Theory of Computation (Fall 2014): Universal Programs

Theory of Computation

Universal Programs

Vladimir KulyukinDepartment of Computer Science

Utah State University

www.vkedco.blogspot.com

http://www.vkedco.blogspot.com/

A movement is accomplished in six stagesAnd the seventh brings return

Syd Barrett, Chapter 24, The Piper at the Gates of Dawn

http://en.wikipedia.org/wiki/The_Piper_at_the_Gates_of_Dawn

Outline● Introduction● Developing Universal Programs in Six Stages

Universality Theorem

Stage 01: Source Code Extraction, Encoding Program State, Initializing In-struction Counter

Stage 02: Establishing Termination Condition

Stage 03: Decoding Next Instruction

Stage 04: Determining Next Instruction's Type

Stage 05: Execution of Conditional Dispatch

Stage 06: Execution of No-Op, Subtraction, Addition

Return

Introduction

Review: Variable & Label Orderings

● The variables are ordered as follows:Y X1 Z1 X2 Z2 X3 Z3 …

● The labels are ordered as follows:A1 B1 C1 D1 E1 A2 B2 C2 D2 E2 A3 …

● #(Y)=1, #(X1)=2, #(X3)=6, #(Z2)=5● #(B1)=2, #(A2)=6, #(C2)=8

Why Universal Programs are Called Universal

Every natural number in the universe is a program awaiting its computer. The program runs if and when its computer appears. Universal programs are one example of such computers. They are called universal, because they turn every natural number into a running program.

Back to The Big Picture

GÖDEL Coder

GÖDEL Decoder

L Program Number Universal Program

Number

???

GÖDEL Coder = CompilerGÖDEL Decoder = Reverse CompilerUniversal Program = Operating System/VM/Interpreter

Developing Universal Programsin Six Stages

Theorem 3.1 (Ch. 4)

.computablepartially is

,,...,function the,0each For 1)( yxxΦn n

n

What does this theorem really say? y is a natural number; y is a compiled program; x

1, …, x

n are inputs to program

compiled as y; the function Φ takes y, runs it on the values of x

1, …, x

n and returns the result iff there is a computation,

which is why Φ is partially computable.

Proof 3.1

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Stage 01

Source Code Extraction, Encoding Program State, Initializing Instruction Counter

Program State & Instruction Counter

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Stage 02

Establishing Termination Condition


• The next thing that Un must do is check for the termination condition

• So, when should Un stop running program y?• The termination condition occurs when the instruction

counter is I+1 or 0, where I=Lt(Z) is the number of instructions in the source code compiled into natural number Z

• The function Lt(Z) computes the length of natural number Z• The instruction counter is 0 when the program is dispatched to

a label that does not exist

Review: Access Function for Gödel Numbers

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Stage 03

Decoding Next Instruction


● The next thing that Un must do is decode the next instruction to execute (decode simply means to extract the source code)

● The source code contains only primitive L instructions: all macros have been expanded properly and variables renamed

● Recall that an instruction is defined by three components: ● the label ● the instruction type● the variable


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Determining Next Instruction's Type

Un Construction: Part 04

● The next thing that Un must do is to determine the type of the next instruction to be executed

● The following labels are used:● N – stands for no-op (do nothing)● A – stands for addition● M – stands for subtraction (minus)● Conditional dispatch needs to go to a specific label


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• However, if we know that the value of the variable is 0, we do not need to subtract 1 from it, because it will remain 0 anyway

• So, before we dispatch to subtraction, we check if the value of the variable is 0; if it is, we do nothing

Program State & Variable Values

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Execution of Conditional Dispatch

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Execution of NO-OP, Subtraction, & Addition


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Return

What We Just Proved

.computablepartially is ,,...,function the

,0each For :Theorem)ity (Universal 3.1 Theorem

1)( yxxΦ

n

nn

Back to The Big Picture

GÖDEL Coder

GÖDEL Decoder

L Program Number Universal Program

Number

???

GÖDEL Coder = CompilerGÖDEL Decoder = Reverse CompilerUniversal Program = Operating System/VM/Interpreter

Universality Theorem

A Practical Interpretation of Universality Theorem

● If you look at the proof of Theorem 3.1 (Ch. 4) as a programmer or a computer scientist, you notice that we have constructed an interpreter (a mini-OS) for L in L itself

● It is feasible to write operating systems that can interpret/execute any program in a specific assembly dialect

● It is also feasible to write ● Java interpreter/compiler in Java● Python interpreter/compiler in Python● Lisp interpreter/compiler in Lisp● C interpreter/compiler in C

Stream, Infinite & Universal

0,run can

,...,,...,,

1

121

nUU

UUUU

nn

nn

Program Simulation● Suppose a universal program runs a program on some inputs● We can expect two mutually exclusive outcomes:–The program terminates (halts) after some finite number of steps and produces an output (a number);–The program does not terminate at all (e.g., it has an infinite loop)● In many situations, we would like to know if a program terminates within a given number of steps without running the program● Enter Simulation which we will look at in the next lecture

References

● Ch. 4, Computability, Complexity, and Languages, 2nd Edition, by Davis, Weyuker, Sigal

● http://en.wikipedia.org/wiki/Kurt_Gödel

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