Explaining interdependent action delays in multiagent plans execution

Abstract

In this paper we address the problem of diagnosing the execution of Multiagent Plans with interdependent action delays. To this end, we map our problem to the Model-Based Diagnosis setting, and solve it by devising a novel modeling and reasoning method to infer preferred diagnoses based on partial observation of the start and end times of plan actions. Interestingly, we show that the kind of problem we address can be seen as an extension to the well known disjunctive temporal problem with preferences, augmented with a (qualitative) Bayesian network that models dependencies among action delays. An extensive set of tests performed with a prototype implementation on two different problem domains proves the feasibility of the proposed methodology.

Proof of Theorem 1

Before presenting the proof of Theorem 1, it is convenient to introduce some definitions and lemmas to simplify the discussion. First of all, remember that given a TMAP $P:\langle T, D, O, M, A, R, BN\rangle$, a current diagnostic hypothesis $H$, and a set of observations $Obs$, it is possible to build a corresponding distance graph $G_{\langle P,H,Obs\rangle }$ encoding all the temporal constraints (see 7.2). Moreover, recall that some actions in $A$ play the role of synchronization points, and that these actions are sorted according to their precedence relations in a list named $SYNC$.

We define reduction as an operation that given a consistent distance graph $G_{\langle P,H,Obs\rangle }$ and a synchronization point $a \in SYNC$, constrains the starting time of $a$ in $G_{\langle P,H,Obs\rangle }$ with the minimal and maximal waiting times of $a$; we thus say that $a$ has been reduced in $G_{\langle P,H,Obs\rangle }$ meaning that the interval associated with its starting time has been tightened. More formally:

Definition 11

Let $a \in SYNC$ be a synchronization point and let $G_{\langle P,H,Obs\rangle }$ be a consistent distance graph, the reduction of $a$ over $G_{\langle P,H,Obs\rangle }$ consists in:

• (1) asserting in $G_{\langle P,H,Obs\rangle }$ the two constraints $mwt(a)\le a.start - z \le MWT(a)$, and

• (2) invoking Johnson’s algorithm to restrict the new constraint network.

where $z$ is the reference point used as the initial starting time.

Note that algorithm consistent applies the reduction operation for each synchronization point in $SYNC$ taken in the order. It follows that whenever a synchronization point $a$ is reduced, any other synchronization point $a'$ preceding $a$ has already been reduced, too. This guarantees that the computation of $mwt(a)$ and $MWT(a)$ take into account all the constraints that have been added during the reductions of the synchronization points preceding $a$. As a shortcut, we will say that a synchronization point is reducible if all the synchronization points preceding it in $SYNC$ have been reduced, too. Of course, the first action in $SYNC$ is always reducible.

Since the reduction operation involves Johnson’s algorithm, after the reduction of a synchronization point we can either be in one of the following situations:

• if the resulting $G_{\langle P,H,Obs\rangle }$ has negative cycles, then the current hypothesis $H$ is not consistent with the observations $Obs$,

• otherwise, it holds the following relation:

  $$\begin{aligned} mwt(a) \le rlb \le a.start - z \le rub \le { MWT}(a) \end{aligned}$$

  (6)

  where $rlb$ and $rub$ stand for reduced lower bound and reduced upper bound, respectively.

Relation (6) can be easily proved. Before constraining $a.start -z$ to be within $[mwt(a), MWT(a)]$, the distance between these two points is already constrained to be in the interval $[lb, ub]$ (possibly $lb=0$ and $ub=+\infty$, if no observation for actions following $a$ is available). Since, by hypothesis, reducing $a$ does not introduce a negative cycle in $G_{\langle P,H,Obs\rangle }$, it must necessarily be true that the two intervals $[lb,ub]$ and $[mwt(a), MWT(a)]$ are not disjoined. It follows that

$$\begin{aligned}{}[rlb, rub] = [lb, ub] \oplus [mwt(a),{ MWT}(a)] \end{aligned}$$

(7)

where $\oplus$ is the intersection between intervals as defined by Dechter et al. in [8]. In particular, it must happen that:

• $rlb = max(lb, mwt(a))$,

• $rub = min(ub, MWT(a))$.

The following proposition states an important property of the time points within interval $[rlb,rub]$.

Proposition 4

Let $a$ be a reducible synchronization point in $SYNC$, and $pre(a)=\{a^1,\ldots ,a^{n}\}$ be its immediate predecessors. Let $G_{\langle P,H,Obs\rangle }$ be the distance graph obtained after having reduced $a$, and assume that it contains no negative cycles, then for all and only the points in $[rlb,rub]$ (i.e., the reduced lower and upper bounds for $a$) there exists an assignment to the temporal variables $a^i.end$ ($a^i \in {\textit{dPred}}(a)$) consistent with the ASAP policy and the observations.

