Nonzero-Sum Games
Among Arbitrarily Many Players
John Thistle
(joint work with Hadi Zibaeenejad)
Electrical and Computer Engineering
University of Waterloo
Dagstuhl Seminar
Nonzero-Sum Games and Control
For details:
M.H. Zibaeenejad and J.G. Thistle, 'Weak invariant
simulation and its application to analysis of
parameterized networks,' IEEE Transactions on
Automatic Control 59 (8), August 2014; pp. 2024-2037.
M.Hadi Zibaeenejad and John G. Thistle, 'Dependency
graph: an algorithm for analysis of generalized
parameterized networks,' 2015 American Control
Conference. (To appear.)
Motivation:
– Computer/communication networks,
transportation networks,
manufacturing systems,
… modelled as interacting, similar, finite state-machines.
– Number of components may be large, variable or unknown.
– How much synthesis/analysis can be done without fixing number?
– e.g., reachability of deadlock
“Parameterized systems”
– Verification generally undecidable Apt/Kozen '86
– Some decidability results for ring networks:
Emerson et al '02,'03
– unidirectional unary token-passing
– or, DP chopstick-style tokens
– logics can't express deadlock.
– Can we check reachability of deadlock under less restrictive
assumptions?
Idea of approach
– Restrict flow of control & information less
severely
… by formulating simulation relations.
– If one process simulates another,
… it does not “block” their shared events
… yet their interaction can be complex.
Invariant simulation
A simulation relation …
that is preserved whenever both processes execute the same
events:
Invariant Simulation (ctd)
Such relations need not be reflexive (!):
Weak Invariant Simulation
– Weak version, in usual sense:
The larger the subalphabet, the stronger the simulation requirement, the weaker the invariance
requirement.
Weak Invariant Simulation and Synchronous
Products
– A weak invariant simulation with respect to all
shared events is preserved under synchronization
…
… if the event subalphabet is larger, the
simulation need not be preserved.
Ring networks
Assume all component processes isomorphic,
… and based on same state set.
Ring networks (ctd)
– Impose a direction of “dependency”:
– Assume that each process is strongly connected ...
… and that each shared event is possible only at a single state in the
upstream neighbour …
Simulation relations
Assume a WIS of each process by its upstream neighbour w.r.t. their set of shared events
… and another w.r.t. all the shared events of the upstream neighbour.
– The simulation requirements of the latter assumption are strong – we'll weaken it when
we look at other topologies.
– But note that the invariance requirement is weak.
– Nevertheless, one can show that at least one such relation holds at every moment.
Lemma:
At no point in the network's evolution is any shared event permanently prevented by
the upstream neighbour.
Define a “dependency relation”:
For any two neighbours, find the reachable component of their synchronous product
… after deleting from the upstream neighbour any transitions shared with its
upstream neighbour …
… and find all states where the only event defined is a shared event of the
downstream neighbour with its other neighbour.
– “dependent on downstream processes”
Theorem:
The reachable deadlocked states of all size
instances of the network are exactly those
corresponding to cycles in this dependency
relation.
– So these states can be encoded as the words of
a regular language.
Branching topology
Specify a network graph:
Nodes = distinguished processes
Edges = linear networks of arbitrarily many isomorphic processes
Branching topology (ctd)
– No leaves
– Unique node with in-degree > 1: input process – has
outdegree 1
– Others are output processes
– Assume that every isolated cycle satisfies same
assumptions as ring network, except “strong” one.
Input process assumption
Assume that there is a weak invariant simulation by the
input process of its downstream neighbour …
… with respect to all shared events of the input process
… which contains all state pairs in the synchronous
product of the input process and its downstream
neighbour.
Also, events shared with different neighbours can't
occur in same state of input process.
Output process assumption
Assume that there is a weak invariant simulation
by any output process of each of its downstream
neighbours …
… with respect to the set of all shared events of
the output process that are not shared with its
upstream neighbour.
Dependency graph
– Considering isolated cycles, define a dependency
relation similar to that of ring networks.
– Use it to construct a dependency graph.
– The circular waits of ring networks generalize to full,
consistent subgraphs of this dependency graph.
Full: accounts for all branches
Consistent: contains at most one state of any
distinguished process
Theorem:
A partial deadlock represented by a full, consistent
subgraph is reachable if and only if the subgraph
is output-reachable.
Output-reachable: Any state of an output process
is reachable within that process by means of local
events and events shared with the upstream
neighbour.
The set of network states represented by full,
consistent, output-reachable subgraphs can be
recognized by a finite automaton.
Conclusion
– Used weak invariant simulation to restrict flow of
control
– Allowed characterization of reachable deadlock
states as regular languages
– First step toward deadlock-free synthesis?