Courtesy Costas Busch - RPI1 Turing’s Thesis

Courtesy Costas Busch - RPI2 Turing’s thesis: Any computation carried out by mechanical means can be performed by a Turing Machine (1930)

Courtesy Costas Busch - RPI3 Computer Science Law: A computation is mechanical if and only if it can be performed by a Turing Machine There is no known model of computation more powerful than Turing Machines

Courtesy Costas Busch - RPI4 Definition of Algorithm: An algorithm for function is a Turing Machine which computes

Courtesy Costas Busch - RPI5 When we say: There exists an algorithm Algorithms are Turing Machines We mean: There exists a Turing Machine that executes the algorithm

Courtesy Costas Busch - RPI6 Variations of the Turing Machine

Courtesy Costas Busch - RPI7 Read-Write Head Control Unit Deterministic The Standard Model Infinite Tape (Left or Right)

Courtesy Costas Busch - RPI8 Variations of the Standard Model Stay-Option Semi-Infinite Tape Off-Line Multitape Multidimensional Nondeterministic Turing machines with:

Courtesy Costas Busch - RPI9 We want to prove: Each Class has the same power with the Standard Model The variations form different Turing Machine Classes

Courtesy Costas Busch - RPI10 Same Power of two classes means: Both classes of Turing machines accept the same languages

Courtesy Costas Busch - RPI11 Same Power of two classes means: For any machine of first class there is a machine of second class such that: And vice-versa

Courtesy Costas Busch - RPI12 a technique to prove same powerSimulation: Simulate the machine of one class with a machine of the other class First Class Original Machine Second Class Simulation Machine

Courtesy Costas Busch - RPI13 Configurations in the Original Machine correspond to configurations in the Simulation Machine Original Machine: Simulation Machine:

Courtesy Costas Busch - RPI14 The Simulation Machine and the Original Machine accept the same language Original Machine: Simulation Machine: Final Configuration

Courtesy Costas Busch - RPI15 Turing Machines with Stay-Option The head can stay in the same position Left, Right, Stay L,R,S: moves

Courtesy Costas Busch - RPI16 Example: Time 1 Time 2

Courtesy Costas Busch - RPI17 Stay-Option Machines have the same power with Standard Turing machines Theorem:

Courtesy Costas Busch - RPI18 Proof: Part 1: Stay-Option Machines are at least as powerful as Standard machines Proof:a Standard machine is also a Stay-Option machine (that never uses the S move)

Courtesy Costas Busch - RPI19 Part 2: Standard Machines are at least as powerful as Stay-Option machines Proof:a standard machine can simulate a Stay-Option machine Proof:

Courtesy Costas Busch - RPI20 Stay-Option Machine Simulation in Standard Machine Similar for Right moves

Courtesy Costas Busch - RPI21 Stay-Option Machine Simulation in Standard Machine For every symbol

Courtesy Costas Busch - RPI22 Example Stay-Option Machine: 12 Simulation in Standard Machine: 123

Courtesy Costas Busch - RPI23 Standard Machine--Multiple Track Tape track 1 track 2 one symbol

Courtesy Costas Busch - RPI24 track 1 track 2 track 1 track 2

Courtesy Costas Busch - RPI25 Semi-Infinite Tape

Courtesy Costas Busch - RPI26 Standard Turing machines simulate Semi-infinite tape machines: Trivial

Courtesy Costas Busch - RPI27 Semi-infinite tape machines simulate Standard Turing machines: Standard machine Semi-infinite tape machine

Courtesy Costas Busch - RPI28 Standard machine Semi-infinite tape machine with two tracks reference point Right part Left part

Courtesy Costas Busch - RPI29 Left partRight part Standard machine Semi-infinite tape machine

Courtesy Costas Busch - RPI30 Standard machine Semi-infinite tape machine Left part Right part For all symbols

Courtesy Costas Busch - RPI31 Standard machine Semi-infinite tape machine Right part Left part Time 1

Courtesy Costas Busch - RPI32 Time 2 Right part Left part Standard machine Semi-infinite tape machine

Courtesy Costas Busch - RPI33 Semi-infinite tape machine Left part At the border: Right part

Courtesy Costas Busch - RPI Semi-infinite tape machine Right part Left part Right part Left part Time 1 Time 2

Courtesy Costas Busch - RPI35 Theorem:Semi-infinite tape machines have the same power with Standard Turing machines

Courtesy Costas Busch - RPI36 The Off-Line Machine Control Unit Input File Tape read-only read-write

Courtesy Costas Busch - RPI37 Off-line machines simulate Standard Turing Machines: Off-line machine: 1. Copy input file to tape 2. Continue computation as in Standard Turing machine

