Summary of Previous Class

1
Summary of Previous Class

• Computational problems. Types of problems.
  Encoding of instances.
• Languages. Equivalence between decision problems
  and languages.
• Algorithms. Church-Turing Thesis. Turing
  machines.

2
Simple examples

• ?0, L000
• ?0,1, L00,11
• ?0,1, L(01)n n?N
• ?a,b,c, Lanbncn n?N

3
Machine that decides Lanbncn n?N in the
alphabet ?a,b,c,

4
Machine that decides L0t?n such that t2n in
?0
5
Machine that decides Lsss?? where ?0,1
6
Recognizable Languages and Decidable Languages

• The set of strings accepted by a Turing machine M
  is denoted by L(M).
• A language L is called recognizable (or
  enumerable) if there exists a Turing machine M
  such that LL(M).
• A language L is called decidable if there exists
  a Turing machine M such that
• M halts on every input (with accept or
  reject)
• LL(M).

7
Composition of Turing Machines

• Union. For any two Turing machines M1 and M2,
  there exists a Turing machine M such that
  L(M)L(M1) ? L(M2)
• Intersection. For any two Turing machines M1 and
  M2, there exists a Turing machine M such that
  L(M)L(M1) ? L(M2)

8
Variants of Turing Machines Robustness

• Left, Right, Stay
• One tape, multiple heads
• Multiple heads, multiple tapes
• Two-dimensional tape
• Non-deterministic
• Enumerator
• Computation of functions

9
Multiple heads and tapes

10
Enumerator

• Two tape Turning Machine that enumerates language
  L has working and output tape
• Starts with empty working tape
• At any point working tape contains
  w1w2...wkv where wi?L and v??
• For any w?L it will eventually be written on the
  output tape preceded and followed by
• For any alphabet ? we can construct enumerator of
  ?.

11
Nondeterministic Turing Machine simulated by 3
tape deterministic

12
Random Access Machines

• RAMs input array T1, T2, T3,
• RAMs registers A finite number of registers
  R0, R1, , Rs, each capable of containing
  an integer.
• RAMs program A finite sequence of instructions.

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Instructions

• READ j R0 ? TRj
• WRITE j TRj ? R0
• STORE j Rj ? R0
• LOAD j R0 ? Rj
• LOAD c R0 ? c
• ADD j R0 ? R0 Rj
• ADD c R0 ? R0 c
• SUB j R0 ? maxR0-Rj, 0
• SUB c R0 ? maxR0-c, 0
• HALF R0 ? floorR0/2
• JUMP t K ? t
• JPOS t if R0gt0 then K ? t
• JZERO t if R00 then K ? t
• HALT

14
Equivalence Between Models

• Informally
• Any model of computation can be simulated by any
  other model with at most a polynomial increase of
  time and space.