Hasenjaeger’s electromechanical small universal Turing machine is time efficient
Rainer Glaschick1, Turlough Neary2, Damien Woods3, Niall Murphy4
1Paderborn, Germany
2 Institute for Neuroinformatics, University of Z¨urichand ETH Z¨urich,Switzerland
3 California Institute of Technology, USA
c a m p u s
4 MONCLOA Universidad Polit´ecnicade Madrid, CEI Campus Moncloa, UCM-UPM, Madrid, Spain
Turing in Context II, October 11, 2012
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Summary
Hasenjaeger, a (near) contemporary of Turing’s Built electromechanical universal Turing machine that was remarkably small Simulated Wang’s B-machine (non-erasing machine) We prove that Wang B-machines are a time efficient model of computation (an exponential improvement) As an immediate corollary we find that Hasenjaeger’s small machine is efficiently universal
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Some context
1936 “On Computable Numbers, with an Application to the Entscheidungs problem” Scholz in M¨unsterand Braithwaite in Cambridge requested preprints Scholz was founding a mathematical logic group in M¨unster Undergraduate Hasenjaeger
Copyright MFO/CC BY-SA 2.0
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Gisbert Hasenjaeger
Scholz got him a place in the High Command cryptography group 1942 Hasenjaeger looked for weaknesses in the Enigma machine After the war Hasenjaeger became Scholz’s PhD student Became professor of Mathematical logic in M¨unster Didn’t learn of Turing’s involvement with Enigma until the Copyright: Reinhold Remmert, Oberwolfach 70’s Photo Collection
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient 1951, Moore gave the first small universal Turing machine 2 regular infinite tapes, 1 circular tape simulates standard TM rules, (qi , 1, 0, R, qk ) simple encoding system 1957, Wang publishes his B-machine non-erasing machine simpler instruction set
R M J(3) L M R L M complicated encoding (powers of primes) 1963, Hasenjaeger built a universal Turing machine using old telephone relays used some ideas from Wang and Moore 3-tapes, 4 states, 2 symbols
Hasenjaeger and the machine
Aspired to build a cheap electromechanical universal Turing machine
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient 1957, Wang publishes his B-machine non-erasing machine simpler instruction set
R M J(3) L M R L M complicated encoding (powers of primes) 1963, Hasenjaeger built a universal Turing machine using old telephone relays used some ideas from Wang and Moore 3-tapes, 4 states, 2 symbols
Hasenjaeger and the machine
Aspired to build a cheap electromechanical universal Turing machine 1951, Moore gave the first small universal Turing machine 2 regular infinite tapes, 1 circular tape simulates standard TM rules, (qi , 1, 0, R, qk ) simple encoding system
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient 1963, Hasenjaeger built a universal Turing machine using old telephone relays used some ideas from Wang and Moore 3-tapes, 4 states, 2 symbols
Hasenjaeger and the machine
Aspired to build a cheap electromechanical universal Turing machine 1951, Moore gave the first small universal Turing machine 2 regular infinite tapes, 1 circular tape simulates standard TM rules, (qi , 1, 0, R, qk ) simple encoding system 1957, Wang publishes his B-machine non-erasing machine simpler instruction set
R M J(3) L M R L M complicated encoding (powers of primes)
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Hasenjaeger and the machine
Aspired to build a cheap electromechanical universal Turing machine 1951, Moore gave the first small universal Turing machine 2 regular infinite tapes, 1 circular tape simulates standard TM rules, (qi , 1, 0, R, qk ) simple encoding system 1957, Wang publishes his B-machine non-erasing machine simpler instruction set
R M J(3) L M R L M complicated encoding (powers of primes) 1963, Hasenjaeger built a universal Turing machine using old telephone relays used some ideas from Wang and Moore 3-tapes, 4 states, 2 symbols
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Anatomy of Hasenjaeger’s machine
Four state processor Unary counter tape
