Decision Problems for Linear Recurrence Sequences Jo el Ouaknine - PowerPoint PPT Presentation

Decision Problems for Linear Recurrence Sequences Jo¨ el Ouaknine Department of Computer Science, Oxford University (Joint work with James Worrell and Matt Daws) Algorithms Workshop Oxford, October 2012

Termination of Simple Linear Programs x := a ; while cond ( x ) do x := M · x + b ;

Termination of Simple Linear Programs x := a ; while cond ( x ) do x := M · x + b ; where cond ( x ) is linear, e.g. ‘ u · x � = 0’ or ‘ u · x ≥ 5’.

Termination of Simple Linear Programs x := a ; while cond ( x ) do x := M · x + b ; where cond ( x ) is linear, e.g. ‘ u · x � = 0’ or ‘ u · x ≥ 5’. Termination Problem Instance: � a ; cond ; M ; b � Question: Does this program terminate?

Termination of Simple Linear Programs Much work on this and related problems in the literature over the last three decades: Manna, Pnueli, Kannan, Lipton, Sagiv, Podelski, Rybalchenko, Cook, Dershowitz, Tiwari, Braverman, Ben-Amram, Genaim, . . . Approaches include: linear ranking functions size-change termination methods spectral techniques . . . Tools include:

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ?

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ? Does there exist T such that, for all n ≥ T Prob(‘being in s k after n steps’) ≥ 1 / 2 ?

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ? Does there exist T such that, for all n ≥ T Prob(‘being in s k after n steps’) ≥ 1 / 2 ? (1 , 0 , 0 , 0)

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ? Does there exist T such that, for all n ≥ T Prob(‘being in s k after n steps’) ≥ 1 / 2 ? (1 , 0 , 0 , 0) · M = (0 , 0 . 5 , 0 . 2 , 0 . 3)

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ? Does there exist T such that, for all n ≥ T Prob(‘being in s k after n steps’) ≥ 1 / 2 ? (1 , 0 , 0 , 0) · M = (0 , 0 . 5 , 0 . 2 , 0 . 3) · M = (0 . 16 , 0 , 0 . 5 , 0 . 34)

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ? Does there exist T such that, for all n ≥ T Prob(‘being in s k after n steps’) ≥ 1 / 2 ? (1 , 0 , 0 , 0) · M = (0 , 0 . 5 , 0 . 2 , 0 . 3) · M = (0 . 16 , 0 , 0 . 5 , 0 . 34) · M = (0 . 318 , 0 . 08 , 0 . 032 , 0 . 57)

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ? Does there exist T such that, for all n ≥ T Prob(‘being in s k after n steps’) ≥ 1 / 2 ? (1 , 0 , 0 , 0) · M = (0 , 0 . 5 , 0 . 2 , 0 . 3) · M = (0 . 16 , 0 , 0 . 5 , 0 . 34) · M = (0 . 318 , 0 . 08 , 0 . 032 , 0 . 57) · M = (0 . 13 , 0 . 159 , 0 . 1436 , 0 . 5674)

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ? Does there exist T such that, for all n ≥ T Prob(‘being in s k after n steps’) ≥ 1 / 2 ? (1 , 0 , 0 , 0) · M = (0 , 0 . 5 , 0 . 2 , 0 . 3) · M = (0 . 16 , 0 , 0 . 5 , 0 . 34) · M = (0 . 318 , 0 . 08 , 0 . 032 , 0 . 57) · M = (0 . 13 , 0 . 159 , 0 . 1436 , 0 . 5674) · M = (0 . 18528 , 0 . 065 , 0 . 185 , 0 . 56472)

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ? Does there exist T such that, for all n ≥ T Prob(‘being in s k after n steps’) ≥ 1 / 2 ? (1 , 0 , 0 , 0) · M = (0 , 0 . 5 , 0 . 2 , 0 . 3) · M = (0 . 16 , 0 , 0 . 5 , 0 . 34) · M = (0 . 318 , 0 . 08 , 0 . 032 , 0 . 57) · M = (0 . 13 , 0 . 159 , 0 . 1436 , 0 . 5674) · M = (0 . 18528 , 0 . 065 , 0 . 185 , 0 . 56472) · M = (0 . 205444 , 0 . 09264 , 0 . 102056 , 0 . 59986)

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ? Does there exist T such that, for all n ≥ T Prob(‘being in s k after n steps’) ≥ 1 / 2 ? (1 , 0 , 0 , 0) · M = (0 , 0 . 5 , 0 . 2 , 0 . 3) · M = (0 . 16 , 0 , 0 . 5 , 0 . 34) · M = (0 . 318 , 0 . 08 , 0 . 032 , 0 . 57) · M = (0 . 13 , 0 . 159 , 0 . 1436 , 0 . 5674) · M = (0 . 18528 , 0 . 065 , 0 . 185 , 0 . 56472) · M = (0 . 205444 , 0 . 09264 , 0 . 102056 , 0 . 59986) · M = (0 . 171 , 0 . 102722 , 0 . 133729 , 0 . 592549)

