## Wednesday, May 11, 2016

### A Turing Machine For Calculating The Fibonacci Sequence

 Input/Output Tape Terms in Series 0b1;1; 1, 1 0b1;1;11; 1, 1, 2 0b1;1;11;111 1, 1, 2, 3 0b1;1;11;111;11111; 1, 1, 2, 3, 5 0b1;1;11;111;11111;11111111; 1, 1, 2, 3, 5, 8

1.0 Introduction

I thought I would describe the program for a specific Turing machine. This Turing machine computes the Fibonacci sequence in tally arithmetic, as illustrated in Table 1 above. The left-hand column shows the tape for the Turing machine for successive transitions into the Start state. (The location of the head is indicated by the bolded character.) The right-hand column shows a more familiar representation of a Fibonacci sequence. This Turing machine never halts for valid inputs. It can calculate other infinite sequences, such as specific Lucas sequences, for other valid inputs.

A Turing machine is specified by the alphabet of characters that can appear on the tape, possible valid sequences of characters for the start of the tape, the location of the head at the beginning of a computation, the states and the state transition rules, and the location of the state pointer at beginning of a computation.

2.0 Alphabet

 Symbol Number OfOccurrences Comments 0 1 Start of tape marker b Potentially Infinite Blank ; Potentially Infinite Symbol for number termination 1 Potentially Infinite A tally x 1 For internal use y 1 For internal use z 1 For internal use

3.0 Specification of Valid Input Tapes

At start, the (input) tape should contain, in this order:

• 0, the start of tape marker.
• b, a blank.
• Zero or more 1s.
• ;, a semicolon.
• One or more of the following:
• Zero or more 1s.
• ;, a semicolon.

The head shall be at a blank or semicolon such that exactly two semicolons exist in the tape to the right of the head. Table 3 provides examples (with the head being at the bolded character).

 0b;; 0b1;; 0b1;1; 0b11;1; 0b1;1;11;111;11111;11111111;

4.0 Definition of State

The states are grouped into two subroutines, CopyPair and Add. Error is the only halting state, to be entered when an invalid input tape is detected. The Turing machine begins the computation with the state pointer pointing to the Start state, in the CopyPair subroutine. Eventually, the Turing machine enters the PauseCopy state. The machine then transitions to the StartAdd state, in the Add subroutine. Another number in the sequence has been successfully appended to the tape when the Turing machine enters the PauseAdd state.

The Turing machine then transitions into the Start state. The CopyPair and Add subroutines are repeated in pairs forever.

4.1 CopyPair

The input tape for the CopyPair subroutine is any valid input tape, as described above. The state pointer starts in the Start tape. Error is the only halting state. The subroutine exits with a transition from the PauseCopy state to the StartAdd state. When the PauseCopy state is entered, the tape shall be in the following configuration:

• The terminal semicolon in the tape, when the Start state was entered, shall be replaced with a z.
• The head shall be at that z.
• The tape to the right of the z shall contain a copy of the character string to the right of the head when the Start state was entered.

This subroutine can be implemented by the states described in Table 4. The detailed implementation of each state is provided in the appendix. Throughout these states, there are transitions to the Error state triggered by encountering on the tape a character that cannot be there in a valid computation.

 State Description Start Moves the head forward one character. ReadFirstChar Replaces first ; or 1 (after position of head when the subroutine was called) with x or y, respectively. WriteFirstSemi Writes a ; at the end of the tape. Transitions to GoToTapeEnd. WriteFirstOne Writes a 1 at the end of the tape. Transitions to GoToTapeEnd. GoToTapeEnd Moves the head backward one character to locate the head at the character that was at the end of the tape when the subroutine was called. MarkTapeEnd Replaces original terminating ; with z. NexChar Replaces the x or y on the tape with ; or 1, respectively. StepForward Moves the head forward one character. ReadChar Replaces the next ; or 1 with x or y, respectively. WriteSemi Writes a ; at the end of the tape. Transitions to NextChar. WriteOne Writes a 1 at the end of the tape. Transitions to NextChar. WriteLastSemi Writes a ; at the end of the tape. Transitions to SetHead. SetHead Moves head to the z on the tape. PauseCopy For noting that last two numbers on the tape, when the subroutine was called, have been copied to the end of the tape.

