Computability in Quaternion Matrix Semigroups

Transcription

1 Computability in Quaternion Matrix Semigroups Paul Bell Complexity Theory and Algorithmics Group Joint work with Dr. I. Potapov The University of Liverpool Workshop on Algorithms on Words p.1/19

2 Introduction & Aims Computability problems for finitely generated semigroups. Workshop on Algorithms on Words p.2/19

3 Introduction & Aims Computability problems for finitely generated semigroups. Survey the boundary of undecidability for quaternion matrix semigroups. Workshop on Algorithms on Words p.2/19

4 Introduction & Aims Computability problems for finitely generated semigroups. Survey the boundary of undecidability for quaternion matrix semigroups. Explore the geometric interpretations of these problems. Workshop on Algorithms on Words p.2/19

5 Introduction & Aims Computability problems for finitely generated semigroups. Survey the boundary of undecidability for quaternion matrix semigroups. Explore the geometric interpretations of these problems. State some new open problems in this area. Workshop on Algorithms on Words p.2/19

6 Quaternions Rational Quaternions are four-dimensional complex numbers and form a division ring H(Q); they are associative but non-commutative: q = a + bi + cj + dk : a,b,c,d Q Workshop on Algorithms on Words p.3/19

7 Quaternions Rational Quaternions are four-dimensional complex numbers and form a division ring H(Q); they are associative but non-commutative: q = a + bi + cj + dk : a,b,c,d Q They are often used to respresent three dimensional rotations in aerospace, robotics and computer graphics. Workshop on Algorithms on Words p.3/19

8 Quaternions Rational Quaternions are four-dimensional complex numbers and form a division ring H(Q); they are associative but non-commutative: q = a + bi + cj + dk : a,b,c,d Q They are often used to respresent three dimensional rotations in aerospace, robotics and computer graphics. Single quaternions can represent three dimensional rotations and a pair of quaternions can represent four dimensional rotations. Workshop on Algorithms on Words p.3/19

9 Quaternion Julia Fractals Workshop on Algorithms on Words p.4/19

10 Quaternion Julia Fractals 2 Workshop on Algorithms on Words p.5/19

11 Words and Quaternions We showed that using two rational quaternions, we may encode a free group. Workshop on Algorithms on Words p.6/19

12 Words and Quaternions We showed that using two rational quaternions, we may encode a free group. This follows from Swierczkowski s result that the subgroup of SO 3 (R) generated by rotations of θ about two perpendicular axes is free when cos(θ) Q iff cos(θ) 0, ± 1 2, ±1. Thus we get an injective homomorphism γ : Σ H(Q). Workshop on Algorithms on Words p.6/19

13 Free Complex Matrices It is well known that a rational complex number can be represented as a 2 2 rational matrix. Workshop on Algorithms on Words p.7/19

14 Free Complex Matrices It is well known that a rational complex number can be represented as a 2 2 rational matrix. Similarly, any rational quaternion can be represented by a 2 2 rational complex matrix. Workshop on Algorithms on Words p.7/19

15 Free Complex Matrices It is well known that a rational complex number can be represented as a 2 2 rational matrix. Similarly, any rational quaternion can be represented by a 2 2 rational complex matrix. Thus the monomorphism γ : Σ H(Q) gives a free group of rational complex matrices, i.e. ζ : Σ C(Q) 2 2 : ζ(a) = ( i i ),ζ(b) = ( ) Workshop on Algorithms on Words p.7/19

16 Semigroup Problems A semigroup, S, is a set, of possibly infinite size, which is closed under a given binary associative operator. Workshop on Algorithms on Words p.8/19

17 Semigroup Problems A semigroup, S, is a set, of possibly infinite size, which is closed under a given binary associative operator. A generator, G, of a semigroup is a subset of S of minimal size where any element of S can be expressed as a product of elements of G. Workshop on Algorithms on Words p.8/19

18 Semigroup Problems A semigroup, S, is a set, of possibly infinite size, which is closed under a given binary associative operator. A generator, G, of a semigroup is a subset of S of minimal size where any element of S can be expressed as a product of elements of G. For example, take the generator G = {1,2,3,5}. With multiplication, this generates the semigroup S = { 2 a 3 b 5 c } where a,b,c 0. Workshop on Algorithms on Words p.8/19