1 1 1 1 Universal Turing 0 0 1 1 1 1 1 0 1 1 machine program 0 1 1 C 1 1 P 0 1 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0
... 1 1 0 1 1 1 0 Read... only tape Non-erasable tape Non-erasing read/write headNon-erasable work tape
Donated by Hasenjaeger’s family to the Heinz Nixdorf MuseumsForum, Paderborn
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Anatomy of Hasenjaeger’s machine
Four state processor Unary counter tape
1 1 1 1 Universal Turing 0 0 1 1 1 1 1 0 1 1 machine program 0 1 1 C 1 1 P 0 1 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0
... 1 1 0 1 1 1 0 Read... only tape Non-erasable tape Non-erasing read/write headNon-erasable work tape
Donated by Hasenjaeger’s family to the Heinz Nixdorf MuseumsForum, Paderborn
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Anatomy of Hasenjaeger’s machine
Four state processor Unary counter tape
1 1 1 1 Universal Turing 0 0 1 1 1 1 1 0 1 1 machine program 0 1 1 C 1 1 P 0 1 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0
... 1 1 0 1 1 1 0 Read... only tape Non-erasable tape Non-erasing read/write headNon-erasable work tape
Donated by Hasenjaeger’s family to the Heinz Nixdorf MuseumsForum, Paderborn
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Rainer Glaschick figured out Hasenjaeger’s Universal machine
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Rainer Glaschick figured out Hasenjaeger’s Universal machine
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Rainer Glaschick figured out Hasenjaeger’s Universal machine
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Hasenjaeger’s Universal machine: Rules
# QPCR PCRQ0 1 q1 1 0 . q1 2 q1 1 1 . q1 3 q1 0 . q2 4 q2 1 0 . . q1 5 q2 0 0 .. q2 6 q2 1 1 .// q1 7 q2 0 1 ./ q3 8 q3 0 0 . q3 9 q3 1 0 . q1 10 q3 0 1 .. q3 11 q3 1 1 .. q4 12 q4 0 1 . q4 13 q4 1 1 ./ q4 14 q4 0 q1
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Circular Wang machines
J(6) M L R R M M R L L L R J(4) R R M R M
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient What do the rules do?
Simulate Wang’s B-machine Using an encoding scheme inspired by Moore
M : 1 : mark: write a 1 to the current tape cell, R : 01 : move head right, L : 001 : move head left, J(x) : 00 0x 1 : if the current cell is 1 (marked) then skip x instructions, otherwise move to the next instruction.
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Hasenjaeger built a small efficient Wang B-machine simulator
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Proving Hasenjaeger’s machine is efficiently universal
TM → Non-Erasing TM → Wang B-machine → Hasenjaeger UTM t O(t3) O(2t ) O(2t )
Efficient!
A → B, indicates A is simulated by B
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Proving Hasenjaeger’s machine is efficiently universal
TM → Non-Erasing TM → Wang B-machine → Hasenjaeger UTM t O(t3) O(2t ) O(2t )
Efficient!
A → B, indicates A is simulated by B
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Turing machine to Non-erasing Turing machine O(t3)
1:
2: write 3: write 4: ... erase
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Turing machine to Non-erasing Turing machine O(t3)
1:
2: write 3: write 4: ... erase
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Turing machine to Non-erasing Turing machine O(t3)
1:
2: write 3: write 4: ... erase
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Turing machine to Non-erasing Turing machine O(t3)
1:
2: write 3: write 4: ... erase
To erase a bit, make a new copy of the work tape.
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Turing machine to Non-erasing Turing machine O(t3)
1:
2: write 3: write 4: ... erase
To erase a bit, make a new copy of the work tape.
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Proving Hasenjaeger’s machine is efficiently universal
TM → Non-Erasing TM → Wang B-machine → Hasenjaeger UTM t O(t3) O(2t ) O(2t )
Efficient!
A → B, indicates A is simulated by B
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Proving Hasenjaeger’s machine is efficiently universal
TM → Non-Erasing TM → Wang B-machine → Hasenjaeger UTM t O(t3) O(2t ) O(2t )
Efficient!
