Simulating Termination Analyzer Not Fooled by Pathological Input

[polcott]

The notion of a simulating termination analyzer is examined at the concrete level of pairs of C functions. This is similar to AProVE: Non-Termination Witnesses for C Programs: The termination status decision is made on the basis of the dynamic behavior of the input. This paper explores what happens when a simulating termination analyzer is applied to an input that calls itself.

Simulating Termination Analyzer HHH correctly simulates its input until:
(a) Detects a non-terminating behavior pattern: abort simulation and return 0.
(b) Simulated input reaches its simulated "return" statement: return 1.

int DD()
{
int Halt_Status = HHH(DD);
if (Halt_Status)
HERE: goto HERE;
return Halt_Status;
}

Can you see that DD correctly simulated by HHH demonstrates the recursive simulation non halting behavior pattern that cannot possibly reach its own "if" statement?

 

[WhiteCube]

HHH tells me that DD does not halt.

When I call DD, it halts.

 

[polcott]

This one page paper addresses that objection:

https://philpapers.org/archive/OLCHPS.pdf

It is the Claude AI analysis of my proposal. ChatGPT and Grok derive
the same results. Their analysis is less succinct.

 

[WhiteCube]

I have read the .pdf

The infinite loop that HHH is detecting, is caused by HHH using simulation.

There is no recursion inside DD. (I'm assuming DD does not know how HHH is coded.)

The line ...=HHH(DD) is just DD asking HHH "Do I halt?"

"My (HHH) simulation won't stop, therefore you (DD) don't stop." is not a valid answer in my book.

"Based only on the fact that you (DD) asked me (HHH) that question, I will conclude that you do not halt."

If DD knows that HHH will use simulation, DD has a bug.

 

[polcott]

The x86utm operating system enables one C function to execute another C function in debug-step mode. HHH simply watches this step by step behavior to see if it matches any non-halting behavior patterns.

It is true that DD correctly simulated by HHH cannot possibly reach its own "return" instruction final halt state. HHH uses the "recursive simulation non-halting behavior pattern" to detect this. This by itself proves that the behavior specified by the input to HHH(DD) is non-halting.

It may be very difficult to understand, yet the behavior of non-inputs does not count. HHH is only accountable for the behavior that its inputs specify.

Correct simulation is a correct measure of this behavior. Correct simulation and direct execution only vary when an input calls its own simulating termination analyzer. In the case it is the input that rules and non-inputs are irrelevant.

GitHub - plolcott/x86utm: Universal Turing Machine (UTM) having the x86 language as its Machine description language.

Universal Turing Machine (UTM) having the x86 language as its Machine description language. - plolcott/x86utm

github.com

 

[WhiteCube]

If simulator HHH detects an infinite loop, and returns 0, that is perfectly reasonable. HHH is a valid, well defined function.

I still maintain that a non-halting function can be written without infinite loops, but that's another story.

Lets say that HHZ is a simulator like HHH, but does not detect infinite loops. It returns 1, or just keeps running.

Lets say DD had the line ...=HHZ(DD)

In that situation, and only that situation, I would accept that HHH(DD)=0

 

[polcott]

If simulator HHH detects an infinite loop, and returns 0, that is perfectly reasonable. HHH is a valid, well defined function.

I still maintain that a non-halting function can be written without infinite loops, but that's another story.

Lets say that HHZ is a simulator like HHH, but does not detect infinite loops. It returns 1, or just keeps running.

Lets say DD had the line ...=HHZ(DD)

In that situation, and only that situation, I would accept that HHH(DD)=0

I call this the recursive simulation non-halting behavior pattern.
Three different LLM systems Claude AI that you just reviewed, ChatGPT and Grok all agree that HHH(DD)==0 is correct. DD is the halting problem counter-example input that has been the basis of the all of the conventional halting problem proofs. Prior to my innovation of a simulating termination analyzer this was thought to be impossible.