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Csr2011 june17 11_30_vyalyi

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  • 1. Orbits of Linear Maps and Regular Languages S. Tarasov, M. Vyalyi Dorodnitsyn Computing Center of RAS CSR 2011S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 1 / 17
  • 2. Contents1 Orbits of linear maps2 Regular realizability (RR)3 Examples of relation between RR and linear algebra S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 2 / 17
  • 3. Chamber hitting problemDefinitionAn orbit OrbΦ x is {Φk x : k ∈ Z+ }, where Φ : Qd → Qd is a linear mapand x ∈ Qd .DefinitionA chamber HS = {x ∈ Qd : sign(hi (x)) = si for 1 i m}, where hi areaffine functions and s ∈ {±1, 0}m is a sign pattern.Chamber hitting problem (CHP)INPUT: Φ, x0 , h1 , . . . , hm , s.OUTPUT: ‘yes’ if OrbΦ x0 ∩ Hs = ∅ and ‘no’ otherwise. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 3 / 17
  • 4. Chamber hitting problemDefinitionAn orbit OrbΦ x is {Φk x : k ∈ Z+ }, where Φ : Qd → Qd is a linear mapand x ∈ Qd .DefinitionA chamber HS = {x ∈ Qd : sign(hi (x)) = si for 1 i m}, where hi areaffine functions and s ∈ {±1, 0}m is a sign pattern.Chamber hitting problem (CHP)INPUT: Φ, x0 , h1 , . . . , hm , s.OUTPUT: ‘yes’ if OrbΦ x0 ∩ Hs = ∅ and ‘no’ otherwise. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 3 / 17
  • 5. Chamber hitting problemDefinitionAn orbit OrbΦ x is {Φk x : k ∈ Z+ }, where Φ : Qd → Qd is a linear mapand x ∈ Qd .DefinitionA chamber HS = {x ∈ Qd : sign(hi (x)) = si for 1 i m}, where hi areaffine functions and s ∈ {±1, 0}m is a sign pattern.Chamber hitting problem (CHP)INPUT: Φ, x0 , h1 , . . . , hm , s.OUTPUT: ‘yes’ if OrbΦ x0 ∩ Hs = ∅ and ‘no’ otherwise. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 3 / 17
  • 6. Special cases: the orbit problemOrbit problemINPUT: Φ, x, y .OUTPUT: ‘yes’ if y ∈ OrbΦ x and ‘no’ otherwise.In this case the chamber is {y }.Theorem (Kannan, Lipton, 1986)There exists a polynomial time algorithm for the orbit problem. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 4 / 17
  • 7. Special cases: the orbit problemOrbit problemINPUT: Φ, x, y .OUTPUT: ‘yes’ if y ∈ OrbΦ x and ‘no’ otherwise.In this case the chamber is {y }.Theorem (Kannan, Lipton, 1986)There exists a polynomial time algorithm for the orbit problem. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 4 / 17
  • 8. Problems reducible to CHPSkolem problemINPUT: a1 , . . . , ad ; b1 , . . . , bd .xn — a linear recurrent sequence d xn = ai xn−i , (n > d), xn = bn (1 n d) i=1OUTPUT: ‘yes’ if xn = 0 for some n and ‘no’ otherwise.Positivity problemINPUT: a1 , . . . , ad ; b1 , . . . , bd ; xn is LRS.OUTPUT: ‘yes’ if xn > 0 for all n and ‘no’ otherwise. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 5 / 17
  • 9. Problems reducible to CHPSkolem problemINPUT: a1 , . . . , ad ; b1 , . . . , bd .xn — a linear recurrent sequence d xn = ai xn−i , (n > d), xn = bn (1 n d) i=1OUTPUT: ‘yes’ if xn = 0 for some n and ‘no’ otherwise.Positivity problemINPUT: a1 , . . . , ad ; b1 , . . . , bd ; xn is LRS.OUTPUT: ‘yes’ if xn > 0 for all n and ‘no’ otherwise. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 5 / 17
  • 10. State of the artOpen questionsIs CHP decidable? Is Skolem problem decidable? Is positivity problemdecidable?Known decidability results for small d d =2 d =3 d =4 d =5 Skolem folklore Vereshchagin, 1985 Halava et al., 2005 Pos. pr. Halava et al., Laohakosol, 2006 Tangsupphathawat, 2009 CHP Sechin, 2011 S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 6 / 17
  • 11. Regular realizability problems (RR)A set L ⊂ Σ∗ is called a filter. Each filter determines a specific regularrealizability problem:L-realizability problemINPUT: a description of a regular language R.OUTPUT: ‘yes’ if R ∩ L = ∅ and ‘no’ otherwise. w R Filter L w ∈R∩L w∈L / S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 7 / 17
