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Solving Games Without Determinization
Nir Piterman
École Polytechnique Fédéral de Lausanne (EPFL)
Switzerland
Joint work with Thomas A. Henzinger
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Nondeterminizing NondeterministicAutomata
Nir Piterman
École Polytechnique Fédéral de Lausanne (EPFL)
Switzerland
Joint work with Thomas A. Henzinger
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What?
• Get a nondeterministic automaton with n states.
• Construct a nondeterministic automaton with 2nn2n states.
• Why?
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Plan of Talk
• Verification.
• Automata on Infinite Words.
• Synthesis.
• Design Synthesis in Action.
• Our solution.
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Verification
• The normal process of development:– Write specifications (informally).– Develop design.– Test.
• Check that the system satisfies the specification.
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Reactive Systems
• We are interested in systems that behave rather than compute (CPU, Operating system).
• Main complexity is in maintaining communication with a user / another program / the environment.
• The system has to be ready for every possible input.
• The system maintains behavior forever.
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What is Behavior?
• The sequence of states the system passes along a computation.
• Nondeterministic systems / many possible inputs produce many possible behaviors.
• For reactive systems the behavior is infinite.
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Automata Theoretic Approach to Verification
• Use automata to reason about systems and specifications.
• Questions like satisfiability and model checking reduce to emptiness of automata.
• Separates logical and algorithmic aspects of problems.
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Automata on Infinite Words
• Introduced by Büchi, McNaughton, Elgot, Trakhtenbrot, Rabin, … in the 60s.
• Basically: take the same machine; run it on infinite words.
• In infinite runs there is no last state. Use the set of recurring states.
• Büchi acceptance: the set of recurring states intersects the set of accepting states.
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Examples
q0 q1
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Examples
q0 q1
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Applications
• Satisfiability of S1S [Buc62] and linear time logics.– A linear time formula characterizes sets of
sequences.– Construct an automaton that accepts the set of
models of the formula.– Is the language of the automaton empty?
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Applications
• Linear-time model checking [VW94].– A linear time formula characterizes sets of
sequences.– Construct an automaton that accepts all non-
models of the formula.– Consider the intersection of the automaton and the
system.– Is the intersection empty?
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Verification
• The normal process of development:– Write specifications (informally).– Develop design.– Test.
• Check that the system satisfies the specification.
• We need a formal way to write specifications: temporal logic.
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Specifications
• We formally write specifications using temporal logic.
• We use automata on infinite words as an intermediate tool to reason about specifications.
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Synthesis
• Can’t we automatically produce the system from the specification?
• Produce systems that are ensured to work correctly.
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Church’s Problem
In 1965 Church posed this problem as:
Given a circuit interface and a behavioral
specification, determine:
1. Does there exist an automaton (circuit) that realizes the specification?
2. Construct an implementing circuit.
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Solutions
• Rabin develops the theory of automata on infinite trees [Rab69].
• Büchi and Landweber propose a reduction to infinite duration games [BL69].
• These are the main two solutions up till today.
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Synthesis as a Game
• System controls internal variables. Environment controls input.
• Moves of system must match all possible future moves of environment.
• System plays against environment. – System tries to satisfy specification.– Environment tries to falsify specification.
• Success of system determined by the outcome of interaction.
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Game Graphs• We represent games as directed graphs.
G=hV,V0,V1,E,v0i
• The vertices are partitioned to those of player 0 (system) and player 1 (environment).
• A play starts with a pebble on v0.
• If the pebble is on v2V0, player 0 chooses an outgoing edge and transfers the pebble.
• If the pebble is on v2V1, player 1 chooses the successor.
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Winning Condition• An infinite play is an infinite sequence of states.
• Winning conditions:– Recurrence / persistence in terms of states of the game. – Linear temporal logic or automata on infinite words
over states of the game.
• Does there exist a winning strategy?
• Use the automaton to follow the play and determine the winner?
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Use Automaton
• Add one pebble on the automaton.• Move the pebble on the automaton according to the
move in the game.• Decide acceptance according to the automaton.