19 Semigroup Problems 2 Matrix Membership Problem: Given a matrix semigroup, S, and a matrix M, is M S? Workshop on Algorithms on Words p.9/19

20 Semigroup Problems 2 Matrix Membership Problem: Given a matrix semigroup, S, and a matrix M, is M S? Many undecidability results are known in this area e.g. mortality, scalar matrices and freeness. Many open problems remain. The decidability depends upon parameters of the input. Workshop on Algorithms on Words p.9/19

21 Quaternion Matrices We considered standard matrix semigroup problems on quaternion matrices. Workshop on Algorithms on Words p.10/19

22 Quaternion Matrices We considered standard matrix semigroup problems on quaternion matrices. Many results undecidable for dimension 2 2. We showed that membership for semigroup of Lipschitz integers is decidable. Workshop on Algorithms on Words p.10/19

23 Quaternion Matrices We considered standard matrix semigroup problems on quaternion matrices. Many results undecidable for dimension 2 2. We showed that membership for semigroup of Lipschitz integers is decidable. But why study quaternion matrices? Why not also octonions etc.? Workshop on Algorithms on Words p.10/19

24 Quaternion Semigroup Results For two-dimensional rational quaternion matrix semigroup S: Membership in S is undecidable. Determining if S is free is undecidable. Determining if any M S is diagonal is undecidable. Workshop on Algorithms on Words p.11/19

25 Quaternion Semigroup Results For two-dimensional rational quaternion matrix semigroup S: Membership in S is undecidable. Determining if S is free is undecidable. Determining if any M S is diagonal is undecidable. We used a new version of Post s Correspondence Problem.. Workshop on Algorithms on Words p.11/19

26 Post s Correspondence Problem Post s correspondence problem (PCP) is a famous undecidable problem which simulates a Turing machine. Workshop on Algorithms on Words p.12/19

27 Post s Correspondence Problem Post s correspondence problem (PCP) is a famous undecidable problem which simulates a Turing machine. We are given an indexed set of pairs of (binary) words {(u 1,v 1 ), (u 2,v 2 ),, (u n,v n )}. Workshop on Algorithms on Words p.12/19

28 Post s Correspondence Problem Post s correspondence problem (PCP) is a famous undecidable problem which simulates a Turing machine. We are given an indexed set of pairs of (binary) words {(u 1,v 1 ), (u 2,v 2 ),, (u n,v n )}. PCP(n): Does there exist a finite sequence (i 1,i 2,...,i k ) such that u i1 u i2 u ik = v i1 v i2 v ik? Workshop on Algorithms on Words p.12/19

29 Post s Correspondence Problem Post s correspondence problem (PCP) is a famous undecidable problem which simulates a Turing machine. We are given an indexed set of pairs of (binary) words {(u 1,v 1 ), (u 2,v 2 ),, (u n,v n )}. PCP(n): Does there exist a finite sequence (i 1,i 2,...,i k ) such that u i1 u i2 u ik = v i1 v i2 v ik? We may reduce this word problem to a matrix problem to show undecidability. Workshop on Algorithms on Words p.12/19

30 An Example of PCP P 1 = [ ] aab a,p 2 = [ ] ba,p 3 = a [ ] ab bbaabb Workshop on Algorithms on Words p.13/19

31 An Example of PCP P 1 = [ ] aab a,p 2 = [ ] ba,p 3 = a [ ] ab bbaabb Now take the sequence P 1 P 2 P 3 P 2 : Workshop on Algorithms on Words p.13/19

32 An Example of PCP P 1 = [ ] aab a,p 2 = [ ] ba,p 3 = a [ ] ab bbaabb Now take the sequence P 1 P 2 P 3 P 2 : aab ba ab ba a a bbaabb a Workshop on Algorithms on Words p.13/19

33 An Example of PCP P 1 = [ ] aab a,p 2 = [ ] ba,p 3 = a [ ] ab bbaabb Now take the sequence P 1 P 2 P 3 P 2 : aab ba ab ba a a bbaabb a It is known that PCP(2) is decidable and PCP(7) is undecidable. Workshop on Algorithms on Words p.13/19