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ? Does there exist T such that, for all n ≥ T Prob(‘being in s k after n steps’) ≥ 1 / 2 ? (1 , 0 , 0 , 0) · M = (0 , 0 . 5 , 0 . 2 , 0 . 3) · M = (0 . 16 , 0 , 0 . 5 , 0 . 34) · M = (0 . 318 , 0 . 08 , 0 . 032 , 0 . 57) · M = (0 . 13 , 0 . 159 , 0 . 1436 , 0 . 5674) · M = (0 . 18528 , 0 . 065 , 0 . 185 , 0 . 56472) · M = (0 . 205444 , 0 . 09264 , 0 . 102056 , 0 . 59986) · M = (0 . 171 , 0 . 102722 , 0 . 133729 , 0 . 592549) · M = (0 . 185374 , 0 . 0855 , 0 . 136922 , 0 . 592204)

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ? Does there exist T such that, for all n ≥ T Prob(‘being in s k after n steps’) ≥ 1 / 2 ? (1 , 0 , 0 , 0) · M = (0 , 0 . 5 , 0 . 2 , 0 . 3) · M = (0 . 16 , 0 , 0 . 5 , 0 . 34) · M = (0 . 318 , 0 . 08 , 0 . 032 , 0 . 57) · M = (0 . 13 , 0 . 159 , 0 . 1436 , 0 . 5674) · M = (0 . 18528 , 0 . 065 , 0 . 185 , 0 . 56472) · M = (0 . 205444 , 0 . 09264 , 0 . 102056 , 0 . 59986) · M = (0 . 171 , 0 . 102722 , 0 . 133729 , 0 . 592549) · M = (0 . 185374 , 0 . 0855 , 0 . 136922 , 0 . 592204)

Reachability and Invariance in Markov Chains M : Markov chain over states s 1 , . . . , s k Is it the case, say, that starting in state s 1 , ultimately I am in state s k with probability at least 1 / 2 ? Does there exist T such that, for all n ≥ T Prob(‘being in s k after n steps’) ≥ 1 / 2 ? Ultimate Invariance Problem Instance: � stochastic matrix M ; r ∈ (0 , 1] �   0 . .   Question: Does ∃ T s.t. ∀ n ≥ T , (1 , 0 , . . . , 0) · M n · .  ≥ r ?     0  1

Positivity of Linear Recurrence Sequences u 0 = 1, u 1 = 1 u n +2 = u n +1 + u n

Positivity of Linear Recurrence Sequences u 0 = 1, u 1 = 1 u n +2 = u n +1 + u n

Positivity of Linear Recurrence Sequences u 0 = 1, u 1 = 1 u n +2 = u n +1 + u n 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .

Positivity of Linear Recurrence Sequences u 0 = 1, u 1 = 1 u n +5 = u n +4 + u n +3 − 1 3 u n 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .

Positivity of Linear Recurrence Sequences u 0 = 1, u 1 = 1, u 2 = 2, u 3 = 3, u 4 = 5 u n +5 = u n +4 + u n +3 − 1 3 u n 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .

Positivity of Linear Recurrence Sequences u 0 = 1, u 1 = 1, u 2 = 2, u 3 = 3, u 4 = 5 u n +5 = u n +4 + u n +3 − 1 3 u n − 10 w n +5 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .

Positivity of Linear Recurrence Sequences u 0 = 1, u 1 = 1, u 2 = 2, u 3 = 3, u 4 = 5 u n +5 = u n +4 + u n +3 − 1 3 u n − 10 w n +5 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . . Positivity Problem Instance: A linear recurrence sequence � u n � Question: Is it the case that ∀ n , u n ≥ 0 ?

Sample Decision Problems Termination Problem for x := a ; Simple Linear Programs while u · x � = 0 do Instance: � a ; u ; M ; b � over Z x := M · x + b ; Question: Does this program terminate? Ultimate Invariance Problem for Markov Chains Instance: A stochastic matrix M over Q   0 . . Question: Does ∃ T s.t. ∀ n ≥ T , (1 , 0 , . . . , 0) · M n ·  .   ≥ 1 2 ?     0  1 Positivity Problem for Linear Recurrence Sequences Instance: A linear recurrence sequence � u n � over Z or Q Question: Is it the case that ∀ n , u n ≥ 0 ?

Linear Recurrence Sequences Definition A linear recurrence sequence is a sequence � u 0 , u 1 , u 2 , . . . � of real numbers such that there exist k and constants a 1 , . . . , a k , such that ∀ n ≥ 0, u n + k = a 1 u n + k − 1 + a 2 u n + k − 2 + . . . + a k u n . k is the order of the sequence