  • 12. Permutation filterDefinitionPB ⊂ {#, 0, 1}∗ consists of permutation words, i.e., words of the form #w1 #w2 # . . . wN #,where wi ∈ {0, 1}∗ are blocks, |wi | = n, i = 1, 2, . . . , N (n is the block rank), N = 2n , n 1, each binary word of length n is a block.Examples #00#11#10#01# ∈ PB #10#11#00#01# ∈ PB #10#01#00#11# ∈ PB S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 8 / 17
  • 13. Orbits vs Regular realizabilityTheorem (Tarasov, Vyalyi, 2010)CHP and PB -realizability problem are Turing equivalent.From PB -realizability to CHPReduction starts from a Q-linear extension of the transition monoid. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 9 / 17
  • 14. Orbits vs Regular realizabilityTheorem (Tarasov, Vyalyi, 2010)CHP and PB -realizability problem are Turing equivalent.From PB -realizability to CHPReduction starts from a Q-linear extension of the transition monoid. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 9 / 17
  • 15. From CHP to PB -realizabilityThe idea is to represent an arithmetic computation in a ‘natural’ form.The main construction R1 , R2 — regular languages. How to check that there exists an integer n such that Card({w : |w | = n ∧ w ∈ R1 }) = Card({w : |w | = n ∧ w ∈ R2 })? (♣) Regular expression E = #((R1 ∩ R2 )#)∗ ((R1 R2 )#(R2 R1 )#)∗ ((R1 ∩ R2 )#)∗ (♣) is equivalent to E ∩ PB = ∅. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 10 / 17
  • 16. From CHP to PB -realizabilityThe idea is to represent an arithmetic computation in a ‘natural’ form.The main construction R1 , R2 — regular languages. How to check that there exists an integer n such that Card({w : |w | = n ∧ w ∈ R1 }) = Card({w : |w | = n ∧ w ∈ R2 })? (♣) Regular expression E = #((R1 ∩ R2 )#)∗ ((R1 R2 )#(R2 R1 )#)∗ ((R1 ∩ R2 )#)∗ (♣) is equivalent to E ∩ PB = ∅. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 10 / 17
  • 17. From CHP to PB -realizabilityThe idea is to represent an arithmetic computation in a ‘natural’ form.The main construction R1 , R2 — regular languages. How to check that there exists an integer n such that Card({w : |w | = n ∧ w ∈ R1 }) = Card({w : |w | = n ∧ w ∈ R2 })? (♣) Regular expression E = #((R1 ∩ R2 )#)∗ ((R1 R2 )#(R2 R1 )#)∗ ((R1 ∩ R2 )#)∗ (♣) is equivalent to E ∩ PB = ∅. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 10 / 17
  • 18. From CHP to PB -realizabilityThe idea is to represent an arithmetic computation in a ‘natural’ form.The main construction R1 , R2 — regular languages. How to check that there exists an integer n such that Card({w : |w | = n ∧ w ∈ R1 }) = Card({w : |w | = n ∧ w ∈ R2 })? (♣) Regular expression E = #((R1 ∩ R2 )#)∗ ((R1 R2 )#(R2 R1 )#)∗ ((R1 ∩ R2 )#)∗ (♣) is equivalent to E ∩ PB = ∅. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 10 / 17
  • 19. More examples of relation between RR and linear algebra undecidable track product of the periodic and permutation filter unknown permutation filter decidable surjective filter injective filter S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 11 / 17
  • 20. Surjective filterDefinitionSB consists of words of the form #w1 #w2 # . . . wN #,where wi ∈ {0, 1}∗ are blocks, |wi | = n, i = 1, 2, . . . , N, n is the block rank, each binary word of length n is a block.Examples #00#00#11#10#01# ∈ SB #10#11#10#00#01# ∈ SB #10#01#00#01#11# ∈ SB S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 12 / 17
  • 21. Injective filterDefinitionIB consists of words of the form #w1 #w2 # . . . wN #,where wi ∈ {0, 1}∗ are blocks, |wi | = n, i = 1, 2, . . . , N, n is the block rank, wi = wj for i = j.Examples #00#10#01# ∈ IB #101#111#001#010# ∈ IB #1000#0110#0000#1111# ∈ IB S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 13 / 17