34 Index Coding PCP Given a binary alphabet Σ = {a,b}, inverse alphabet Σ = {a,b} and a finite set of pairs of words: {(u i,v i ) 1 i n} (Σ Σ) (Σ Σ). Does there exist a finite sequence s = (s 1,s 2,...,s k ) such that exactly one s j = n and u s1 u s2 u sk = v s1 v s2 v sk = ε? Workshop on Algorithms on Words p.14/19

35 Index Coding PCP 2 Instead of storing two words u,v and testing if they are equal, we encode both words in word u (using inverse elements for v) and use the second word only to store the index. Workshop on Algorithms on Words p.15/19

36 Index Coding PCP 2 Instead of storing two words u,v and testing if they are equal, we encode both words in word u (using inverse elements for v) and use the second word only to store the index. We proved in an earlier paper that this problem is undecidable and it has been useful in several problems. Workshop on Algorithms on Words p.15/19

37 Index Coding PCP 2 Instead of storing two words u,v and testing if they are equal, we encode both words in word u (using inverse elements for v) and use the second word only to store the index. We proved in an earlier paper that this problem is undecidable and it has been useful in several problems. Our construction used 30 pairs of words but this was recently improved to just 14 by Halava, Harju and Hirvensalo. Workshop on Algorithms on Words p.15/19

38 Quaternionic Rotation Let (v 1,v 2,v 3 ) Q 3 be a rotation axis such that (v 1,v 2,v 3 ) = 1 and define quaternion: q = θ + v 1 i + v 2 j + v 3 k H(Q) Workshop on Algorithms on Words p.16/19

39 Quaternionic Rotation Let (v 1,v 2,v 3 ) Q 3 be a rotation axis such that (v 1,v 2,v 3 ) = 1 and define quaternion: q = θ + v 1 i + v 2 j + v 3 k H(Q) Now, let P = (p 1,p 2,p 3 ) Q 3 be a three-dimensional point and R = 0 + p 1 i + p 2 j + p 3 k H(Q) Workshop on Algorithms on Words p.16/19

40 Quaternionic Rotation Let (v 1,v 2,v 3 ) Q 3 be a rotation axis such that (v 1,v 2,v 3 ) = 1 and define quaternion: q = θ + v 1 i + v 2 j + v 3 k H(Q) Now, let P = (p 1,p 2,p 3 ) Q 3 be a three-dimensional point and R = 0 + p 1 i + p 2 j + p 3 k H(Q) Then the rotation operator is given by qrq. Workshop on Algorithms on Words p.16/19

41 Quaternionic Rotation 2 Point Rotation Problem (PRP(n)): Given two points x,y Q n on the unit length (n 1)-sphere, and a semigroup S of n-dimensional rotations. Does there exist any M S such that M rotates x to y?. Workshop on Algorithms on Words p.17/19

42 Quaternionic Rotation 2 Point Rotation Problem (PRP(n)): Given two points x,y Q n on the unit length (n 1)-sphere, and a semigroup S of n-dimensional rotations. Does there exist any M S such that M rotates x to y?. Not difficult to show PRP(2) is decidable. Workshop on Algorithms on Words p.17/19

43 Quaternionic Rotation 2 Point Rotation Problem (PRP(n)): Given two points x,y Q n on the unit length (n 1)-sphere, and a semigroup S of n-dimensional rotations. Does there exist any M S such that M rotates x to y?. Not difficult to show PRP(2) is decidable. We showed that PRP(4) is undecidable using quaternions. Workshop on Algorithms on Words p.17/19

44 Open Problems Is membership decidable for semigroups of rational quaternions? Alternatively, is the Point Rotation Problem decidable in three dimensions (PRP(3))? Workshop on Algorithms on Words p.18/19

45 Open Problems Is membership decidable for semigroups of rational quaternions? Alternatively, is the Point Rotation Problem decidable in three dimensions (PRP(3))? Is the freeness of a semigroup of quaternions decidable? Workshop on Algorithms on Words p.18/19

46 End of Talk Any questions or comments? Workshop on Algorithms on Words p.19/19