The Church-Turing Thesis Jos Baeten Centrum Wiskunde & - - PowerPoint PPT Presentation

The Church-Turing Thesis

Jos Baeten

Centrum Wiskunde & Informatica, Amsterdam, and LoCo, ILLC

Midsummernight Colloquium, Amsterdam June 17, 2015

Alonzo Church and Alan Turing

1903 - 1995 1912 - 1954

Informatics = Information + Computation

Computation: Church-Turing computation theory (1936)

Church defines computable function by means of

λ-calculus; non-computability

Turing defines computable function by means of

machine model; non-computability of halting problem

Computation Theory

A Turing machine is an adequate abstract model of a

computer

Church-Turing thesis (strong): anything that can be

done with a computer, now or in the future, can also be done with a Turing machine, given enough time and memory.

Turing Machine

Automaton Input Output Tape

Turing machine for unary addition

[+/| ]R [|/| ]R [|/| ]R [/]L [|/]L || + ||

Turing machine for unary addition

[+/| ]R [|/| ]R [|/| ]R [/]L [|/]L || + || || + ||

Turing machine for unary addition

[+/| ]R [|/| ]R [|/| ]R [/]L [|/]L || + || || + || ||+||

Turing machine for unary addition

[+/| ]R [|/| ]R [|/| ]R [/]L [|/]L || + || || + || ||+|| |||||

Turing machine for unary addition

[+/| ]R [|/| ]R [|/| ]R [/]L [|/]L || + || || + || ||+|| ||||| ||||| |||||

Turing machine for unary addition

[+/| ]R [|/| ]R [|/| ]R [/]L [|/]L || + || || + || ||+|| ||||| ||||| ||||| |||||

Turing machine for unary addition

[+/| ]R [|/| ]R [|/| ]R [/]L [|/]L || + || || + || ||+|| ||||| ||||| ||||| ||||| ||||

What is a computation?

Turing machine gives a function transforming input

string on tape to output string

Models a computer of the ’70s (program, CPU, RAM) Criticism possible on suitability as a theoretical model

• f a modern-day computer

Reactive Systems

“A Turing machine cannot drive a car, but a real computer can!”

Controversy 20 years ago

Peter Wegner: Church-Turing thesis is wrong! Turing machine can be fixed in several ways. Most elegantly: it denotes function on streams

Interaction

User interaction: not just initial, final word on the tape. Make interaction between control and memory explicit. . . . a theory of concurrency and interaction requires a new conceptual framework, not just a refinement of what we find natural for sequential computing.

Robin Milner, Turing Award Lecture, 1993

Concurrency

Transition systems Bisimulation (Park, Van Benthem) Structural operational semantics (Plotkin) Process algebra (Milner, Hoare, Bergstra, Klop) Comparative concurrency semantics (Van Glabbeek)

Executability Theory

Computability + Concurrency Real integration, aim is not to increase the computational power of the traditional model nor to investigate the extra expressivity of interaction

Reactive Turing Machine

τ[/]L τ[/]R i[/1]R

• [1/]L

i i i τ τ τ

• τ

i τ

The stack

i?0[/]R i?0[n/n]R

• !∅[/]L
• !n[n/]L

i?1[/]R i?1[n/n]R τ[/0]R τ[/]L τ[/1]R τ[/]L

Behavior of the stack

i?0

• !0

i?1

• !1
• !∅

i?0

• !0

i?1

• !1

i?0

• !0

i?1

• !1

Executable process

An executable process is a branching bisimilarity class

• f transition systems containing one of an RTM

Results

Transition system defined by an RTM is computable Every finitely branching computable transition system

is executable

The parallel composition of two executable transition

systems is executable

There is a universal RTM There is a good grammar for executable processes,

that makes interaction between control and memory explicit

Robustness

Bas Luttik, Fei Yang: Executability characterized by a simple variant of the π-calculus (with replication) up to branching bisimulation.

Conclusion

Executability = Computability + Concurrency Unified framework for computation and interaction Upcoming course in Master of Logic:

Computability and Interaction