Environment
System
Game Automaton
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Simple Game
1 0 1
Visit finitely many 0’s
Environment
System
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Nondeterminism is bad
1 0 1Environment
System
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What’s the Problem?
• The opponent chooses between (infinitely) many different paths.
• A guess should match all possible paths.
• Deterministic automata don’t guess!
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Determinization
• Need stronger acceptance conditions [Lan69].
• Starting with NBW with n states:– DRW with 22n states [McN66]. – DRW with (12)nn2n states and 2n index [Saf88].– DPW with n2n+2 states and 2n index [Pit06].
• Lower bound nO(n) [Mic88,Yan06]
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Back to Games
• Games:– The opponent chooses between many different paths.
– A deterministic automaton enables monitoring the goal of the game.
• Games with LTL/NBW goals:– Convert LTL to NBW, convert NBW to DPW.
– Create product of game and DPW.
• Reasoning about general games reduces to reasoning about parity games.
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The End?!
Not really …
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In Practice
• Determinization is extremely complex.
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Safra’s Construction
• Have a tree of subset constructions.
• Whenever a node (subset) visits F, create a new son with the states in F.
• If a node is removed – flash red light.
• If a node equals its sons – flash green light.
• The Rabin condition has a pair for every node. Node flashes red – bad. Node flashes green – good.
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Deterministic State
• Ordered tree.
• Nodes are elements in {1,…,n}.
• Every node is labeled by a subset of the states.
• Every node is colored green, red, or white.
• Unused names are colored red.
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Deterministic TransitionThe transition of d is the result of the following
transformations.
• Replace node label by labels of successors (subset construction).
• Spawn new sons with accepting states.
• Move states to ‘best’ nodes.
• Remove empty nodes.
• Nodes that equal their sons colored green.
0,1,3
3 1
1
42
0,1,3
3 1
1
421
1
3
5
0,1,3
3 1
1
4
15
20,1,3
3 1
1
4
0,1,3,4
4 12
1
4
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What about your variant?
• Recently, improvement of Safra:– Safra: NBW(n) ! DRW(12nn2n,n)– Variant: NBW(n) ! DPW(n2n+2,2n)
• But: still trees, and everything else.
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Or abcdefghij
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In Practice
• Determinization is extremely complex.
• First implementation in CIAA05.
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OmegaDet [STW05]
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In Practice
• Determinization is extremely complex.
• First implementation in CIAA05.
• No way to implement symbolically.
• All or nothing.
• Resort to other solutions.
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• Restrict attention to a subset of LTL. – Safety / reachability – linear time [RW89,AMPS98].– Recurrence / persistance – quadratic time [AMPS98].– Boolean combinations of safety / reachability [AT04].– Generalized Reactivity(1) – cubic time [PPS06].
Practical Solution 1
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Practical Solution 2 [JGB05,HRS05]
• Heuristics that use the NBW.
• Works? Good.
• Does not work?
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Nondeterminism
• Nondeterministic automata cannot be used for game monitoring.
• Or can they?
• They just have to be built correctly…
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Good for Games Automata• Automata that can be controlled in a step-wise
fashion.
• Defined via a game on the structure of the automaton.
• Can be used for game monitoring.Environment
System
Game Automaton
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Definition
• Define the monitor game played on the structure of the automaton:– Start from the initial state.– Opponent chooses a letter.– We choose successor.– We win if:
• The resulting word is not in the language
• The resulting run is accepting
• An automaton is GFG if we win from initial state.
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1 1 1 1 1 1 1 · · · · 1 1 0 1 1 1 1 1 1 1 · · · ·
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21 3
0,1
1 10,1
0
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Use for Game Monitoring
• Given a GFG we combine the game with the GFG.
• Player 0 chooses how to advance the GFG.
Environment
System
Game Automaton
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Where do I get one?
• Prove that an automaton is good for games if it fair-simulates another good for games.
• Deterministic automata are trivially good for games. So start from the deterministic automaton.
• We show how to construct one.
