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Methods for Knowledge Based Controlling of Distributed Systems Saddek Bensalem, Marius Bozga, Susanne Graf, Doron Peled, Sophie Quinton

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3 trains: TGV, regular, freight, compete on tracks to enter the station. We want to implement the system such that each track decides “ locally ”, without a centralized control.

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Main problem Controlling a concurrent system that needs to satisfy some global restrictions. In general this is an undecidable problem. Decidable approximation: allow synchronizing actions as needed by the analysis (at the limit it becomes a sequential control problem). Actions are controlled based on the knowledge of being in a certain collection of global states. This “knowledge” is calculated in advance (model checking), and then the program is transformed to exploit it.

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Transition system with priorities We are given a transition system, e.g., a Petri Net. We want to impose that each pair (s, ) of a state and transition fired from it must be from a given allowed set . Running example: Priorities - a partial order between the transitions. At each global state, we can execute a transition that has a maximal priority among the enabled transitions.

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A (safe) Petri Net p1p1 p7p7 p3p3 t1t1 t 1 – a transition. With input places p 1 and p 7, and output place p 3. A transition fires executes if the input transition have tokens and the output transitions do not have tokens (is empty). A Petri Net is safe if when the input places to a transition have tokens, the output transitions do not have tokens.

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A (safe) Petri Net p1p1 t4t4 p7p7 p6p6 p5p5 p4p4 p3p3 p2p2 t2t2 t3t3 t1t1 A state is a set of places, e.g., {p 1, p 2, p 7 }. Can be written as p 1 /\ p 2 /\ ¬ p 3 /\ ¬ p 4 /\ ¬ p 5 /\ ¬ p 6 /\ p 7.

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A (safe) Petri Net p1p1 t4t4 p7p7 p6p6 p5p5 p4p4 p3p3 p2p2 t2t2 t3t3 t1t1

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p1p1 t4t4 p7p7 p6p6 p5p5 p4p4 p3p3 p2p2 t2t2 t3t3 t1t1 p 7 is a “semaphore”!

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A (safe) Petri Net p1p1 t4t4 p7p7 p6p6 p5p5 p4p4 p3p3 p2p2 t2t2 t3t3 t1t1

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p1p1 t4t4 p7p7 p6p6 p5p5 p4p4 p3p3 p2p2 t2t2 t3t3 t1t1 Transitions are independent (concurrent) if they do not share input and out places

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A (safe) Petri Net with priorities: Partial order between transitions. p1p1 t4t4 p6p6 p5p5 p4p4 p3p3 p2p2 t2t2 t3t3 t1t1 t 1 <

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A (safe) Petri Net with priorities. Partial order between transitions. p1p1 t4t4 p6p6 p5p5 p4p4 p3p3 p2p2 t2t2 t3t3 t1t1 t 1 <

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A (safe) Petri Net with priorities. Partial order between transitions. p1p1 t4t4 p6p6 p5p5 p4p4 p3p3 p2p2 t2t2 t3t3 t1t1 t 1 <

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Problem: how to guarantee execution according to priorities? Execute only maximally enabled transitions. Make a distributed (local) decision. Is it even possible? Can we check when it is possible? What if we cannot make the distributed decision?

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So, what can we do? What do we know in each local state? Use logic of knowledge to state (and check) if a process knows that its enabled transition has maximal priority. Transform the program to act based on the knowledge. There are variations of knowledge (based on the ability to recall our local history); tradeoff on price of transformation.

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Some definitions A process is a set of transitions.[No two transitions of a process can execute independently at any moment]. The processes cover all the transitions (there can be additional constraints). The neighborhood of a process is the set of all input and output places of it. p1p1 t4t4 p7p7 p6p6 p5p5 p4p4 p3p3 p2p2 t2t2 t3t3 t1t1 The local information of a process in a state, is the state limited to the neighborhood of the process. E.g., for the left process, in state { p 1, p 2, p 7 }, it is {p 1,p 7 }. (=“local state”).

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We are ready to define “ knowledge ” Given a “local information” as a process, a process knows everything that is consistent with it. Easy to calculate what a process knows. That is: I know of any property that holds in all global states with the same “local information”. This is a standard definitions (e.g., [FMHV]). Later: “knowledge with perfect recall”. p1p1 t4t4 p7p7 p6p6 p5p5 p4p4 p3p3 p2p2 t2t2 t3t3 t1t1 When left process is given that p 1 /\ p 7, it knows that p 4. (p 1 /\ p 7 ) K l p 4

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Calculating what a process knows given some local information We can write a formula that represents “all reachable states”: reach We can write a formula that represents “a given local information x”: x Intersect the above two formulas, and obtain “the states given some local information”: reach /\ x. We know, when x holds, whatever is implied from this intersection ( reach /\ x ) . The standard notation is K and we can embed this within temporal logic.

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Combined knowledge of several processes The knowledge of several processes is the properties that hold in all the global states consistent with the combined local information of these processes. Since we combine the local information, there are fewer consistent states. So, together, the processes know more. However, to utilize this combined knowledge, the processes need to coordinate with each other.

