t. gregory bandyinteraction machines seminarfebruary 21, 2003 - 1 union college - computer science...

T. Gregory Bandy Interaction Machines Seminar February 21, 2003 - 1Union College - Computer Science Graduate Program

Interaction Machines

Are they more powerful than Turing Machines?

T. Gregory Bandy


Today's Presentation

● A New Paradigm● Turing Machines● Interaction Machines● Understanding the Issues● Extending Turing Machines● Observations● Questions


A New Paradigm● Peter Wegner● 1997 - Interaction Machines are

a more powerful class of computability than TuringMachines.

● More powerful than algorithms.● Interaction with the external world and

responding to new inputs.● Not inductive.


An Interaction Machine?


An Interaction Machine?

networks

cell phones

object-oriented programming

artificialintelligence


Turing Machine● Alan Turing (1912-1954)● The Standard Turing MachineThe Turing Machine has an input-output tape that is infinite in both

directions and allows an infinite number of left and right moves.

Commands:● Read tape● Move tape left● Move tape right● Write 0 on tape● Write 1 on tape● Jump to another command● Halt


Turing Machine● The Standard Turing Machine

– TM starts with all input already on tape. Finite amount of input.

– TM result written to tape when TM halts.– State transitions specified when / before TM starts.– Deterministic or Nondeterministic.– TM's may be combined.– Halting is important.


Turing Machine● Standard Turing Machine variations

– Semi-infinite tape– Off-line input– Multi-tape– Multi-dimensional tape


Turing Machine● The Universal Turing Machine

– Can simulate any other TM.– Multi-tape TM with 3 tapes

● Description of other TM.● Other TM's tape.● Other TM's internal state during execution.

– A reprogrammable TM.– UTM designed with 2 state and 7 colors.


Turing Machine History● Alonzo Church (1903-1995) - recursive

functions define 'computability.'● Kurt Gödel (1906-1978) - reached similar

conclusion.● Alan Turing - demonstrated TM's compute

recursive functions.● S.C. Kleene, et. al - recognized 'effective

methods' must be recursive.

Church

Post

Kleene

Gödel


Interaction Machine

Turing machinesextended by

adding input and output actions that support dynamic interaction

with an external environment are called

interaction machines.


Interaction Claims● Interaction is not expressible by algorithms.● Interaction Machines cannot be modeled by

Turing Machines.● Interaction is more powerful than algorithms.● Interaction cannot be specified by first-order

logic or state-transition semantics.● Interaction permits computing that is not

algorithmically describable.


Interaction Machine● A Turing Machine, extended . . .

– initial input is not fully defined– input and output are streams, not strings– specified by interaction histories, rather than

algorithms– receives external input during execution– output remembered between executions– no halting state


Interaction Streams● Streams are

– dynamic - not fixed in advance– evolving - changing independent of the IM– partial structures - not defined in advance– may depend on time, adversaries, oracles, and

protocols of interaction - again, changing independent of the IM

● Streams may be produced by multiple unspecified IMs


Interaction Histories

interface specification for

one event

time stamp

Interaction History: sequences of tracesinput

stream

Histories provide the means to observe what the IM does.

Each trace reveals the interface being used and the informationprovided. Traces may be simple or complex in structure and content.

a trace


Interaction Interfaces

Operations share state they cannot algorithmically control.

Operations remember state from previous executions.

Operations implement an interface.

Interfaces permits partial specification.

Behavior of the interface is observable.

A machine is defined by observable

interactions rather than algorithmic transformations.


Understanding the Issues


Understanding the IssuesComputability

Classic Computability– all inputs to an algorithm yield a valid result.– effective computability yields the result in finite steps and with

no outside help– measured by classes of complexity

Interactive Computation– provides a service over time.– values of inputs and outputs are interdependent– measured by expressiveness


Understanding the IssuesChurch-Turing Thesis

Classic CT equates the effective computability of functions and algorithms with Turing Machines.– This maps the intuitive notion of computability by algorithms

to the formal notion of Turing Machines.– The TM formal model dominates.

Interaction CT would correlate interactive computing with coinductive models of computing using coalgebras.– This maps the formal notion of coinduction to the intuitive

notion of interactive computing.– The interactive computing notion dominates.


Understanding the IssuesPowerfulness

Classic Powerfulness– Power means the ability to solve more complex classes of

problems. I.e., computational limits.– Push-Down Automata (PDA) are more powerful or expressive

than Finite State Automata (FSA).

Interaction Expressiveness– the ability of observers to make observational distinctions.– observations permit evaluating algorithms and

interactions.– better way to reason about a process (Milner)

Milner


Understanding the IssuesMathematics & Logic

Classic computability theory is dominated by First-Order Logic.– basic to algorithms and computability theory– shown to be incomplete.– uses least-fixpoint semantics

Interactive computation is expressed using coinduction, coalgebras, and bisimulation.– accommodates incompleteness.– coalgebras define steps of observation– uses greatest-fixpoint semantics in logic which permit

operating on infinite input streams.

Goldin


The PTM is a perpetual nondeterministic Turing Machine with 3 tapes that continuously reads input, computes, and outputs results.– Extends the existing nondeterministic 3 tape TM

(N3TM) ( ≡ Standard single tape TM).– The work tape captures the N3TM state and persists

between computations for later use.– The input and output are a coinductively defined pair

(win, wout) that define the computation performed.– Infinite sequences of pairs that get their source from

and produce output to an external environment.

Extending Turing MachinesPersistent Turing Machine

GoldinSmolkaWegner


The ITMA extends the Turing Machine with advice, with infinite computations, and accommodates interaction.– Infinite computations and interaction are existing TM

extensions.– Advice is a noncomputable alteration that affects and

improved the computing capability.– Turing proposed choice machines as well as

automated machines that we know as TMs.

Extending Turing MachinesInteractive Turing Machine with Advice

van LeeuwenWiedermann


Observations● Excellent testimony for the value of academia.

Vigorous discussions, critiques, and rebuttals.● Interaction is widely observed, but is hard to

formalize. Co- mathematics hold promise.● Interaction appeals to the intuitive notion that not

all computing is algorithmic. ● Confusion of terms hamper evaluating IMs.● Unclear whether IMs are indeed more powerful

than TMs.


Questions● Can IMs be thought of as sophisticated combinable

TMs?● Can the relative power of algorithms and coinduction be

evaluated in terms of their observational behavior?● How far can TMs be legitimately extended?● Does IM theory deserve being taught along with

automata theory?● Union College's Mathematics Department course listing

does not appear to teach coinduction, coalgebra, or bisimulation. Should such courses be offered?

t. gregory bandyinteraction machines seminarfebruary 21, 2003 - 1 union college - computer science...

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