Proof

Let us first show that each point in $[rlb, rub]$ is consistent with ASAP and $Obs$. First of all, since $G_{\langle P,H,Obs\rangle }$ has no negative cycles by our hypothesis, each point in $[rlb,rub]$ is trivially consistent with the observations. Thus, we have just to show that each point in this interval satisfies the ASAP policy as defined in Definition 10. To show this, it is sufficient to demonstrate that for each point in $[rlb,rub]$ there exists at least one action whose ending time can correspond to the starting time of action $a$.

By definition of $mwt(a)$, there must exist an action $a^i \in {\textit{dPred}}(a)$ such that ${ shortest\_makespan(a^i) } = mwt(a)$. In other terms, ${ shortest\_makespan }(a^i)$ corresponds to the lower bound of the distance $a^i.end-z$, we will denote such a value as $low(a^i.end)$. Similarly, by definition of $MWT(a)$, there must exist an action $a^j \in {\textit{dPred}}(a)$ such that $MWT(a) = { longest\_makespan }(a^j)$; this corresponds to the upper bound of the distance $a^j.end - z$, that we will denote as $up(a^j.end)$. In case $a^i$ and $a^j$ coincide, then it is easy to see that each point in $[mwt(a),MWT(a)]$ is consistent with the ASAP policy, in fact there exists an action (i.e., $a$ itself) for which the ending time can always correspond to the start of action $a$. Since $[rlb,rub]$ is a subinterval of $[mwt(a),MWT(a)]$, and since we have already shown that each point in $[rlb, rub]$ is consistent with $Obs$, we have that each point in such an interval is consistent both with the ASAP policy and with $Obs$.

In case $a^i$ and $a^j$ are not the same action, then it must be true that the lower bound of the distance $a^j.end-z$, denoted as $low(a^j.end)$ is lesser than $low(a^i.end)$, otherwise we would be in the previous case. But then, action $a^j$ can terminate in any point within $[mwt(a), MWT(a)]$. As above, it follows that each point in $[rlb,rub]$ is consistent both with the ASAP policy and $Obs$.

We have now to show that no other point outside $[rlb,rub]$ is consistent both with ASAP and $Obs$. Let us assume by contradiction that such a point $x \notin [rlb, rub]$ exists. Since $x$ must be consistent with $Obs, x$ must belong to $[lb, ub]$ (i.e., the interval constraining $a.start-z$ before reducing $a$). At the same time, since $x$ must be consistent with the ASAP policy, $x$ must belong to $[mwt(a),MWT(a)]$. Since $[rlb,rub]$ is the intersection between $[lb,ub]$ and $[mwt(a),MWT(a)], x$ can’t be outside $[rlb,rub]$. Contradiction. $\square$

We will also need the following lemma.

Lemma 1

Let $a$ be a reducible synchronization point, and let ${\textit{dPred}}(a)=\{a^1,\ldots ,a^{n}\}$ be its direct predecessors. Then, for no action $a^i \in {\textit{dPred}}(a)$ the reduction of $a$ can impact on the distance $a^i.end-z$. In other terms, if $lb \le a^i.end - z \le ub$ holds in $G_{\langle P, H, Obs\rangle }$ before the reduction of $a$, then the same constraint is preserved in the distance graph obtained after the reduction of $a$.

Proof

In order to affect an action $a^i$ in ${\textit{dPred}}(a)$, the reduction of $a$ should introduce in the current distance graph $G_{\langle P, H, Obs\rangle }$ a constraint that is more restrictive than the one already defined for $a^i$, let say

$$\begin{aligned} lb \le a^i.end - z \le ub. \end{aligned}$$

But this could happen only in case the reduction of $a$ would impose the constraint

$$\begin{aligned} a.start - z \le ub^* \end{aligned}$$

such that

$$\begin{aligned} ub^* < ub. \end{aligned}$$

However, the reduction of $a$ corresponds to asserting in the current distance graph $G_{\langle P, H, Obs\rangle }$, the constraint

$$\begin{aligned} mwt(a)\le a.start -z \le { MWT}(a). \end{aligned}$$

Since $mwt(a)$ and $MWT(a)$ are determined as the shortest and the longest makespans, respectively, of the actions in ${\textit{dPred}}(a)$, it follows that neither of them can ever be more restrictive than the constraints already encoded in $G_{\langle P, H, Obs\rangle }$. In fact, for each action $a^i$ in ${\textit{dPred}}(a)$, such that the constraint $lb \le a^i.end - z \le ub$ is defined in $G_{\langle P, H, Obs\rangle }$ before the reduction of $a$, the following conditions are always satisfied by definition:

$$\begin{aligned} lb \le mwt(a) \qquad ub \le { MWT}(a). \end{aligned}$$

It follows that the reduction of $a$ cannot change the ending intervals of its immediate predecessors, and as a consequence, of any other action preceding them. $\square$

Now we are in the position for presenting the proof of Theorem 1.