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What is known, during execution can be model checked: 1 Each process knows which of its enabled transitions have maximal priorities. 2 Each process knows about at least one of its enabled transitions that has a maximal priority. 3 In each non-deadlock state, at least one process knows about at least one enabled maximal priority transition. Maybe even 3 does not hold…

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The supporting process policy A transition can belong to multiple processes (e.g., communication). If enabled, and at least one process knows of it having maximal priority, then it can fire. Related to a problem in control theory: we are seeking a distributed disjunctive controller with limited observability (this is, in general: undecideable). Build for each process a “support table”: for each local information, which transition, if any, is supported.

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Some important points: The knowledge is calculated based on the states of the original (priorityless) program. Then transitions are being controlled, which generates a different program. The transformation adds conditions to the enabledness conditions, based on the support table. Knowledge about the original program remains known in the transformed program, since the reachable states can only reduce reachable states and executions!

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And what if neither of these invariants hold? Combine processes. The “knowledge of several processes” is bigger than of single processes. Compromise, and allow transitions that are not maximal, to complete support table, such that in any reachable state, at least one transition is enabled. Experimentally measure how far is a compromised execution from a perfectly prioritized one. Switch to ”knowledge of perfect recall”.

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t 3 <

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t 3 <

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t 3 <

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t 3 <

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t 3 <

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t 3 <

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t 3 <

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Knowledge of perfect recall Knowledge is based on the history that a process can sense, up to any point in its computation. A process can sense changes in its neighborhood, including changes to the neighborhood by other processes. This way we can distinguish between two local states (local information) with different histories. So, we know more! But don’t we need infinite amount of information?

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Knowledge of perfect recall: finite information. Ron Van der Meyden has shown that finite amount of information is sufficient. Given some history of computation, the rest of the system can be in a finite number of states consistent with it (a “subset construction”). If a process makes a single transition, or some process changes the neighborhood, the system can be in any global state that is obtained by making that transition, plus any sequence of (independent) transitions that do not change the neighborhood. For the transformation, we can keep the information as an automaton. This can become a gigantic automaton.

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Another solution When there are states where no one knows which transition as higher priority: offer interactions between processes that together have the knowledge. Interaction is offered through a coordinating algorithm called -core. The problem becomes decidable, since at the limit we get a full product that is controlled globally.

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The -core algorithm Perez, Corchuelo, Toro, Concurrency – Practice and Experience 16(12):1173-1206, 2004. There are several kinds of coordinations, each between a set of processes. Each coordination is controlled by a separate process. A process may have several choices between coordinations. Program processes and coordination processes exchange messages until coordination is established. P1P1 P3P3 P4P4 P2P2 C1C1 C2C2 P 4 has multiple choice at some state.

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We can mark each local state of a process p as follows: 1.K 1 : p knows he has a transition with maximal priority. 2.K 2 : p does not know …, but knows that some other process knows … (not necessarily the same process for each state consistent with the local state). 3.K 3 : Both above cases do not hold. 3 : for each global state, there is a local state where 1 holds

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Calculating K 2 We first mark local states by K 1 according to the previous algorithm. Then K 2 holds in a local state, if all the global states containing it have a local state where K 1 holds. If a local state of some process satisfies K 2, then the process “shouldn’t worry” – “someone else will do the job”.

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What if 3 does not hold? We set up a table that tells us about combined knowledge. Useless local states to worry about satisfy k 3. Take some tuples of them (start with pairs), and check whether they have a combined knowledge of maximal priority. Make a table of such tuples. Each tuple is a coordination! Add tuples until each global state includes such a tuple. Each tuple suggests a coordination where combined knowledge about who has maximal priority exists.

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Minimize the number of interactions

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Minimize the number of interactions (NP-Complete)

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Calculate the support table according to the executions satisfying the constraint property! a<<{c,d}<**
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The rule: It is safe to calculate the knowledge based on the states reachable in executions that satisfy the needed restriction . Intuition: a transition is only allowed from a state such that as a pair they satisfy . So it is safe. One still needs to check that no new deadlocks are introduced.

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Another principle: check reachability of potential deadlocks Analyzing knowledge according to the full state space, we obtain that in the state {p 4,p 10 } and in {p 6,p 10 }, there is no supported transition! – a potential deadlock. This is because the left process does not know whether j, with the higher priority than k, is now enabled. But b is never supported and also c from the initial state (since c<**
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The principles are independent Use just “constrainted” executions. Check for unreachable “deadlocks”. Use knowledge of perfect recall. Examples in the paper.

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Conclusions We can employ model checking of knowledge properties to implement priorities in a distributed way. May apply to other problem of distributed implementation, while solving some global constraints. Solves a distributed controllability with limited observability problem. Does not provide a perfect solution (not always the least constrained solution)! Space/performance tradeoff in using logic of perfect recall. Can combine knowledge of several processes temporarily through a coordination algorithm. Apply various compromises when knowledge is not enough, e.g., allow further synchronization (makes the problem decidable). Analysis should be based on the set of constrained executions, not the executions of the original. Potential deadlocks in support tables should be checked for reachability rather than first look for additional communication.

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