Computing single-piece unifiers and their aggregation

We callsingle-piece aggregatorthe rewriting operator that computes the set of one-step rewritings of a queryQby considering all the most general single-piece aggregated unifiers ofQ.

Theorem 7 The single-piece aggregator is sound, complete and prunable.

Proof:Soundness comes from Property 9 and from the fact that for any set of rulesR, let the ruleRbe its aggregation, one hasR |=R. Completeness and prunability rely on the fact that the piece-based rewriting operator fulfills these properties and the fact that for any queriesQandQ⁰and any ruleR, ifQ⁰=β(Q, R, µ), whereµis a piece-unifier, then the queryQ⁰⁰obtained with the single-piece aggregator corresponding to µis more general thanQ⁰, as expressed by Property 10.

7 Implementation and Experiments

As explained in Section 6, we now restrict our focus to rules with an atomic head.

We first detail algorithms for computing all the most general single-piece unifiers of a query Qwith a rule R and explain how we use them to compute all single-piece aggregators. Then we report first experiments.

7.1 Computing single-piece unifiers and their aggregation

When a ruleRhas an atomic head, it holds that every atom inQparticipates inat most onemost general single-piece unifier ofQwithR(up to bijective variable renaming).

This is is a corollary of the next property.

Property 11 Let Rbe an atomic-head rule andQbe a BCQ. For all atoma ∈ Q, there is at most oneQ⁰ ⊆Qsuch thata∈Q⁰andQ⁰ is a piece for a piece-unifier of QwithR.

Proof: We prove by contradiction that two single-piece unifiers cannot share an atom ofQ. Assume there areQ⁰₁ ⊆QandQ⁰₂⊆Qsuch thatQ⁰₁ 6=Q⁰₂andQ⁰₁∩Q⁰₂6=∅, andµ₁ = (Q⁰₁, H, P_u¹)andµ₂ = (Q⁰₂, H, P_u²)two single-piece-unifiers ofQwithR,

withH =head(R). SinceQ⁰₁6=Q⁰₂, one hasQ⁰₁\Q⁰₂6=∅orQ⁰₂\Q⁰₁ 6=∅. Assume Q⁰₁\Q⁰₂ 6= ∅. LetA = Q⁰₁∩Q⁰₂ andB = Q⁰₁\A. There is at least one variable x ∈ vars(A)∩vars(B)such that there is an existential variableeofhead(R)in the class ofP_u¹containingx(otherwiseµ1has more than one piece). SinceH is atomic, there is a unique way of associating any atom withH, thus the class ofP_u²containing xcontains alsoe. It follows thatQ⁰₂is not a piece since one atom ofAand one atom of B sharexunified with an existential variable inµ2whileAis included inQ⁰₂andB

is not.

To compute most general single-piece unifiers, we first introduce the notion of the unification of a set of atoms with the head of a rule. This notion is an adaptation of the classical logical unification that takes existential variables into account. To define a piece-unifier, the set of atoms has to satisfy an additional constraint on its separating variables.

Definition 17 (Partition by Position) LetAbe a set of atoms with the same predicate p. Thepartition by positionassociated withA, denoted byPp(A), is the partition on terms(A)such that two terms ofAappearing in the same positioni(1≤i≤arity(p)) are in the same class ofPp(A).

Definition 18 (Unifiability) LetRbe an atomic head rule and letAbe a set of atoms with same predicatepashead(R).AisunifiablewithRif no class ofPp(A∪head(R)) contains two constants, or contains two existential variables ofR, or contains a con-stant and an existential variable ofR, or contains an existential variable ofRand a frontier variable ofR.

Definition 19 (Sticky Variables) LetQbe a BCQ,Rbe an atomic head rule andQ⁰ be a subset of atoms inQwith the same predicatepashead(R). Thesticky variables ofQ⁰with respect toQandR, denoted bysticky(Q⁰), are the separating variables of Q⁰that occur in a class ofPp(Q⁰∪head(R))containing an existential variable ofR.

