Variations on Turing Machines

1. CS 3240 – Chapters Variations on Turing Machines

2. Mechanisms equivalent to Standard TMs • PDAs with 2 stacks or 1 queue • TMs with 1-way infinite tape • TMs with n tapes or n heads or n-dim tapes • TMs with various “move” options • Move by a number of cells (including 0), random cell access by position • Adding non-determinism to a Standard TM • Church's Lambda Calculus (LISP, Haskell) • Unrestricted Grammars • Matrix Grammars • Post Systems • Markov Algorithms • Structured Programming with unlimited memory • Any abstract machine yet imagined by man!

3. Queue Machines • Queues pop at front, push at back • 2 I/O points gives more flexibility • Equivalent to a TM! • Begin by pushing all initial data • Can simulate “moving around” by cycling through the data • circular shift via pop and push of same character • use a delimiter character ($) to keep track of the start of data

4. Queue Machine for anbncn

5. Tracing the Queue Machine $aabbcc aabbcc$ abbcc$X bbcc$Xa bcc$XaY cc$XaYb c$XaYbZ $XaBbZc XaYbZc$ aYbZc$X YbZc$XX bZc$XXY Zc$XXYY c$XXYYZ $XXYYZZ XXYYZZ$ YYZZ$XX ZZ$XXYY $XXYYZZ XXYYZZ$ <accept>

6. Multi-track TM’sSection 10.2 Just store the tuple (a,b,c) as the “symbol” in a single cell. Change part or all of the data as needed.

7. Using a 1-way Infinite Tape Just “wrap around the fold” to simulate a 2-way tape

8. Figure 10.04:

9. Figure 10.05:

10. Using Multiple Tapes Just place in different sections of a single tape

11. Figure 10.11:

12. Non-determinism = DeterminismSection 10.3 Code the machine so it replicates and calls itself for each choice. (A form of backtracking). Do a breadth-first search for a halting configuration.

13. Special-Purpose Computers • The TM we’ve seen so far have been “special-purpose computers” • they implement only one algorithm, or • they accept only one language • But TMs can take another TM as input, and simulate (run) it • “stored program computer” • general purpose computer as we know it

14. Universal Turing MachinesSection 10.4 A TM that simulates other TMs. Mu takes M as input, along with data input. See page 267 for a sample encoding for TMs.

15. How can one TM be input for another TM? • The input TM must be encoded as a string • States, transitions, etc. • One way: • Order the tape alphabet, Γ = {a1, a2, a3…} • a1 = ☐, a2 = L, a3 = R, a4 = 0, a5 = 1, … • Encode each state by its index+1 • q0= 1, q1 = 11 … • Encode each transition in δ • Use 0 as the separator everywhere

16. Encoding a Transition • δ(q0,1) = (q0,1,R) • Consider it as the quintuple (q0,1,q0,1,R) • Which encodes as: • (1,5,1,5,3)= 1 11111 1 11111 111= 1011111010111110111

17. UTM Configuration • Notation: TU = the UTM; TM = the input machine • TU has a memory location for the current state of TM • TU reserves a portion of the tape for the encoding of the TM • TU reserves an infinitely large portion of the tape for the working memory of the TM (in one direction), initialized with the input to TM • And keeps track of the read/write position • TU has its own infinitely large memory area (in the other direction)

18. Just “Trust Us” • We won’t investigate the inner workings of UTMs further • We’ll just assume they exist • (If they didn’t, neither would computers :-) • What we will do is examine the consequences of the existence of UTMs • especially the consequences of encoding TMs as strings

19. An Online UTM • http://ironphoenix.org/tril/tm/ • Here are the instructions for the swap machine: • 1,a,1,b,> • 1,b,1,a,> • 1,_,2,_,< • 2,a,2,a,< • 2,b,2,b,< • 2,_,H,_,>

20. TMs and Strings • Every TM can be encoded as a bitstring • Not all strings represent TMs. TM form: • (11*0)50((11*0)50)* • Some TMs can have more than one string representation • e.g., the name(number) of the state is immaterial • How many strings are there? • How many TMs are there?

21. Countability • Some infinite sets are countable • like the set of positive even numbers • {0, 2, 4, …} = {2n | n ∈ N}; function: p(i) = 2i • There is a one-to-one mapping between the set’s elements and the Natural numbers • Others are not countable: • e.g., the set of real numbers • Consequence: |(0,1)| = |(-∞,∞)| !!!

22. Negative Evens • -2, -4, -6, … • Maps to 1, 2, 3, … • By the function: n(i) = -2i

23. All Evens • e(0) = 0 • e(1) = 2 • e(2) = -2 • e(3) = 4 • e(4) = -4 • e(5) = 6 • e(6) = -6 • The “formula”:If i is evene(i) = -ielsee(i) = i + 1

24. Positive Rationals are Countable

25. Enumerating Σ* • Take a cue from number systems’positional notation • 123 = 1*102 + 2*10 + 3 • aba = v(a)*n2 + v(b)*n + v(a), wheren = |Σ|, v(a) = 1, v(b) = 2, f(λ) = 0 • So f(aba) = 1*22 + 2*2 + 1 = 9 • aba is the 9th string (0-based) in Σ*:{λ,a,b,aa,ab,ba,bb,aaa,aab,aba,…}

26. How “Big” is a Language?Countable or Uncountable? • A Language is a subset of Σ* • We can enumerate the elements of Σ* (or any subset thereof) in proper order • lexicographically in groups by increasing length • Therefore, there is a first one, a second one, etc. • So, the strings of a language are enumerable • ⇒ Every language is a countable set of strings

27. The Totality of TMs isCountable • aka enumerable • They can be arranged in a sequence • TM0, TM1, TM2, etc. • Why? • Because they can be encoded as strings • Strings over any alphabet can be enumerated in proper order • by length groups, lexicographically • So there you have it!

28. Linear Bounded AutomataSection 10.5 • Some TMs only need a fixed size of working storage • never grows beyond (a factor of the) input size • Example: anbncn • Doesn’t expand working storage (remember?) • These are called Linear Bounded Automata • LBA; they use end-markers: [<data goes here>] • Why do we care? • We don’t, much; will use briefly in Chapter 11