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Construct a GFG Automaton
• Replace the tree structure by nondeterminism.• Follow nondeterministically n subsets of
states.• Ensure that all the runs followed by some
subset visit accepting states infinitely often.• Wrong guess? Change your mind!• Intuition:
- first set is the subset construction.- other n-1 sets follow subsets of first set.
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Construct a GFG
• Let’s start with details on determinization.
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Determinization in Detail
0,11
1a b aba
1
1
0
0
Subset Construction
• There are infinitely many runs that reach an accepting state a finite number of times.
• Somehow these runs have to be separated.
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Determinization Construction
• Have a tree of subset constructions.
• Whenever a node (subset) visits F, create a new son with the states in F.
• If a node is removed – flash red light.
• If a node equals its sons – flash green light.
• The parity condition follows the minimal node that flashed red/green infinitely often.
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What is a state
• A tree.
• Nodes are elements in {1,…,n}.
• Every node is labeled by a subset of the states.
• G2{1,...,n+1} - the least node colored green.
• R2{1,…,n+1} – the least node that got erased.
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Transition
• Replace label by the set of successors (subset construction).
• Create youngest son with subset of accepting states.
• Move double states to older brothers.• If node equal to union of sons, remove sons
and color green.• Remove empty nodes.• Compact names.
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0,1,3,4
4
0,3 0,3
b
0,1,3,4
4
subset construction
0,1,3,4
4 4,1
spawn sons
4
0,1,3,4
4
move to older sons
4
1
0,1,3,4
4
Handle full nodes
1
remove empty nodes
c
1
2
1
2
1
1
2
1
3
4 2
1
3
4 2
1
4
0,1,3
3 1
1
4
subset construction
2
0,1,3
3 1
1
42
spawn sons
1
1
3
5
0,1,3
3 1
1
4
15
move to older sons
2
0,1,3
3 1
1
4
Handle full nodes
a
subset construction
2
2
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From OmegaDet [STW05]
1
0
1
0
1
10
0
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Safra from a node’s point of view
• I follow some states.
• Some of them may disappear.
• If all visit acceptance set, I raise a green flag.
• If all disappear I die.
• After I die, I can be revived with a new set.
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Our ConstructionA State
• Up to n subsets of the states of the NBW.
• Every state in a subset is either marked or unmarked.
• If a subset is empty all subsets above it are empty.
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Our ConstructionA Transition
• Replace every set with a subset of the possible successors.
• Successors of marked states are marked; accepting states are marked.
• If all are marked, remove marking.
• An empty set can load a subset of the first set.
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Advantages
• Very simple construction.
• Amenable to symbolic implementation.
• Natural incremental structure leading to complete solution.
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A Range of Constructions
• We can get closer / further from the deterministic automaton.
• The number of states goes between n2n and n3n.
• It all depends on the symbolic implementation…
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Incremental Construction
• We don’t always need n sets.
• An automaton with i+1 sets ‘monitors fully’ more games than an automaton with i sets.
• It depends on the game itself.
• It is not related (directly) to memory.
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Summary• Replace deterministic automata by
nondeterministic automata.• Definition of GFG automata.• Construction of GFG automata. • Simple, amenable to symbolic implementation.• Incremental structure leading to the full solution.• Initial enumerative implementation.• Lower bound.
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Safraless Decision Procedures [KV05]• Emptiness of alternating parity tree automata by
rank computation.
• Requires determinization for the upper bound.
• Reduces to Büchi games instead of parity.
• Complexity may be quadratically worse.
• Strategy may be exponentially worse.
• Enables solution of games with LTL winning conditions. Does not apply for NBW winning conditions. Does not apply to infinite structures.
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Future Work
• Implementation.
• Reuse work done in increments.
• Understand better the incremental structure.
• Automata for the complement language.
• Lower bound on the index.
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Going Both Ways
• It would be nice to find both winning and losing states fast.
• Starting from LTL it is easy.– Build NBW N for .– Build NBW N: for :.– Combine the game incrementally with GFG for N.– Combine the game incrementally with GFG for N: .
• Starting from NBW?– Build GFG for N.
– Build KV ranks for N.
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Thank You