the subset of head declaration.Definition 2. An ILP task is a tuple 〈P,B,M〉 where P is a set of conjunc-tions of literals, called properties, B is a normal program, called backgroundtheory, and M is a set of mode declarations. A theory H, called hypothesis, isan inductive solution for the task 〈P,B,M〉, if (a) H ⊆ s(M), and (b) P is truein all the stable models of B ∪H.The above notion is an instance of the learning from entailment setting [2],where positive and negative examples are our properties. In the following we omitthe additional control on the type of the arguments in the mode declarations forsimplicity and readability. The extension of the definitions and proofs with typecheck is straightforward, they can be added to the bodies of the hypotheses and ofthe rules of the additional theories introduced using a standard pre-processingprocedure used in ASP. Furthermore, our definitions and proposition will begiven for ground programs. Rules with variables will be used as a shorthand forthe set of their ground instances. We define a strong completeness requirement,compared to alternative formulations where completeness only requires that asolution is found if one exists.Given a mode declaration id : modeh(s) (or modeb(s)), id is the (unique)identifier for the mode declaration and sc(id) denotes the schema s of the modedeclaration.Definition 3. s is the literal obtained from s by replacing all placemarkers withdifferent variables var(s, s) = (X1, ..., Xn), called a variable list and it is calledthe variabilisation of s. con(s, s) = (C1, ..., Cc) is the constant list of variablesin s that replace only constant placemarkers in s. inp(s, s) = (I1, ..., Ii) andout(s, s) = (O1, ..., Oo) are defined similarly for input and output placemarkers.A variable is in the variable list if and only if is one of the constant, input oroutput list. When s is clear from the context we omit the second argument fromvar(s, s), con(s, s), inp(s, s) and out(s, s).Given a set of mode declarations M , a top theory > = t(M) is constructedas follows:– For each head declaration modeh(s), with unique identifier id, the following ruleis in >s← rule(RId, (id, con(s), ()),rule id(RId),body(RId, 1, inp(s))(1)– For each body declaration modeb(s), with unique identifier id the following clauseis in >body(RId, L, I)← rule(RId, L, (id, con(s), Links)),link(inp(s), I, Link),s,append(I, out(s), O),body(RId, L + 1, O)(2)ASPAL. Proof of soundness and completeness. 3– The following rules are in > together with a standard definition for the appendpredicate.body(RId, L, )← rule(RId, L, last) (3)link((X), (X), 1).link((X), (X, ), 2).link((X), ( , X), 2).link((X,Y ), (X,Y ), (1, 2)).link((X,Y ), (Y,X), (2, 1)).link((X), (X, , ), 1).link((X), ( , X, ), 2)....(4)rule id(1)....rule id(rn).(5)rule id(rid) is true whenever 1≤ rid≤ rn where rn is the maximum number ofnew rules allowed. link((a1, ..., am), (b1, ..., bn), (o1, ..., om)) are true if for eachelement in the first list ai, there exists an element in the second list bj such thatai unifies with bj and oi = j. Given the top theory, we seek a set of rule atoms∆, such that P is true in all models of B ∪ > ∪∆.∆ has a one-to-one mapping to a set of rules H = u(∆,M). Intuitively, eachabduced atom represents a literal of the rule identified by the first argument. Thesecond argument collects the constant used in the literal and the third definesthe variable linking.∆rid denotes the subset of ∆ of all the rule abducibles with rid as a firstargument. Each ∆rid corresponds to a rule r in u(∆,M), r = u(∆rid,M). For agiven∆rid ={rule(rid, (idh, conh),rule(rid, 1, (id1, con1, links1)),...,rule(rid, 1, (idn, conn, linksn))}r is a rule h← b1, ..., bn such that, given a list of variables defined as follows:– avar0 is inp(sc(idh))– avari is such that append(avari−1, out(sc(idi)), avari) is true for each i =1, ..., n for the standard definition of appendeach atom is constructed as follows:– h = sc(idh) and con(sc(idh)) is unified with the list of constant conh of thesame length4 Domenico Corapi, Alessandra Russo– bi = sc(idi), con(sc(idi)) is unified with the list of constant coni of thesame length and link(inp(sc(idi)), avari−1, linksi) is true, according to thedefinition of link provided in the top theory for each i = 1, ..., nProposition 1. Let H be a theory and M a set of mode declarations. ThenH ∈ s(M) if and only if there exists a ∆ such that H = u(∆,M).Proof. The proof is immediate as the transformation itself can be used to defines(M). In fact the transformation requires that the head corresponds to the vari-abilisation of a schema of a head declaration where all the constants are replacedby arbitrary constants. Each other condition i in the body can only bind inputto variables in avari−1 by definition and the constants are also replaced by a listof arbitrary constants. utIn the proof we will use the unfolding transformation or generalised principleof partial evaluation (GPPE)[1].Definition 4. Let P be a theory, h ← b1, ..., bi−1, bi, bi+1, ..., bn ∈ P a rule rand bi a positive literal. We say we apply a GPPE transformation to r on bi ifP ′ is the theory obtained from P by replacing rule r with the p rulesh← b1, ..., bi−1, bi+1, ..., Ajwhere Aj is the conjunction of literals in bi ← Aj ∈ P , j = 1..p.Definition 5. Let P and Q be theories. We say P is equivalent to Q if, for