An Actor Application Can Be Hundreds of Times Faster Than a Parallel Lambda Expression

Communication between Turing Machines and between lambda-expressions was crucially omitted thereby crippling them as a foundation for Computer Science. Actors [Hewitt, et. al 1973] remedied the foundations to provide scalable computation. Church’s lambda calculus crucially omitted concurrency with the consequence that an Actors machine can be hundreds of times faster than a parallel lambda expression. The reason that in practice that an Actor can be hundreds of times faster is that in order to carry out a concurrent computation, the parallel lambda expression must in parallel gather together the results of the computation step of each core and then in parallel redistribute the communicated messages from each core to other cores. The slowdown comes from the overhead of a lambda expression spanning out from a core to all the other cores, performing a step on each core, and then gathering and sorting the results of the steps (cf. [Strachey 2000]).

Actor Speed-Up Theorem. There are practical Actor computations of time t which in the parallel lambda calculus require time a significant multiple of t*log[n], where n is the number of cores on a machine.

 

For more information, see the following:

Epistemology Cyberattacks

https://papers.ssrn.com/abstract=3603021

 

 

The Church/Turing Thesis is False

The Church/Turing Thesis [cf. Turing 1936] was developed to that effect that a nondeterministic Turing Machine can perform any computation. The Thesis has been the overwhelming consensus since the 1936.

 

Global states are central to the Church/Turing model of computation. In this model, there is a global state with nondeterministic transition to the next state. A state transition can be initiated by a computation step that specifies that one of a number of specified steps can be executed next. A global-state model of computation has the property of bounded nondeterminism. Bounded nondeterminism means that there is a bound determined in advance for the number of possibilities that a system can explore given that it must always come back with an answer. In the development of a theory of concurrent computation, Dijkstra remained committed to a global state model of computation. This commitment gave rise to the “unbounded nondeterminism” controversy. Digital systems can be physically indeterminate by making use of arbiters in order to resolve reception order in communication between systems. Because of physical indeterminacy, a digital system can have the property of unbounded nondeterminism in that no bound exists when it is started on the number of possibilities that an always-responding digital system can explore. Being restricted to bounded nondeterminism is one the reasons that the Church/Turing model of computation is inadequate for digital computation. Consequently, there are digital computations that cannot be performed by a nondeterministic Turing Machine.

 

See the following for more information:

"Epistemology Cyberattacks"

https://papers.ssrn.com/abstract=3603021

Gödel failed to prove inferential undecidablity or incompleteness in foundations

Hilbert believed every proposition is either provable or disprovable every mathematical proposition is, that is, inferentially decidable {(⊢Ψ) ∨ (⊢¬Ψ)} [Hilbert 1930]. Others doubted inferential decidability held. It would mean that whether a proposition is a theorem could be algorithmically decided by enumerating theorems until the proposition occurs or its negation occurs [von Plato 2018].

[Gödel 1931] seemed to have proved undecidability using the proposition I'mUnprovable, such that ⊢(I’mUnprovable ⇔⊬I’mUnprovable).

However, existence of I’mUnprovable  would enable the following cyberattack [cf. Wittgenstein 1937]:

Theorem: Existence of I’mUnprovable  of leads to inconsistency in foundations.

Proof: Ever since Euclid, it has been a fundamental principle that a theorem can be used in a proof (the principle of  TheoremUse), that is, {⊢((⊢Ψ)⇒Ψ)} [cf. Artemov and Fitting 2019]. However by [Gödel 1931], 

 ⊢(¬I’mUnprovable ⇔⊢I’mUnprovable). Consequently,  

⊢(¬I’mUnprovable ⇒I’mUnprovable)  by TheoremUse.

• Therefore ⊢I’mUnprovable  using ProofBySelfContradiction 
  {⊢((⊢(¬Ψ⊢Ψ)) ⇒ ⊢Ψ)} with Ψ being I’mUnprovable. 
  

• Thus, I’mUnprovable using TheoremUse {⊢((⊢Ψ)⇒Ψ)} with Ψ being I’mUnprovable. Consequently, ⊬I’mUnprovable using
  ⊢(I’mUnprovable ⇔⊬I’mUnprovable)

 Having both ⊢I’mUnprovable  and ⊬I’mUnprovable is a contradiction in foundations.

 

Strong types prevent construction of I’mUnprovable using the following recursive definition:

I’mUnprovable:Proposition viw≡⊬I’mUnprovable. Note that (⊬I’mUnprovable):Proposition vi+1w in the right-hand side of the definition because I’mUnprovable is a propositional variable of I’mUnprovable :Proposition viw. Consequently,

I’mUnprovable:Proposition<i>⇒I’mUnprovable:Proposition<i+1>, which is a contradiction.

 

The crucial issue with the proofs in [Gödel 1931] is that the Gödel number of a proposition does not capture its order. Because of orders of propositions, the Diagonal Lemma [Gödel 1931] fails to construct the proposition I’mUnprovable.

information see the following:

See the following article for more information:

https://papers.ssrn.com/abstract=3603021