A → B, indicates A is simulated by B
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Transforming a non-erasing Turing machine to a Wang machine
q0 (q0, 0 : 0, R, q1) (q0, 1 : 1, R, q2) (q1, 0 : 1, R, q3) (q1, 1 : 1, R, q1) Circular q q (q1, 0 : 0, R, q2) 3 Wang 1 (q2, 0 : 0, R, q2) Program (q2, 1 : 1, R, q3) (q3, 1 : 1, R, q3) (q3, 0 : 1, L, q0) q2
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient qi , 0 : 0, R , qj qi , 1 : 1, R , qk qi , 0 : 0, R , qj qi , 1 : 1, L , qk qi , 0 : 0, L , qj 0 : 0, R qi , 1 : 1, R , qk 0 : 0, L q , 0 : 0, L , q 0 : 1, R i j q , 1 : 1, L , q 0 : 1, L qj i k qi , 0 : 1 , R , qj qi qi , 1 : 1, R , qk
1 : 1, R qk qi , 0 : 1 , R , qj 1 : 1, L qi , 1 : 1, L , qk qi , 0 : 1 , L , qj qi , 1 : 1, R , qk qi , 0 : 1 , L , qj qi , 1 : 1, L , qk
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Idea: The encoding on the input tape
Wang machines only jump on 1’s h1i = 11 h0i = 01
1 1 0 1 0 1 1 1 0 1
h1i h0i h0i h1i h0i
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Jumping on reading a 0 or 1
Circular Wang program sections
NE Turing Machine rules ...... 0 : 0, R qj
qi qi 1 : 1, R qk qk
qj
Work Tape . ... 0 1 1 1 0 1 0 1 0 1 ... . . h0i h1i h0i h0i h0i . . .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Jumping on reading a 0 or 1
Circular Wang program sections
NE Turing Machine rules ...... 0 : 0, R qj R qi qi J(xk ) R 1 : 1, R qk R R qk J(.)
J(xj ) qj J(.) Work Tape . ... 0 1 1 1 0 1 0 1 0 1 ... . . h0i h1i h0i h0i h0i . . .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Jumping on reading a 0 or 1
Circular Wang program sections
NE Turing Machine rules ...... 0 : 0, R qj R qi qi J(xk ) R 1 : 1, R qk R R qk J(.)
J(xj ) qj J(.) Work Tape . ... 0 1 1 1 0 1 0 1 0 1 ... . . h0i h1i h0i h0i h0i . . .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Jumping on reading a 0 or 1
Circular Wang program sections
NE Turing Machine rules ...... 0 : 0, R qj R qi qi J(xk ) R 1 : 1, R qk R R qk J(.)
J(xj ) qj J(.) Work Tape . ... 0 1 1 1 0 1 0 1 0 1 ... . . h0i h1i h0i h0i h0i . . .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Jumping on reading a 0 or 1
Circular Wang program sections
NE Turing Machine rules ...... 0 : 0, R qj R qi qi J(xk ) R 1 : 1, R qk R R qk J(.)
J(xj ) qj J(.) Work Tape . ... 0 1 1 1 0 1 0 1 0 1 ... . . h0i h1i h0i h0i h0i . . .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Jumping on reading a 0 or 1
Circular Wang program sections
NE Turing Machine rules ...... 0 : 0, R qj R qi qi J(xk ) R 1 : 1, R qk R R qk J(.)
J(xj ) qj J(.) Work Tape . ... 0 1 1 1 0 1 0 1 0 1 ... . . h0i h1i h0i h0i h0i . . .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Jumping on reading a 0 or 1
Circular Wang program sections
NE Turing Machine rules ...... 0 : 0, R qj R qi qi J(xk ) R 1 : 1, R qk R R qk J(.)
J(xj ) qj J(.) Work Tape . ... 0 1 1 1 0 1 0 1 0 1 ... . . h0i h1i h0i h0i h0i . . .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Jumping on reading a 0 or 1
Circular Wang program sections
NE Turing Machine rules ...... 0 : 0, R qj R qi qi J(xk ) R 1 : 1, R qk R R qk J(.)
J(xj ) qj J(.) Work Tape . ... 0 1 1 1 0 1 0 1 0 1 ... . . h0i h1i h0i h0i h0i . . .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Jumping on reading a 0 or 1
Circular Wang program sections
NE Turing Machine rules ...... 0 : 0, R qj R qi qi J(xk ) R 1 : 1, R qk R R qk J(.)
J(xj ) qj J(.) Work Tape . ... 0 1 1 1 0 1 0 1 0 1 ... . . h0i h1i h0i h0i h0i . . .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Jumping on reading a 0 or 1
Circular Wang program sections
NE Turing Machine rules ...... 0 : 0, R qj R qi qi J(xk ) R 1 : 1, R qk R R qk J(.)
J(xj ) qj J(.) Work Tape . ... 0 1 1 1 0 1 0 1 0 1 ... . . h0i h1i h0i h0i h0i . . .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Read a 0/write a 1
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Read a 0/write a 1
Circular Wang program sections
. . . NE Turing Machine rules . . .
