
Physical Indeterminacy in Digital Computation - ProfHewitt
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3459566
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ProfHewitt
The Actor Model moved the foundations of digital computing beyond the work of
Alan Turing and Alonzo Church, who developed equivalent models of computation
based on the concept of an algorithm, which by definition is provided an input
from which it is to compute a value without external interaction. After
physical computers were constructed, they soon diverged from computing only
algorithms meaning that the Church/Turing theory of computation no longer
applied to computation in practice because computer systems are highly
interactive as they compute with incremental new input affecting future
computation. The divergence inspired development of the Actor Model in 1972 to
characterize all digital computation. There are simple digital computations
that cannot be performed by a nondeterministic Turing Machine. The Actor Model
has been axiomatized up to a unique isomorphism thereby removing all ambiguity
from the most fundamental theory of digital computation. As part of the
axiomatization, Actor theory has an induction axiom for computation events
that is suitable for proving all kinds of properties (including
specifications) for Actor systems. Event induction is much more powerful and
practical than the methods developed by Turing [1949], Hoare [1969], and
Lamport [2018] because it can express many more properties and because proofs
are more intuitive. Concepts of the Actor Model have been deployed at scale at
Ericsson, Erlang Solutions, Lightbend, Microsoft, PayPal, Twitter, and many
other companies. Actors are being incorporated into the foundations for many-
core intelligent systems that will be the foundation for the future of
computing.

Strong types for the Actor model overturned an assumption beginning with
Euclid that has persisted for millennia including Hilbert, Gödel, Church,
Turing, and von Neumann. The assumption was that the theorems of a theory must
be algorithmically enumerable by beginning with axioms and applying rules of
inference. However, in order to characterize computation up to a unique
isomorphism, it was necessary to develop an event induction axiom with
uncountable instances. Because there are uncountable instances of the event
induction axiom, it cannot possibly be the case that theorems of Actor theory
are algorithmically enumerable because, of course, each axiom instance is a
theorem of the theory. However, the theory of Actors is nevertheless effective
because proof checking is algorithmically decidable. Consequently the
foundational theory of digital computation is algorithmically inexhaustible.
Furthermore, if a mathematical theory of digital computation is consistent,
then the theory must be inexhaustible.

Strong types for the Actor model also exposed inadequacies in Gödel’s proof of
the incompleteness (i.e. inferential undecidability) theorem using his
proposition _I’mUnprovable_. Using strong types, the construction of _I
'mUnprovable_ is blocked because the mapping Ψ↦⊬Ψ has no fixed point because
⊬Ψ has order one greater than the order of Ψ since Ψ is a propositional
variable. Consequently, some other way had to be found to prove inferential
undecidability without using Gödel’s proposition _I 'mUnprovable_. A
complication is that theorems of strongly-typed theories of computation are
not computationally enumerable, as mentioned above. The complication was
overcome using special cases of the induction axioms to prove inferential
undecidabilty.

