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Matthew Hague
with Roland Meyer, Sebastian Muskalla,and Martin Zimmermann
Parity to Safety in Polynomial Time for Pushdown and Collapsible - - PowerPoint PPT Presentation
Parity to Safety in Polynomial Time for Pushdown and Collapsible - - PowerPoint PPT Presentation
Parity to Safety in Polynomial Time for Pushdown and Collapsible Pushdown Games Matthew Hague Royal Holloway, University of London with Roland Meyer, Sebastian Muskalla,and Martin Zimmermann Abstract Safety games for pushdown systems:
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Abstract
Safety games for pushdown systems: n-EXPTIME-complete Parity games for pushdown systems: n-EXPTIME-complete (for order-n collapsible pushdown) We give a "natural" parity->safety reduction
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Abstract
Safety games for pushdown systems: n-EXPTIME-complete Parity games for pushdown systems: n-EXPTIME-complete (for order-n collapsible pushdown) We give a "natural" parity->safety reduction
(i.e. not parity algorithm -> TM -> safety)
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Motivation
Safety is easy to reason about (Parity is hard) Parity is more expressive To know the relationship
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Ideas We Start With
Finite-state parity to safety using counters (Bernet et al, 2002)
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Ideas We Start With
Finite-state parity to safety using counters (Bernet et al, 2002) Pushdown parity to safety with large counters (Fridman and Zimmermann, 2012)
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Ideas We Start With
Finite-state parity to safety using counters (Bernet et al, 2002) Pushdown parity to safety with large counters (Fridman and Zimmermann, 2012) Reduction order-n -> order-(n-1) Rank awareness (H, Murawski, Ong, Serre, 2008)
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Ideas We Start With
Finite-state parity to safety using counters (Bernet et al, 2002) Pushdown parity to safety with large counters (Fridman and Zimmermann, 2012) Reduction order-n -> order-(n-1) Rank awareness (H, Murawski, Ong, Serre, 2008) Encoding large counters in a pushdown stack (Cachat and Walukiewicz, 2007)
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Contributions
Generalise counter encoding to collapse Direct proof based on commutativity of Counters encoding Stack removal Counters behave like a stack
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Parity Game
Elvis Anarchist 4 1 2 4
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Parity Game
Elvis Anarchist 4 1 2 4 Play: 4
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Parity Game
Elvis Anarchist 4 1 2 4 Play: 4 1
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Parity Game
Elvis Anarchist 4 1 2 4 Play: 4 1 2
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Parity Game
Elvis Anarchist 4 1 2 4 Play: 4 1 2 4
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Parity Game
Elvis Anarchist 4 1 2 4 Play: 4 1 2 4 4
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Parity Game
Elvis Anarchist 4 1 2 4 Play: 4 1 2 4 4 1 4 4 1 2 4...
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Parity Game
Elvis Anarchist 4 1 2 4 Play: 4 1 2 4 4 1 4 4 1 2 4...
Winner: Elvis if least infinitely
- ccuring rank is even
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller)
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4:
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4: There is a loop! (Hit 1 five times over 4 states)
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Parity with Counters
4 1 2 4 Count: number of each rank (without seeing anything smaller) Counters: 1: 2: 3: 4: There is a loop! (Hit 1 five times over 4 states) The smallest rank is odd: Elvis loses!
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Parity to Safety
Reduce parity to safety game (Bernet et al): Keep counters in state (n * n^k states) Elvis loses if odd counter is high
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Parity to Safety
Reduce parity to safety game (Bernet et al): Keep counters in state (n * n^k states) Elvis loses if odd counter is high Exponential blow up!
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Pushdown Games
a a b b b b b b p1 c p2 c p3 c p4 c Each state has: Control state (from finite set) Stack of characters Model recursive programs
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Pushdown Parity to Safety
Pushdown Parity Finite-state Parity Finite-State Safety Pushdown Safety
(Walukiewicz 1996)
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Pushdown Parity to Safety
Pushdown Parity Finite-state Parity Finite-State Safety Pushdown Safety
(Walukiewicz 1996) (Bernet et al 2002)
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Pushdown Parity to Safety
Pushdown Parity Finite-state Parity Finite-State Safety Pushdown Safety
(Walukiewicz 1996) (Bernet et al 2002) (Fridman and Zimmermann 2012)
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Pushdown Parity to Safety
Pushdown Parity Finite-state Parity Finite-State Safety Pushdown Safety
(Walukiewicz 1996) (Bernet et al 2002) (Fridman and Zimmermann 2012) Commutes!