0 :1 , R qj R qi qi J(xk ) R
qk J(.) 1 : 1, R qk M R R qj J(.) M . Work Tape . ... 0 1 1 1 0 1 0 0 ... . J(xj ) . h0i h1i h0i h0i
. .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Read a 0/write a 1
Circular Wang program sections
. . . NE Turing Machine rules . . .
0 :1 , R qj R qi qi J(xk ) R
qk J(.) 1 : 1, R qk M R R qj J(.) M . Work Tape . ... 0 1 1 1 1 1 0 0 ... . J(xj ) . h0i h1i h1i h0i
. .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Read a 0/write a 1
Circular Wang program sections
. . . NE Turing Machine rules . . .
0 :1 , R qj R qi qi J(xk ) R
qk J(.) 1 : 1, R qk M R R qj J(.) M . Work Tape . ... 0 1 1 1 1 1 0 0 ... . J(xj ) . h0i h1i h1i h0i
. .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Read a 0/write a 1
Circular Wang program sections
. . . NE Turing Machine rules . . .
0 :1 , R qj R qi qi J(xk ) R
qk J(.) 1 : 1, R qk M R R qj J(.) M . Work Tape . ... 0 1 1 1 1 1 0 0 ... . J(xj ) . h0i h1i h1i h0i
. .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Read a 0/write a 1
Circular Wang program sections
. . . NE Turing Machine rules . . .
0 :1 , R qj R qi qi J(xk ) R
qk J(.) 1 : 1, R qk M R R qj J(.) M . Work Tape . ... 0 1 1 1 1 1 0 0 ... . J(xj ) . h0i h1i h1i h0i
. .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Read a 0/write a 1
Circular Wang program sections
. . . NE Turing Machine rules . . .
0 :1 , R qj R qi qi J(xk ) R
qk J(.) 1 : 1, R qk M R R qj J(.) M . Work Tape . ... 0 1 1 1 1 1 0 0 ... . J(xj ) . h0i h1i h1i h0i
. .
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Moving left
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Moving Left
Circular Wang program sections ......
NE Turing Machine rules qi J(xk ) L L 0 : 0, L qj R L L
qi M L L 1 : 1,L qk J(xj ) L L . . L R Work Tape R R ... 0 1 1 1 0 1 0 1 ... R qk J(.) h0i h1i h0i h0i q . j J(.) . . . Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Moving Left
Circular Wang program sections ......
NE Turing Machine rules qi J(xk ) L L 0 : 0, L qj R L L
qi M L L 1 : 1,L qk J(xj ) L L . . L R Work Tape R R ... 0 1 1 1 0 1 0 1 ... R qk J(.) h0i h1i h0i h0i q . j J(.) . . . Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Moving Left
Circular Wang program sections ......
NE Turing Machine rules qi J(xk ) L L 0 : 0, L qj R L L
qi M L L 1 : 1,L qk J(xj ) L L . . L R Work Tape R R ... 0 1 1 1 0 1 0 1 ... R qk J(.) h0i h1i h0i h0i q . j J(.) . . . Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Moving Left
Circular Wang program sections ......
NE Turing Machine rules qi J(xk ) L L 0 : 0, L qj R L L
qi M L L 1 : 1,L qk J(xj ) L L . . L R Work Tape R R ... 0 1 1 1 0 1 0 1 ... R qk J(.) h0i h1i h0i h0i q . j J(.) . . . Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Moving Left
Circular Wang program sections ......
NE Turing Machine rules qi J(xk ) L L 0 : 0, L qj R L L
qi M L L 1 : 1,L qk J(xj ) L L . . L R Work Tape R R ... 0 1 1 1 0 1 0 1 ... R qk J(.) h0i h1i h0i h0i q . j J(.) . . . Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Moving Left
Circular Wang program sections ......
NE Turing Machine rules qi J(xk ) L L 0 : 0, L qj R L L
qi M L L 1 : 1,L qk J(xj ) L L . . L R Work Tape R R ... 0 1 1 1 0 1 0 1 ... R qk J(.) h0i h1i h0i h0i q . j J(.) . . . Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Moving Left
Circular Wang program sections ......
NE Turing Machine rules qi J(xk ) L L 0 : 0, L qj R L L
qi M L L 1 : 1,L qk J(xj ) L L . . L R Work Tape R R ... 0 1 1 1 0 1 0 1 ... R qk J(.) h0i h1i h0i h0i q . j J(.) . . . Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Moving Left
Circular Wang program sections ......