Theorem 1 (Consistent correctness) The algorithm consistent (Fig. 4) returns $true$ iff there exists a temporal profile $\tau _P$ for the TMAP $P$ satisfying the ASAP policy and consistent with the observations $Obs$.

Proof

To prove the theorem it is convenient to restate it in order to relate the existence of a temporal profile $\tau _P$ to the absence of a negative cycle in the distance graph $G_{\langle P,H,Obs\rangle }$ after the invocation of the Johnson’s algorithm:

In function consistent, the distance graph $G_{\langle P,H,Obs\rangle }$ never contains a negative cycle after an invocation to the Johnson’s algorithm iff there exists a temporal profile $\tau _P$ for the TMAP $P$ satisfying the ASAP policy and consistent with the observations $Obs$.

The proof then proceeds as follows.

($\Rightarrow$) Let us first assume that, during the consistent function, the distance graph $G_{\langle P,H,Obs\rangle }$ never contains a negative cycle, and show that there exists a temporal profile $\tau _P$ consistent with the observations $Obs$ and the ASAP policy.

By Proposition 4, if the reduction of a synchronization point $a$ in $SYNC$ does not induce a negative cycle in the distance graph $G_{\langle P, H, Obs\rangle }$, then the reduction of $a$ constrains the starting time as $rlb \le a.start-z \le rub$; in addition, for each point within $[rlb, rub]$, there exists an assignment of values to the temporal variables $a^i.end$, where $a^i \in pred(a)$, that is consistent both with $Obs$ and with the ASAP policy. By Lemma 1 we know that the reduction of a synchronization point $a \in SYNC$ cannot affect any of the actions preceding $a$ in $P$; that is, constraints defined over actions preceding $a$ are preserved by the reduction of $a$. It follows that, after having reduced a synchronization point $a$, there exists at least one partial temporal profile $\tau _P[a]$ which is consistent both with $Obs$ and with ASAP in the portion of plan $P$ from the beginning to action $a$. Thus, after having considered the last synchronization point of the plan, there exists at least a complete temporal profile $\tau _P$ consistent with $Obs$ and ASAP. To find one of such temporal profiles, it is sufficient to reconsider the synchronization points backwards, from the last reduced to the first one, and for each of them, say $a$, it is sufficient to substitute the constraint $rlb \le a.start -z \le rub$ in the distance graph $G_{\langle P, H, Obs\rangle }$ with a constraint $p \le a.start -z \le p$, where $p$ is any point within $[rlb, rub]$, and then invoke Johnson’s algorithm to propagate the choice in the constraints network.

($\Leftarrow$) Let us now assume that there exists a complete temporal profile $\tau _P$ which is consistent with $Obs$ and ASAP and show that consistent never finds a negative cycle, and hence returns $true$.

Note that a negative cycle can arise in $G_{\langle P, H, Obs\rangle }$ only when consistent invokes Johnson’s algorithm during the reduction of a synchronization point $a \in SYNC$. Thus, a negative cycle could only arise when the interval $[mwt(a),MWT(a)]$, that is asserted by the reduction operation, is disjoint from the interval $[lb,ub]$ constraining $a.start$ in $G_{\langle P, H, Obs\rangle }$ before the reduction operation. However, on the one hand a temporal profile $\tau _P$ consistent with $Obs$ and ASAP exists by hypothesis; on the other hand, by Proposition 4, when the reduction of $a$ does not induce a negative cycle, the resulting constraint $rlb \le a.start - z \le rub$ contains all and only the temporal points in which $a.start$ is consistent with $Obs$ and with ASAP, including the point $\langle a.start, t\rangle$ mentioned in $\tau _P$. Therefore, none of the reductions performed by consistent can ever produce a negative cycle since, for each synchronization point $a$, at least one point (i.e., the point mentioned in $\tau _P$) is both in the interval $[lb, ub]$ (constraining $a.start$ before the reduction), and in the interval $[mwt(a), MWT(a)]$ (asserted by the reduction). $\square$