When the PauseAdd state is entered, the tape shall be in the following configuration:

• The semicolon between the z and the last semicolon, when the StartAdd state is entered, shall be replaced by a 1, if there is at least one 1 between this character and the terminating semicolon.
• The semicolon at the end of the tape, when the StartAdd state is entered, shall be erased (replaced by a blank).
• The character before the erased semicolon shall be replaced by a semicolon.
• The z shall be replaced by a semicolon.
• The head shall be at a semicolon such that two semicolons exist to the right of the head.

 State Description StartAdd Moves the head forward one character. FindSemiForDele Replaces the ; mid-number with 1. FindSumEnd Erases terminating ;. EndSum Writes terminating ; at the tape position one character backwards. FindSumStart Replaces z with ;. StepBackward Moves the head backwards one character. ResetHead Set head to previous ;, before the ; just written. PauseAdd For noting next number in Fibonacci series.

5.0 Length of Tape and the Number of States

After three run-throughs of this Turing machine, five numbers in the Fibonacci sequence will be calculated. And the tape will contain 19 characters. As shown in Table 6, the number of states is 22. For the group activity I have defined for simulating a Turing machine, 42 people are needed. (One more person is needed, in computing the next number in the sequence, to be erased from the tape than ends up as characters on the tape.) I suppose one could get by with 36 people, if one is willing to some represent two states, one in each subroutine.

Appendix A: State Transition Tables

A.1: The CopyPair Subroutine
 Start ReadFirstChar 0 0 Error 0 0 Error b Forwards ReadFirstChar b b Error ; Forwards ReadFirstChar ; x WriteFirstSemi 1 1 Error 1 y WriteFirstOne x x Error x x Error y y Error y y Error z z Error z z Error
 WriteFirstSemi WriteFirstOne 0 0 Error 0 0 Error b ; GoToTapeEnd b 1 GoToTapeEnd ; Forwards WriteFirstSemi ; Forwards WriteFirstOne 1 Forwards WriteFirstSemi 1 Forwards WriteFirstOne x Forwards WriteFirstSemi x Forwards WriteFirstOne y Forwards WriteFirstSemi y Forwards WriteFirstOne z z Error z z Error
 GoToTapeEnd MarkTapeEnd 0 0 0 Error b b b Error ; Backwards MarkTapeEnd ; z NextChar 1 Backwards MarkTapeEnd 1 1 Error x x x Error y y y Error z z z Error
 NextChar StepForward 0 0 Error 0 b b Error b ; Backwards NextChar ; Forwards ReadChar 1 Backwards NextChar 1 Forwards ReadChar x ; StepForward x y 1 StepForward y z Backwards NextChar z
 ReadChar WriteSemi 0 0 Error 0 0 Error b 1 Error b ; NextChar ; x WriteSemi ; Fowards WriteSemi 1 y WriteOne 1 Forwards WriteSemi x x Error x Forwards WriteSemi y y Error y Forwards WriteSemi z z WriteLastSemi z Forwards WriteSemi
 WriteLastSemi SetHead 0 0 Error 0 0 Error b ; SetHead b b Error ; Forwards WriteLastSemi ; Backwards SetHead 1 Forwards WriteLastSemi 1 Backwards SetHead x Forwards WriteLastSemi x x Error y Forwards WriteLastSemi y y Error z Forwards WriteLastSemi z z PauseCopy
 WriteOne PauseCopy 0 0 Error 0 b 1 NextChar b ; Forwards WriteOne ; 1 Forwards WriteOne 1 x Forwards WriteOne x y Forwards WriteOne y z Forwards WriteOne z z StartAdd