A model of computer memory Ben Fairbairn Birkbeck, University of - PowerPoint PPT Presentation

A model of computer memory Ben Fairbairn Birkbeck, University of London Groups St Andrews, August 2013

Joint work with Peter Cameron and Max Gadouleau

An application of group theory to genetics Ben Fairbairn Birkbeck, University of London Groups St Andrews, August 2013

The basic problem GAP, Version 4.6.4 of 04-May-2013 (free software, GPL) http://www.gap-system.org Architecture: i686-pc-cygwin-gcc-default32 Libs used: gmp, readline . . . Error, exceeded the permitted memory (‘-o’ command line option)

Swapping two variables: undergraduate method x y t action 1 2 - -

Swapping two variables: undergraduate method x y t action 1 2 - - 1 2 1 x ← t

Swapping two variables: undergraduate method x y t action 1 2 - - 1 2 1 x ← t 2 2 1 y ← x

Swapping two variables: undergraduate method x y t action 1 2 - - 1 2 1 x ← t 2 2 1 y ← x 2 1 1 t ← y

Swapping two variables: the “XOR” trick x y action 1 2 -

Swapping two variables: the “XOR” trick x y action 1 2 - 3 2 x + y ← x

Swapping two variables: the “XOR” trick x y action 1 2 - 3 2 x + y ← x 3 1 x − y ← y

Swapping two variables: the “XOR” trick x y action 1 2 - 3 2 x + y ← x 3 1 x − y ← y 2 1 x − y ← x

Main definitions (Gadouleau & Riis)

Main definitions (Gadouleau & Riis) ◮ Let A = { 0 , . . . , q − 1 } be a set and let n be a positive integer.

Main definitions (Gadouleau & Riis) ◮ Let A = { 0 , . . . , q − 1 } be a set and let n be a positive integer. ◮ A bijection f : A n → A n is an instruction if

Main definitions (Gadouleau & Riis) ◮ Let A = { 0 , . . . , q − 1 } be a set and let n be a positive integer. ◮ A bijection f : A n → A n is an instruction if for ( x 1 , . . . , x n ) ∈ A n such that f : ( x 1 , . . . , x n ) �→ ( y 1 , . . . , y n )

Main definitions (Gadouleau & Riis) ◮ Let A = { 0 , . . . , q − 1 } be a set and let n be a positive integer. ◮ A bijection f : A n → A n is an instruction if for ( x 1 , . . . , x n ) ∈ A n such that f : ( x 1 , . . . , x n ) �→ ( y 1 , . . . , y n ) we have that x j = y j for all j � = i for some 1 ≤ i ≤ j .

Main definitions (Gadouleau & Riis) ◮ Let A = { 0 , . . . , q − 1 } be a set and let n be a positive integer. ◮ A bijection f : A n → A n is an instruction if for ( x 1 , . . . , x n ) ∈ A n such that f : ( x 1 , . . . , x n ) �→ ( y 1 , . . . , y n ) we have that x j = y j for all j � = i for some 1 ≤ i ≤ j . ◮ We say that f updates the i th register .

Main definitions (Gadouleau & Riis) ◮ Let A = { 0 , . . . , q − 1 } be a set and let n be a positive integer. ◮ A bijection f : A n → A n is an instruction if for ( x 1 , . . . , x n ) ∈ A n such that f : ( x 1 , . . . , x n ) �→ ( y 1 , . . . , y n ) we have that x j = y j for all j � = i for some 1 ≤ i ≤ j . ◮ We say that f updates the i th register . ◮ We call elements of A n states .

An example If A = { 0 , 1 } (so q = 2) and n = 4 then A n consists of . . .

An example If A = { 0 , 1 } (so q = 2) and n = 4 then A n consists of . . . 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0

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Questions Note that each of the Coxeter generators are instructions.

Questions Note that each of the Coxeter generators are instructions. ◮ Is this the smallest set of instructions we need to generate Sym( A n )?

