On Wednesday, January 23, 2019 at 5:52:01 AM UTC-6, Bruno Marchal wrote:
>
>
> On 22 Jan 2019, at 01:49, Philip Thrift <cloud...@gmail.com <javascript:>> 
> wrote:
>
> One of the oddest of things is when physicists use the language of 
> (various) theories of physics to express what can or cannot be the case. 
> It's just a language, which is probably wrong.
>
> There is a sense in which the Church/Turing thesis is true: All out 
> languages are Turing in their syntax and grammar. What they refer to is 
> another matter (pun intended).
>
>
> They refer to the set of computable functions, or to the universal machine 
> which understand that language. But not all language are Turing universal. 
> Only the context sensitive automata (in Chomski hierarchy) are Turing 
> universal. Simple languages, like the “regular” one are typically not 
> Turing universal. Bounded loops formalism cannot be either.
>
> But the notion of language is ambiguous with respect to computability, and 
> that is why I prefer to avoid that expression and always talk about 
> theories (set of beliefs) or machine (recursively enumerable set of 
> beliefs), which avoids ambiguity. 
> For example, is “predicate calculus” Turing universal? We can say yes, 
> given that the programming language PROLOG (obviously Turing universal) is 
> a tiny subset of predicate logic. But we can say know, if we look at 
> predicate logic as a theory. A prolog program is then an extension of that 
> theory, not something proved in predicate calculus.
> Thus, I can make sense of your remark. Even the language with only one 
> symbol {I}, and the rules that “I” is a wff, and if x is wwf, then Ix is 
> too, can be said Turing universal, as each program can be coded by a 
> number, which can be coded by a finite sequence of I. But of course, that 
> makes the notion of “universality” empty, as far as language are concerned. 
> Seen as a theory, predicate calculus is notoriously not universal. Even 
> predicate calculus + the natural numbers, and the law of addition, 
> (Pressburger arithmetic) is not universal. Or take RA with its seven 
> axioms. Taking any axiom out of it, and you get a complete-able theory, and 
> thus it cannot be Turing complete.
>
> Bruno
>
>
>
Here's an example of a kind of "non-digital" language:

*More Analog Computing Is on the Way*
https://dzone.com/articles/more-analog-computing-is-on-the-way



*The door on this new generation of analog computer programming is 
definitely open. Last month, at the Association for Computing Machinery’s 
(ACM) conference on Programming Language Design and Implementation, 
a paper  <https://people.csail.mit.edu/sachour/res/pldi16_arco.pdf>was 
presented that described a compiler that uses a text based, high-level, 
abstraction language to generate the necessary low-level circuit wiring 
that defines the physical analog computing implementation. This research 
was done at MIT’s Computer Science and Artificial Intelligence Laboratory 
(CSAIL) and Dartmouth College. The main focus of their investigation was to 
improve the simulation of biological systems. *


*Configuration Synthesis for ProgrammableAnalog Devices with Arco*
https://people.csail.mit.edu/sachour/res/pldi16_arco.pdf

*Programmable analog devices have emerged as a powerful*
*computing substrate for performing complex neuromorphic*
*and cytomorphic computations. We present Arco, a new*
*solver that, given a dynamical system specification in the*
*form of a set of differential equations, generates physically*
*realizable configurations for programmable analog devices*
*that are algebraically equivalent to the specified system.*
*On a set of benchmarks from the biological domain, Arco*
*generates configurations with 35 to 534 connections and 28*
*to 326 components in 1 to 54 minutes.*


- pt
 

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