Courtesy Costas Busch - RPI38 1. Copy input file to tape Input File Tape Standard machine Off-line machine

Courtesy Costas Busch - RPI39 2. Do computations as in Turing machine Input File Tape Standard machine Off-line machine

Courtesy Costas Busch - RPI40 Standard Turing machines simulate Off-line machines: Use a Standard machine with four track tape to keep track of the Off-line input file and tape contents

Courtesy Costas Busch - RPI41 Input File Tape Off-line Machine Four track tape -- Standard Machine Input File head position Tape head position

Courtesy Costas Busch - RPI42 Input File head position Tape head position Repeat for each state transition: Return to reference point Find current input file symbol Find current tape symbol Make transition Reference point

Courtesy Costas Busch - RPI43 Off-line machines have the same power with Standard machines Theorem:

Courtesy Costas Busch - RPI44 Multitape Turing Machines Control unit Tape 1Tape 2 Input

Courtesy Costas Busch - RPI45 Time 1 Time 2 Tape 1Tape 2

Courtesy Costas Busch - RPI46 Multitape machines simulate Standard Machines: Use just one tape

Courtesy Costas Busch - RPI47 Standard machines simulate Multitape machines: Use a multi-track tape A tape of the Multiple tape machine corresponds to a pair of tracks Standard machine:

Courtesy Costas Busch - RPI48 Multitape Machine Tape 1Tape 2 Standard machine with four track tape Tape 1 head position Tape 2 head position

Courtesy Costas Busch - RPI49 Repeat for each state transition: Return to reference point Find current symbol in Tape 1 Find current symbol in Tape 2 Make transition Tape 1 head position Tape 2 head position Reference point

Courtesy Costas Busch - RPI50 Theorem:Multi-tape machines have the same power with Standard Turing Machines

Courtesy Costas Busch - RPI51 Same power doesn’t imply same speed: Language Acceptance Time Standard machine Two-tape machine

Courtesy Costas Busch - RPI52 Standard machine: Go back and forth times Two-tape machine: Copy to tape 2 Leave on tape 1 Compare tape 1 and tape 2 ( steps)

Courtesy Costas Busch - RPI53 MultiDimensional Turing Machines Two-dimensional tape HEAD Position: +2, -1 MOVES: L,R,U,D U: up D: down

Courtesy Costas Busch - RPI54 Multidimensional machines simulate Standard machines: Use one dimension

Courtesy Costas Busch - RPI55 Standard machines simulate Multidimensional machines: Standard machine: Use a two track tape Store symbols in track 1 Store coordinates in track 2

Courtesy Costas Busch - RPI56 symbols coordinates Two-dimensional machine Standard Machine

Courtesy Costas Busch - RPI57 Repeat for each transition Update current symbol Compute coordinates of next position Go to new position Standard machine:

Courtesy Costas Busch - RPI58 MultiDimensional Machines have the same power with Standard Turing Machines Theorem:

Courtesy Costas Busch - RPI59 NonDeterministic Turing Machines Non Deterministic Choice

Courtesy Costas Busch - RPI60 Time 0 Time 1 Choice 1 Choice 2

Courtesy Costas Busch - RPI61 Input string is accepted if this a possible computation Initial configurationFinal Configuration Final state

Courtesy Costas Busch - RPI62 NonDeterministic Machines simulate Standard (deterministic) Machines: Every deterministic machine is also a nondeterministic machine

Courtesy Costas Busch - RPI63 Deterministic machines simulate NonDeterministic machines: Keeps track of all possible computations Deterministic machine:

Courtesy Costas Busch - RPI64 Non-Deterministic Choices Computation 1

Courtesy Costas Busch - RPI65 Non-Deterministic Choices Computation 2

Courtesy Costas Busch - RPI66 Keeps track of all possible computations Deterministic machine: Simulation Stores computations in a two-dimensional tape

Courtesy Costas Busch - RPI67 Time 0 NonDeterministic machine Deterministic machine Computation 1

Courtesy Costas Busch - RPI68 Computation 1 Choice 1 Choice 2 Computation 2 NonDeterministic machine Deterministic machine Time 1

Courtesy Costas Busch - RPI69 Repeat Execute a step in each computation: If there are two or more choices in current computation: 1. Replicate configuration 2. Change the state in the replica

Courtesy Costas Busch - RPI70 Theorem: NonDeterministic Machines have the same power with Deterministic machines

Courtesy Costas Busch - RPI71 Remark: The simulation in the Deterministic machine takes exponential time compared to the NonDeterministic machine