Path: ...!news.nobody.at!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: I have always been correct about emulating termination analyzers --- PROOF
Date: Wed, 23 Oct 2024 10:28:30 +0300
Organization: -
Lines: 50
Message-ID: <vfa8iu$1ulea$1@dont-email.me>
References: <ves6p1$2uoln$1@dont-email.me> <3232d8a0cc7b5d4bba46321bf682c94573bf1b7c@i2pn2.org> <vesemu$2v7sh$1@dont-email.me> <a9fb95eb0ed914d0d9775448c005111eb43f2c5b@i2pn2.org> <veslpf$34ogr$1@dont-email.me> <647fe917c6bc0cfc78083ccf927fe280acdf2f9d@i2pn2.org> <vetq7u$3b8r2$1@dont-email.me> <522ecce215e636ddb7c9a1f75bff1ba466604cc5@i2pn2.org> <veuvt9$3hnjq$1@dont-email.me> <87634d01e18903c744d109aaca3a20b9ce4278bb@i2pn2.org> <vev8gg$3me0u$1@dont-email.me> <eb38c4aff9c8bc250c49892461ac25bfccfe303f@i2pn2.org> <vf051u$3rr97$1@dont-email.me> <e3f28689429722f86224d0d736115e4d1895299b@i2pn2.org> <vf1hun$39e3$1@dont-email.me> <dedb2801cc230a4cf689802934c4b841ae1a29eb@i2pn2.org> <vf1stu$8h0v$1@dont-email.me> <592109c757262c48aaca517a829ea1867913316b@i2pn2.org> <vf37qt$fbb3$1@dont-email.me> <vf5430$sjvj$1@dont-email.me> <vf5mat$v6n5$4@dont-email.me> <vf7jbl$1cr7h$1@dont-email.me> <vf8b8p$1gkf5$3@dont-email.me>
MIME-Version: 1.0
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Wed, 23 Oct 2024 09:28:30 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="bd0d3d30685943dedb9ac685b035bfaa";
	logging-data="2053578"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1972Ayq0q4xyxE96pTCaYQl"
User-Agent: Unison/2.2
Cancel-Lock: sha1:YvWPeqXjpBGYZMUtsOi1jFE3Yak=
Bytes: 3599

On 2024-10-22 14:02:01 +0000, olcott said:

> On 10/22/2024 2:13 AM, Mikko wrote:
>> On 2024-10-21 13:52:28 +0000, olcott said:
>> 
>>> On 10/21/2024 3:41 AM, Mikko wrote:
>>>> On 2024-10-20 15:32:45 +0000, olcott said:
>>>> 
>>>>> The actual barest essence for formal systems and computations
>>>>> is finite string transformation rules applied to finite strings.
>>>> 
>>>> Before you can start from that you need a formal theory that
>>>> can be interpreted as a theory of finite strings.
>>> 
>>> Not at all. The only theory needed are the operations
>>> that can be performed on finite strings:
>>> concatenation, substring, relational operator ...
>> 
>> You may try with an informal foundation but you need to make sure
>> that it is sufficicently well defined and that is easier with a
>> formal theory.
>> 
>>> The minimal complete theory that I can think of computes
>>> the sum of pairs of ASCII digit strings.
>> 
>> That is easily extended to Peano arithmetic.
>> 
>> As a bottom layer you need some sort of logic. There must be unambifuous
>> rules about syntax and inference.
>> 
> 
> I already wrote this in C a long time ago.
> It simply computes the sum the same way
> that a first grader would compute the sum.
> 
> I have no idea how the first grade arithmetic
> algorithm could be extended to PA.

Basically you define that the successor of X is X + 1. The only
primitive function of Peano arithmetic is the successor. Addition
and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but
can be defined recursively from the successor function and the
order relation is defined similarly.

Anyway, the details are not important, only that it can be done.

-- 
Mikko