The following property follows from the definitions:

Property 12 LetQbe a BCQ,Rbe an atomic head rule, andQ⁰a subset of atoms in Qwith the same predicatepashead(R). Thenµ= (Q⁰,head(R), P_p(Q⁰∪head(R))) is a piece-unifier ofQwithRiffQ⁰is unifiable withhead(R)andsticky(Q⁰) =∅.

The fact that an atom fromQparticipates in at most one most general single-piece unifier suggests an incremental method to compute these unifiers. Assume that the head of R has predicate p. We start from each atoma ∈ Qwith predicate pand compute the subset of atoms fromQthat would necessarily belong to the same piece asa; more precisely, at each step, we buildQ⁰such thatQ⁰andhead(R)can be unified, then check ifsticky(Q⁰) = ∅. If there is a piece-unifier ofQ⁰ built in this way with head(R), all atoms inQ⁰can be removed fromQfor the search of other single-piece unifiers; otherwise,ais removed fromQfor the search of other single-piece unifiers but the other atoms inQ⁰ still have to be taken into account. Note that in both cases, the notion of separating variables is still relative to the originalQ.

Example 12 LetR =q(x) →p(x, y)andQ =p(u, v)∧p(v, t). Let us start from p(u, v): this atom is unifiable withhead(R)andp(v, t)necessarily belongs to the same piece-unifier (if any) becausev ∈ sticky({p(u, v)})(v is in the same class that the existential variable y); however,{p(u, v), p(v, t)} is not unifiable withhead(R) be-cause, sincevoccurs at the first and at the second position of apatom,xandyshould

be unified, which is not possible sinceyis an existential variable; thusp(u, v)does not belong to any piece-unifier withR. However,p(v, t)still needs to be considered.

Let us start from it:p(v, t)is unifiable withhead(R)and forms its own piece because sticky({p(v,t)}) is empty (tis in the same class that the existential variableybut is not shared with another atom). There is thus one (most general) piece-unifier ofQwithR, namely({p(v, t)},{p(x, y)},{{v, x},{t, y}}).

More precisely, Algorithm 2 first builds the subsetAof atoms inQwith the same predicate ashead(R). WhileAhas not been emptied, it initializes a setQ⁰by picking an atomainA, then repeats the following steps:

1. check ifQ⁰is unifiable withhead(R); else, the attempt withafails;

2. check ifsticky(Q⁰) =∅; if so, it is a single-piece unifier and all the atoms inQ⁰ are removed fromA;

3. otherwise, the algorithm tries to extendQ⁰with all the atoms inQcontaining a variable fromsticky(Q⁰); if these atoms are inA,Q⁰ can grow, otherwise the attempt withafails.

Algorithm 2:Computation of all most general single-piece unifiers Data: a CQQand an atomic-head ruleR

Result: the set of most general single-piece unifiers ofQwithR begin

U ← ∅;// resulting set

A← {a∈Q|predicate(a) =predicate(head(R))};

whileA6=∅do

a←choose an atom inA; Q⁰ ← {a};

whileQ⁰ ⊆Aandunif iable(Q⁰,head(R))andsticky(Q⁰)6=∅do Q⁰ ←Q⁰∪ {a⁰∈Q|a⁰contains a variable in sticky(Q⁰)}; ifQ⁰ ⊆Aandunif iable(Q⁰,head(R))then

U ←U∪ {(Q⁰,head(R), Pp(Q⁰∪head(R)))}; A←A\Q⁰

else

A←A\ {a}

returnU

Now, to compute the set of single-piece aggregators ofQwithR, we proceed as follows:

1. Compute all (most general) single-piece unifiers ofQwithR:

U1={µ1, . . . , µk};

2. Forifrom2to the greatest possible rank (as long asUiis not empty): letUibe the set of alli-unifiers obtained by aggregating ani−1-unifier fromUi−1and a single-piece unifier fromU1.

3. Return the union of all theU_iobtained.