Recomendaciones

In this section, we give formal definitions for the data structures used in the distributed algo- rithm. As previously mentioned, a state may contain hypotheses and constraints that need to be checked by all agents at least once. In order to record who has checked what, we introduce the concept of tagging for literals or denials:

Definition 4.3 (Tagging). A tag is a pair (L, S), where L is either a literal or a denial, and S is a (possibly empty) set of agent identifiers. Given a set τ of tags, a literal or denial L is tagged by an agent α (or α has tagged L) if and only if (L, S) ∈ τ and α ∈ S.

Sometimes we also use LS _{to denote that a literal or denial L is tagged by the agents in S.}

Tags are used during distributed abduction in one of the following ways:

• A goal may be tagged by the agents who have delayed it. Such information can be used to prevent an agent from delaying the same goal more than once, and hence avoid the non-progressive cyclic state-passing between agents. For example, suppose a system has two agents α and β. Given a query containing only one goal p, we need to stop the agent interactions cycling between α delays p then sends a state to β, and β delays p then sends a state to α.

• An abducible may be tagged by the agents who have not checked its consistency with respect to their local integrity constraints.

• A denial (collected consistency constraint) may be tagged by agents who have not checked its satisfiability.

Examples of these usages will be given in Section 4.3.3.

Definition 4.4 ((DAREC) Computational State). A (DAREC) computational state (or

• each element in G is a remaining goal, and can be either a literal or a denial of the form

∀ ~X. ← φ1, . . . , φn (n > 0) where ~X is the set of universally quantified variables of the

denial 1_;

• each element in Gd _{is a delayed goal and must be a non-abducible;}

• ST is a tuple of four stores (∆, N , E, C), where – ∆ is a set of abducibles;

– N is a set of denials ∀ ~X. ← φ1, . . . , φn (n > 0), where the ordering of φ1, . . . , φn

matters and φ1 is either an abducible or non-abducible;

– E is a set of (in-)equalities; – C is a set of CLP constraints;

• τ is a set of tags. All free variables appearing in the state Θ are existentially quantified within the scope of the whole state.

A denial in N is called a consistency constraint, and its first body literal is called its constrained literal. More specifically, such a denial is called an abducible (consistency) constraint (on φ) if its constrained literal φ is an abducible, or is called a non-abducible (consistency) constraint (on φ) if φ is a non-abducible. Intuitively, an abducible constraint must be satisfied by all the agents so that for every assumed instance of its constrained abducible, the rest of the denial body must not be provable by any agent. For example, let ∀X. ← a(X), p(X) be an abducible constraint. If the abducible a(1) is assumed by the agents, then p(1) must not be provable by any agent. A non-abducible constraint must be satisfied by all the agents so that for every instance of its constrained non-abducible that is provable by some agent, the rest of the denial body must not be provable by any of the agents. For example, let ∀X. ← p(X), q(X) be a non-abducible constraint. If p(X) is provable by some agent with X = 1, then q(1) must not be provable by any agent.

According to the cases in which tags can be used, only positive non-abducible goals, collected abducibles, and non-abducible constraints of a state may be tagged. Note that other types

of goals, such as abducibles and arithmetic constrains can be reduced by any agent regardless of their background knowledge, and hence we do not allow them to be delayed. Note also that negative goals are always converted into denial goals (as described in the corresponding

local inference rules in Section 4.3.3), and denial goals cannot be delayed as they are in fact

constraints.

Note that a DAREC computational state is similar to an ASystem computational state, with only the following differences:

• an ASystem state does not contain delayed goals;

• the denial set in an ASystem state contains only abducible constraints; • nothing in an ASystem state can be tagged.

As mentioned earlier, a DAREC computational state can be of 3 different types. These are defined as follows.

Definition 4.5 (Initial State). The initial state of a global abductive task for a query Q is h(Q, ∅), (∅, ∅, ∅, ∅), ∅i.

To simplify notation, sometimes we use ST∅ to denote the four empty stores of a state, i.e.,

ST∅ _{= (∅, ∅, ∅, ∅).}

Definition 4.6 (Solved State). A state Θ = (G, Gd_{), (∆, N , E , C), τ}

is a solved state if

G = Gd_{= ∅, E ∪ C is consistent and no element in ∆ ∪ N is tagged according to τ .}

After a solved state h(∅, ∅), (∆, N , E , C), τ i is obtained during distributed abduction, we can extract the DAREC answers h∆, θi from it where θ is the set of variable substitutions induced by E ∪ C.

Definition 4.7 (Transferable State). A state Θ =(G, Gd_{), (∆, N , E , C), τ is a transferable}

state of an agent α’s local abduction if one of the following conditions is satisfied:

• an abducible in ∆ is tagged by some agent(s) according to τ ;

• a non-abducible constraint in N is tagged by some agent(s) according to τ .

Note that a transferable state can contain both delayed goals and tagged abducibles/constraints. Also a transferable state may be further processed by the same agent (in contrast to passing it to another agent) if it still contains remaining goals. The decision of whether to pass or to process that state is based on the agent interaction strategy being adopted by the agent.

The search space of an agent’s local abduction, given a goal selection strategy and an agent interaction strategy, can be described as a local abductive derivation tree:

Definition 4.8 (Local Abductive Derivation Tree). Given a goal selection strategy Ξ and an agent interaction strategy Υ, the local abductive derivation tree for a local abduction by an agent α a tree where:

• the root node is the root state of the local abduction,

• the children nodes are the states obtained by the application of a local inference rule to the parent node according to Ξ and Υ.

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