NE Turing Machine rules qi J(xk ) L L 0 : 0, L qj R L L
qi M L L 1 : 1,L qk J(xj ) L L . . L R Work Tape R R ... 0 1 1 1 0 1 0 1 ... R qk J(.) h0i h1i h0i h0i q . j J(.) . . . Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Moving Left
Circular Wang program sections ......
NE Turing Machine rules qi J(xk ) L L 0 : 0, L qj R L L
qi M L L 1 : 1,L qk J(xj ) L L . . L R Work Tape R R ... 0 1 1 1 0 1 0 1 ... R qk J(.) h0i h1i h0i h0i q . j J(.) . . . Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Moving Left
Circular Wang program sections ......
NE Turing Machine rules qi J(xk ) L L 0 : 0, L qj R L L
qi M L L 1 : 1,L qk J(xj ) L L . . L R Work Tape R R ... 0 1 1 1 0 1 0 1 ... R qk J(.) h0i h1i h0i h0i q . j J(.) . . . Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Moving Left
Circular Wang program sections ......
NE Turing Machine rules qi J(xk ) L L 0 : 0, L qj R L L
qi M L L 1 : 1,L qk J(xj ) L L . . L R Work Tape R R ... 0 1 1 1 0 1 0 1 ... R qk J(.) h0i h1i h0i h0i q . j J(.) . . . Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Key Ideas: Moving Left
Circular Wang program sections ......
NE Turing Machine rules qi J(xk ) L L 0 : 0, L qj R L L
qi M L L 1 : 1,L qk J(xj ) L L . . L R Work Tape R R ... 0 1 1 1 0 1 0 1 ... R qk J(.) h0i h1i h0i h0i q . j J(.) . . . Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Mapping from the table to the rules
qi , 0, 0, R , qj L, L, L, L, L, R, R, J(qk -2) , R, M, J(qj -1) qi , 1, 1, R , qk qi , 0, 0, R , qj L, L, L, L, L, R, R, J(qk -6) , R, M, J(qj -1) qi , 1, 1, L , qk qi , 0, 0, L , qj L, L, L, L, L, R, R, J(qk -2) , R, M, J(qj -7) qi , 1, 1, R , qk qi , 0, 0, L , qj L, L, L, L, L, R, R, J(qk -6) , R, M, J(qj -7) qi , 1, 1, L , qk qi , 0, 1 , R , qj L, L, L, L, L, R, R, J(qk -2) , M , R, M, J(qj -1) qi , 1, 1, R , qk qi , 0, 1 , R , qj L, L, L, L, L, R, R, J(qk -6) , M , R, M, J(qj -1) qi , 1, 1, L , qk qi , 0, 1 , L , qj L, L, L, L, L, R, R, J(qk -2) , M , R, M, J(qj -7) qi , 1, 1, R , qk qi , 0, 1 , L , qj L, L, L, L, L, R, R, J(qk -6) , M , R, M, J(qj -7) qi , 1, 1, L , qk Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Final circular Wang program
J(0) R R L L L L L L q0 L L L L R Circular R J(3) q3 Wang q1 J(1) R Program R L L L L q2 L L L L L L J(2) R R
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient 4 States, 2 symbols, 3 tapes is a remarkably small machine
Conclusion
TM → Non-Erasing TM → Wang B-machine → Hasenjaeger UTM t O(t3) O(2t ) O(2t ) O(t3) O(t3)
A → B, indicates A is simulated by B
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Conclusion
TM → Non-Erasing TM → Wang B-machine → Hasenjaeger UTM t O(t3) O(2t ) O(2t ) O(t3) O(t3)
4 States, 2 symbols, 3 tapes is a remarkably small machine
A → B, indicates A is simulated by B
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient Thanks to . . .
Benedikt L¨owe and the Isaac Newton Institute, Cambridge, UK, “Semantics and Syntax: A Legacy of Alan Turing” 2012 R. Glaschick cooperates with the Heinz Nixdorf MuseumsForum, Germany T. Neary is supported by Swiss National Science Foundation Grant 200021-141029 D. Woods is supported by National Science Foundation Grant 0832824, the Molecular Programming Project, and NSF grants CCF-1219274 and CCF-1162589 N. Murphy is supported by The Campus de Excelencia Internacional Moncloa, PICATA Program and Spanish National project C09101514.
Glaschick, Neary, Woods, Murphy Hasenjaeger’s universal Turing machine is time efficient