  • 22. Decidability resultsTheoremIB -realizability problem is decidable.SB -realizability problem is decidable.Proofs are based on converting IB -realizability problem (resp.,SB -realizability problem) to a problem about orbits. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 14 / 17
  • 23. Decidability resultsTheoremIB -realizability problem is decidable.SB -realizability problem is decidable.Proofs are based on converting IB -realizability problem (resp.,SB -realizability problem) to a problem about orbits. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 14 / 17
  • 24. An undecidable problemZero in the Upper Right Corner Problem (ZURC)INPUT: A1 , . . . , AN are D × D integer matrices.OUTPUT: ‘yes’ if there exists a sequence j1 , . . . , j such that (Aj1 Aj2 . . . Aj )1D = 0 and ‘no’ otherwise.Theorem (Bell, Potapov, 2006)The ZURC problem is undecidable for N = 2 and D = 18.The ZURC problem is reduced to the regular realizability problem for thetrack product of periodic and permutation filters. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 15 / 17
  • 25. An undecidable problemZero in the Upper Right Corner Problem (ZURC)INPUT: A1 , . . . , AN are D × D integer matrices.OUTPUT: ‘yes’ if there exists a sequence j1 , . . . , j such that (Aj1 Aj2 . . . Aj )1D = 0 and ‘no’ otherwise.Theorem (Bell, Potapov, 2006)The ZURC problem is undecidable for N = 2 and D = 18.The ZURC problem is reduced to the regular realizability problem for thetrack product of periodic and permutation filters. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 15 / 17
  • 26. An undecidable problemZero in the Upper Right Corner Problem (ZURC)INPUT: A1 , . . . , AN are D × D integer matrices.OUTPUT: ‘yes’ if there exists a sequence j1 , . . . , j such that (Aj1 Aj2 . . . Aj )1D = 0 and ‘no’ otherwise.Theorem (Bell, Potapov, 2006)The ZURC problem is undecidable for N = 2 and D = 18.The ZURC problem is reduced to the regular realizability problem for thetrack product of periodic and permutation filters. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 15 / 17
  • 27. Track productFor languages L1 ⊂ ({#} ∪ Σ1 )∗ , L2 ⊂ ({#} ∪ Σ2 )∗ the track productL1 L2 ⊂ ({#} ∪ Σ1 × Σ2 )∗ .Projections a1 a2 a3 ... # # ... b 1 b2 b3 π1 π2 . . . # a1 a2 a3 # . . . . . . # b 1 b 2 b3 # . . .Definition of L1 L2 L1 L2 = {w ∈ ({#} ∪ Σ1 × Σ2 )∗ | π1 w ∈ L1 ; π2 w ∈ L2 } S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 16 / 17
  • 28. Track productFor languages L1 ⊂ ({#} ∪ Σ1 )∗ , L2 ⊂ ({#} ∪ Σ2 )∗ the track productL1 L2 ⊂ ({#} ∪ Σ1 × Σ2 )∗ .Projections a1 a2 a3 ... # # ... b 1 b2 b3 π1 π2 . . . # a1 a2 a3 # . . . . . . # b 1 b 2 b3 # . . .Definition of L1 L2 L1 L2 = {w ∈ ({#} ∪ Σ1 × Σ2 )∗ | π1 w ∈ L1 ; π2 w ∈ L2 } S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 16 / 17
  • 29. Track product of periodic and permutation filtersDefinitions Periodic filter PerΣ ⊂ ({#} ∪ Σ)∗ consists of words of the form #w #w # . . . w #, where w ∈ {0, 1}∗ . Definition of the permutation filter PΣ over the alphabet {#} ∪ Σ is similar to the binary case.TheoremZURC ≤m (PerΣ1 PΣ2 )-regular realizability for |Σ1 | = 2, |Σ2 | = 648.Informally, the periodic part is to represent a sequence of matrices and thepermutation part is to encode the condition that the URC entry is 0. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 17 / 17
  • 30. Track product of periodic and permutation filtersDefinitions Periodic filter PerΣ ⊂ ({#} ∪ Σ)∗ consists of words of the form #w #w # . . . w #, where w ∈ {0, 1}∗ . Definition of the permutation filter PΣ over the alphabet {#} ∪ Σ is similar to the binary case.TheoremZURC ≤m (PerΣ1 PΣ2 )-regular realizability for |Σ1 | = 2, |Σ2 | = 648.Informally, the periodic part is to represent a sequence of matrices and thepermutation part is to encode the condition that the URC entry is 0. S. Tarasov, M. Vyalyi (CCAS) Orbits of linear maps and regular languages CSR 2011 17 / 17