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A Survey of Program Termination: Practical and Theoretical - - PowerPoint PPT Presentation
A Survey of Program Termination: Practical and Theoretical - - PowerPoint PPT Presentation
A Survey of Program Termination: Practical and Theoretical Challenges Jo el Ouaknine Department of Computer Science, Oxford University VTSA 2014 Luxembourg, October 2014 Instructive Example Consider the following order-5 recurrence: u n +5
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Instructive Example
Consider the following order-5 recurrence: un+5 = − 19
25un+4 − 114 125un+3 + 114 125un+2 + 19 25un+1 + un
This is simple, with characteristic roots 1, λ1, λ1, λ2, λ2, where λ1 = −3 + 4i 5 and λ2 = −7 + 24i 25
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Instructive Example
Consider the following order-5 recurrence: un+5 = − 19
25un+4 − 114 125un+3 + 114 125un+2 + 19 25un+1 + un
This is simple, with characteristic roots 1, λ1, λ1, λ2, λ2, where λ1 = −3 + 4i 5 and λ2 = −7 + 24i 25 For suitably chosen initial values we have un = 33
8 + λn 1 + λn 1 + 2λn 2 + 2λn 2
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Orbits of Characteristic Roots
{λn
1 : n ∈ N} and {λn 2 : n ∈ N} are both dense in T.
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Orbits of Characteristic Roots
{λn
1 : n ∈ N} and {λn 2 : n ∈ N} are both dense in T.
{(λn
1, λn 2) : n ∈ N} not dense in T2 due to relation λ2 1λ2 = 1.
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Orbits of Characteristic Roots
{λn
1 : n ∈ N} and {λn 2 : n ∈ N} are both dense in T.
{(λn
1, λn 2) : n ∈ N} not dense in T2 due to relation λ2 1λ2 = 1.
{(λn
1, λn 2) : n ∈ N} dense in helix {(z1, z2) ∈ T2 : z2 1z2 = 1}.
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Orbits of Characteristic Roots
{λn
1 : n ∈ N} and {λn 2 : n ∈ N} are both dense in T.
{(λn
1, λn 2) : n ∈ N} not dense in T2 due to relation λ2 1λ2 = 1.
{(λn
1, λn 2) : n ∈ N} dense in helix {(z1, z2) ∈ T2 : z2 1z2 = 1}.
Point (−1, −1) does not lie on helix.
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Example
Critical Point! ( − 1
8 + √ 63i 8 , − 31 32 + √ 63i 32 )
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Example
Critical Point! ( − 1
8 + √ 63i 8 , − 31 32 + √ 63i 32 )
For (λn
1, λn 2) near this point,
un := 33
8 + λn 1 + λn 1 + 2λn 2 + 2λn 2
is close to 0.
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Example
Critical Point! ( − 1
8 + √ 63i 8 , − 31 32 + √ 63i 32 )
For (λn
1, λn 2) near this point,
un := 33
8 + λn 1 + λn 1 + 2λn 2 + 2λn 2
is close to 0. un is ultimately positive—just.
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