Questions Note that each of the Coxeter generators are instructions. ◮ Is this the smallest set of instructions we need to generate Sym( A n )? We need at least one for each register, so we should really ask . . .

Questions Note that each of the Coxeter generators are instructions. ◮ Is this the smallest set of instructions we need to generate Sym( A n )? We need at least one for each register, so we should really ask . . . ◮ Can we generate Sym( A n ) with n instructions?

Questions Note that each of the Coxeter generators are instructions. ◮ Is this the smallest set of instructions we need to generate Sym( A n )? We need at least one for each register, so we should really ask . . . ◮ Can we generate Sym( A n ) with n instructions? ◮ What about Alt( A n )?

Questions Note that each of the Coxeter generators are instructions. ◮ Is this the smallest set of instructions we need to generate Sym( A n )? We need at least one for each register, so we should really ask . . . ◮ Can we generate Sym( A n ) with n instructions? ◮ What about Alt( A n )? If q is a prime power then we can view A n as a finite vector space and restrict our attention to instructions that are linear maps.

Questions Note that each of the Coxeter generators are instructions. ◮ Is this the smallest set of instructions we need to generate Sym( A n )? We need at least one for each register, so we should really ask . . . ◮ Can we generate Sym( A n ) with n instructions? ◮ What about Alt( A n )? If q is a prime power then we can view A n as a finite vector space and restrict our attention to instructions that are linear maps. ◮ Can we generate GL n ( q ) with n instructions?

Questions Note that each of the Coxeter generators are instructions. ◮ Is this the smallest set of instructions we need to generate Sym( A n )? We need at least one for each register, so we should really ask . . . ◮ Can we generate Sym( A n ) with n instructions? ◮ What about Alt( A n )? If q is a prime power then we can view A n as a finite vector space and restrict our attention to instructions that are linear maps. ◮ Can we generate GL n ( q ) with n instructions? ◮ What about SL n ( q )?

Some results Theorem (Cameron, F., Gadouleau, 201?) Unless n = q = 2 , the group Sym(A n ) can be generated by n instructions.

Some results Theorem (Cameron, F., Gadouleau, 201?) Unless n = q = 2 , the group Sym(A n ) can be generated by n instructions. Theorem (Cameron, F., Gadouleau, 201?) Unless q = 2 and n = 2 or 3 , the group Alt(A n ) can be generated by n instructions.

Some results Theorem (Cameron, F., Gadouleau, 201?) Unless n = q = 2 , the group Sym(A n ) can be generated by n instructions. Theorem (Cameron, F., Gadouleau, 201?) Unless q = 2 and n = 2 or 3 , the group Alt(A n ) can be generated by n instructions. Theorem (Cameron, F., Gadouleau, 201?) dito GL n ( q ) and SL n ( q ) .

Sketch proof for Sym( A n ) Theorem (Cameron, F., Gadouleau, 201?) Unless n = q = 2 , the group Sym(A n ) can be generated by n instructions. “Proof”

Sketch proof for Sym( A n ) Theorem (Cameron, F., Gadouleau, 201?) Unless n = q = 2 , the group Sym(A n ) can be generated by n instructions. “Proof” It’s convenient to consider different cases separately:

Sketch proof for Sym( A n ) Theorem (Cameron, F., Gadouleau, 201?) Unless n = q = 2 , the group Sym(A n ) can be generated by n instructions. “Proof” It’s convenient to consider different cases separately: ◮ | A | = 2 ;

Sketch proof for Sym( A n ) Theorem (Cameron, F., Gadouleau, 201?) Unless n = q = 2 , the group Sym(A n ) can be generated by n instructions. “Proof” It’s convenient to consider different cases separately: ◮ | A | = 2 ; ◮ | A | > 2 and n is odd;

Sketch proof for Sym( A n ) Theorem (Cameron, F., Gadouleau, 201?) Unless n = q = 2 , the group Sym(A n ) can be generated by n instructions. “Proof” It’s convenient to consider different cases separately: ◮ | A | = 2 ; ◮ | A | > 2 and n is odd; ◮ | A | > 2 and n is even;