Approaches to Reactive System Synthesis

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Approaches to Reactive System Synthesis

• J.-H. Roland Jiang

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Road Map
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Functional Synthesis

• First-order specification "x y. f(x,y)
• Find program p(x) such that "x. f(x,p(x))
• Extract programs from proofs

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Road Map
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Reactive Synthesis

• Characterized by sequential behavior
• Specification f is temporal

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Closed System Synthesis

• x y. f(x,y)
• Construct two components C1 and C2 which can
  modify x and y, respectively, such that the
  running values of x and y satisfy f(x,y)

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Open System Synthesis

• "x y. f(x,y)
• C1, which modifies x, represents the environment
  over which the implementor has no control, while
  C2 , which modifies y, is the body of the system
  itself
• Find f(x) such that "x. f(x, f(x))

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Churchs Problem Church 62

• Summary of early digital synthesis and
  verification
• Specification C(X Iw, Y Jw) in restricted
  recursive arithmetic
• Find operator f Iw Jw such that "X.C(X, f(X))
  is valid
• Requirements
• f may not depend on the future
• f may not depend on the far past

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Solution 1 Tree Automata

• "x y. f(x, y)
• X-player chooses branch Y-player chooses
  labeling
• Realizability Non-emptiness
• Extract deterministic transducer from model
• Complexity 2EXPTIME

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Specification a Tree Automata
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Solution 2 Game Automata

• Infinite games played on finite graphs
• G (Q0, Q1, E0, E1, d, W)
• Specification Í (Q0 Q1)w
• Sequential games Ì Borel games
• All Borel games are determined

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Specification a Game
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Solution 3 Control Synthesis

• Discrete event system P
• S controllable actions
• Q system states
• q0 initial state
• dP Q ? S 2Q transition function
• L Q 2Prop state labeling
• Find controller
• C á M, m0, dC M S ? M ñ
• such that C ? P ² f

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Specification a Controller
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Inter-reductions

• Tree automaton a game
• Game a controller
• Controller a tree automaton

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Tree Automaton a Game
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Game a Controller
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Controller a Tree Automaton
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Implementability Problem

• Prior formulations mainly focused on the
  implementability problem
• Asks if there exists a solution
• Largest solution in language equation vs. most
  permissive strategy in game
• Most permissive strategy only exist for safety
  games
• A strategy is permissive if it allows all the
  behaviors of all memoryless winning strategies in
  the game
• For every game there is a permissive strategy
  with finite memory
• Support design refinement

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Supervisory Control Synthesis

• Controllability
• S Sc È Su
• Controllable events can be disabled at any time
  uncontrollable events are always enabled
• Observability
• Partial observation can be see as a projection
  p S So È e
• Natural projection (So Í S)
• Signal hiding Kupferman Vardi 97

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Wining Strategies and Controller Synthesis

• Most permissive strategies
• Exist only for safety games

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Control of Synchronous Systems

• de Alfaro Henzinger Mang 00
• Non-blocking
• Every state should have at least one successor
  state
• Typing
• Prevents combinational loops

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Research Directions

• Language equation solving with general partial
  observations
• Connection between S1 p S2 and X1 p X2 (for the
  same F)
• Game formulation of the unknown component problem
• Connection between permissive strategies in games
  and largest solutions in language equations

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References

• J. Bernet, D. Janin and I. Walukiewicz.
  Permissive strategies from parity games to
  safety games. RAIRO, 2002.
• N. Bjorner. A survey of reactive synthesis.
  Slides for DIMACS, 1996.
• L. de Alfaro, T. Henzinger and F. Mang. The
  control of synchronous systems. In Proc. CONCUR,
  2000.
• O. Kupferman and M. Vardi. Synthesis with
  incomplete information. In Proc. Intl Conf.
  Temporal Logic, 1997.
• A. Pnueli and R. Rosner. On the synthesis of a
  reactive module. In Proc. POPL, 1989.
• P. Ramadge, W. Wonham. A control of discrete
  event systems. Proceedings of the IEEE, 1989.