allinterpretation I, I is a stable model for P if and only if I is a stable model forQ.Proposition 2. [1] Let P be a theory. If P ′ is the theory obtained applyingGPPE to any rule in P then P ′ and P are equivalent.Proposition 3. Let B be a normal program, H be a hypothesis and M a set ofmode declarations such that H = u(∆,M) and > = t(M). Let α denote B ∪Hand β denote B ∪ > ∪ ∆. Let K = I ∪ J be an interpretation for β such thatI ⊆ Bα, J ⊆ (Bβ \ Bα). Then K is a stable model for β if and only if I is astable model for α.Proof. Consider rules (1) of the top theory. Applying GPPE to them on body(RId, 1, inp(s))we obtain a theory >1 in which rules (1) are replaced by the following:sc(idh)← rule(RId, (idh, con(sc(idh)), ()),rule id(RId),rule(RId, 1, (id1, con(sc(id1)), Links1)),link(inp(sc(id1)), inp(sc(idh)), Links1),sc(id1),append(inp(sc(idh)), out(sc(id1)), O1),body(RId, 2, O1)(6)ASPAL. Proof of soundness and completeness. 5for all idh such that idh ∈ Mh and for all id1 ∈ Mb. We used the rules (2)where the second argument is 1. Using (3) in the transformation the followingrules are included in >1sc(idh)← rule(RId, (idh, con(sc(idh)), ()),rule id(RId),rule(RId, 1, last)(7)We can now apply the transformation again on body(RId, 2, O1)) and rules(2) and (3), and again on body(RId, i+ 1, Oi) for each i ≤ nc. nc the maximumnumber of conditions allowed is enforced by an additional condition on the secondargument of the body predicate, omitted for ease of exposition. So we obtain atheory >′ that contains the following rules for each m = 1..nc.sc(idh)← rule(RId, (idh, con(sc(idh)), ()),rule id(RId),rule(RId, 1, (id1, con(sc(id1)), Links1)),link(inp(sc(id1)), inp(sc(idh)), Links1),sc(id1),append(inp(sc(idh)), out(sc(id1)), O1),...,rule(RId, n, (idm, con(sc(idm)), Linksm)),link(Om−1, inp(sc(idm)), Linksm),sc(idm),append(Om−1, out(sc(idm)), Om),rule(RId,m+ 1, last),(8)and for all id such that id ∈Mh and idi ∈Mb, i = 1, ..,m.Of these rules the only rules whose body can be true are those that containexactly the atoms in ∆rid for some rid. Consider h← b1, ..., bn = u(∆rid,M)∆rid ={rule(rid, (idh, conh, ())),rule(rid, 1, (id1, con1, links1)),...,rule(rid, 1, (idn, conn, linksn))}we can consider all the rules that contain the atoms in such ∆rid and ap-ply GPPE on conditions rule id and append. Oi as result of the append willcorrespond to avari as defined previously:6 Domenico Corapi, Alessandra Russosc(idh)← rule(rid, (idh, conh, ()))rule(rid, 1, (id1, con1, links1)),link(inp((s)), avar0, links1),sc(id1),...,rule(rid, n, (idm, conm, linksn)),link(inp(sc(idn)), avarn−1, linkn),sc(idn),rule(rid,m+ 1, last)(9)Using the truth values defined in J we can see that rules (9) correspond torules h← b1, ..., bn. We can then say B ∪>∪∆ is equivalent to B ∪H ∪>− ∪∆where >− is the original > without rules (1) that have been transformed intoH. Now >− ∪∆ define a disjoint set of predicates from B ∪H. The set J onlydefines those predicates that appear in > and ∆ but not in B, thus it mustdefine the predicates link, rule id, append, rule and body appearing in the headof rules in >− ∪∆. The first four predicates are defined extensionally by facts.The definition of body predicates is given by a stratified theory (as a result of theincreasing value of the integer used as a second variable of body) and completesthe definition of J . I is thus a stable model for the new theory if and only if itis a stable model for B ∪H that proves the proposition. utProposition 4. Given an ILP task 〈P,B,M〉, H is an inductive solution if andonly if there is a ∆ such that H = u(∆,M), > = t(M) and P is true in all stablemodels of B ∪ > ∪∆.Proof. The proposition is a consequence of the Propositions 3 and 1. H is asolution if and only if (a) H ⊆ s(M), and (b) P is true in all the stable modelsof B ∪ H. H ⊆ s(M) if and only if there exists a ∆ such that H = u(∆,M)according to Proposition 1. P is true in all answer sets of B ∪H. But all suchanswer sets have a corresponding answer set for β, according to Proposition 3that “agrees” on P , thus proving the proposition. utASPAL is an ILP system that is based on the transformation introduced andon an ASP solver used abductively to derive ∆. The soundness (if a ∆ is foundthan u(∆,M) is an inductive solution) and completeness (if H is an inductivesolution than a ∆ is found such that H = u(∆,M)) of ASPAL follow fromProposition 1 and the soundness and completeness properties of the underlyingASP solver that ensures that for any given theory T and conjunction of literalsP , any ∆ such that P is true in a stable model of T ∪ ∆ is an output of thesystem.References1. Stefan Brass and Jürgen Dix. Characterizations of the stable semantics by partialevaluation. In Logic Programming and Non-Monotonic Reasoning, Proceedings ofthe Third International Conference, LNCS 928, pages 85–98. Springer, 1995.ASPAL. Proof of soundness and completeness. 72. Luc De Raedt. Logical settings for concept-learning. Artif. Intell., 95(1):187–201,1997.3. J. W. Lloyd. Foundations of logic programming. Springer-Verlag New York, Inc.,New York, NY, USA, 1984.4. S. Muggleton. Inverse entailment and progol. New Gen. Comp., 13(3&4):245–286,1995.

ASPAL. Proof of soundness and completeness

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ASPAL. Proof of soundness and completeness

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