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Pushdown Parity to Safety
a a b b b b b b p1 c p2 c p3 c p4 c (a, c1, c2) (b, c1, c2) (b, c1, c2') p1 (c, c1, c2) p2 (c, c1, c2) From: To:
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Pushdown Parity to Safety
a a b b b b b b p1 c p2 c p3 c p4 c (a, c1, c2) (b, c1, c2) (b, c1, c2') p1 (c, c1, c2) p2 (c, c1, c2) From: To: Counters
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Exponential Blow Up?
(a, c1, c2') p (b, c1, c2) Counter values: can be exponential (Pushdown->finite-state has 2^n states) Number of stack characters: (2^n)^k (For k ranks)
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Exponential Blow Up?
(a, c1, c2') p (b, c1, c2) Counter values: can be exponential (Pushdown->finite-state has 2^n states) Number of stack characters: (2^n)^k (For k ranks)
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Reducing Size
(a, c1, c2') p (b, c1, c2) a c2' c1 b c2 p c1 From: To:
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Reducing Size
(a, c1, c2') p (b, c1, c2) a c2' c1 b c2 p c1 From: To: Number of characters: 2^n
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Updating Counters
a c2' c1 b c2 p c1 Increment c1
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Updating Counters
a c2' c1 b c2 p c1 Increment c1 ( , a)
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Updating Counters
a c2' c1 b c2 p c1 Increment c1 ( , a)
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Updating Counters
a c2' c1 b c2 p c1 Increment c1 ( , a) Lost c2' (It will be reset)
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Updating Counters
a c2' c1 b c2 p c1 Increment c1 ( , a) Lost c2' (It will be reset)
'
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Updating Counters
a c2' c1 b c2 p c1 Increment c1 ( , a) Lost c2' (It will be reset)
'
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Updating Counters
a c2' c1 b c2 p c1 Increment c1 ( , a) Lost c2' (It will be reset)
'
a
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Polynomial No. of Characters
a c2' c1 b c2 p c1 Counters up to 2^n Alphabet exponential
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Polynomial No. of Characters
a c2' c1 b c2 p c1 Counters up to 2^n Alphabet exponential
a 1 1 1 . . p .
c2' in binary c1 in binary
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Polynomial No. of Characters
a c2' c1 b c2 p c1 Counters up to 2^n Alphabet exponential
a 1 1 1 . . p .
c2' in binary c1 in binary Pushdowns handle length-n binary Alphabet polynomial!
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Pushdown Parity to Safety
Pushdown parity game -> pushdown safety game Naively exponential Use stack discipline of counters Binary counter encoding of pushdowns Polynomial time reduction
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Collapsible Pushdown Systems
Pushdown Systems = First-Order Recursion Collapsible Pushdown Systems = Higher-Order Recursion
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Higher-Order Programming: Niche?
Almost all modern languages support it Scala, Go, JavaScript, Python, ... Retro-fitted to C++ and Java Asynchronous programs/callbacks Map/Reduce
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Collapsible Pushdown Systems
Higher-order program: functions of functions Higher-order pushdown: stack of stacks a b p c
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Collapsible Pushdown Systems
Higher-order program: functions of functions Higher-order pushdown: stack of stacks a b p c a b c
b c
d a c p
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Collapsible Pushdown Systems
Higher-order program: functions of functions Higher-order pushdown: stack of stacks a b p c a b c
b c
d a c p a b b a a c c c b a b a a c c p
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Collapsible Pushdown Systems
Higher-order program: functions of functions Higher-order pushdown: stack of stacks a b p c a b c
b c
d a c p a b b a a c c c b a b a a c c p (+links)
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Generalising Parity->Safety
Collapsible Pushdown -> Finite-State (n-Exponential blow up) n-Exponential Counters Can encode in binary on order-n stack!
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Summary
Parity -> Safety Counters look out for loops (Even with infinite states) Reduction to finite-state Counters behave like stacks Counter values match limits of system Polynomial-time encoding
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