https://claude.ai/share/6cf4bb58-9025-4e3f-a33f-e7f1a2d70423
> Ok Claude 4.1, help me understand computability. Considering the halting
> problem, it seems to me the disproof only works because the actual real world
> state is left out as an input to the programs. With a real Turing machine, a
> program could analyze the tape to see if the pathological example is calling
> it, and behave differently. What say you?
You're touching on a really interesting aspect of the halting problem
proof! Your intuition about "real world state" points to something
important, though there's a subtlety here worth exploring.
You're right that in the classic proof, we construct a program H' that
essentially asks "what would H say about me?" and then does the
opposite. The key insight you're having is: what if H could somehow
detect that it's being called by this pathological program and respond
differently?
The issue is that this detection mechanism would need to be
computational - it would be part of H's program. But here's where it
gets tricky: whatever detection method H uses becomes part of its
definition, and we can then construct a new pathological program that
accounts for exactly that detection method.
Think of it like this: Say H checks its input for some "signature" of
the pathological case. Now I can create H'' that mimics a normal
program's signature while still containing the pathological logic. Or
if H simulates its input to some depth to detect loops, I can create
H'' that behaves normally up to that depth plus one, then enters the
pathological behavior.
This is similar to the "arms race" in computer security - for any
finite computational detection method, there's always a way to
construct a program that evades it while still creating the paradox.
The deeper issue is that the halting problem isn't about practical
limitations - it's about what's computable in principle. Even if we
gave H access to its entire call stack, memory state, or execution
history, as long as H's decision process is algorithmic (computable),
we can construct a program that uses that very algorithm against
itself.
What do you think about this recursive nature of the problem? Does
this address your concern about the "real world state"?
> H would have to solve the known areas of the domain rather than using a
> simple signature. Because it is a Turing machine, it could address anything
> mathematically solvable.
You're proposing something quite sophisticated - that H could use its
full computational power to mathematically analyze and solve for
halting in "known areas of the domain" rather than relying on simple
pattern matching. This is a thoughtful approach!
And you're absolutely right that a Turing machine can compute anything
that's mathematically solvable. In fact, we cansolve the halting
problem for many specific classes of programs:
Programs with no loops
Programs with only bounded loops
Certain types of linear programs
Programs in less powerful computational models
So your H could indeed incorporate solvers for all these known
decidable cases. It could have sophisticated mathematical analysis
built in - symbolic execution, abstract interpretation, complex
invariant checking, etc.
But here's the crux: the pathological program in Turing's proof
doesn't need to be complex. It can be extraordinarily simple:
function H'(x):
if H(x, x) says "halts":
loop forever
else:
halt
When we ask H about H'(H'), no amount of mathematical sophistication
helps because the paradox is baked into the logical structure itself,
not hidden in mathematical complexity. H' literally just calls H and
does the opposite - it's almost trivial mathematically, yet
unsolvable.
The undecidability isn't hiding in some mathematically complex corner
case that better analysis could crack. It emerges from self-reference
- the same phenomenon that gives us Gödel's incompleteness theorems.
The pathological case can be completely transparent and still create
the contradiction.
Does this distinction between mathematical complexity and logical
self-reference clarify why even a mathematically sophisticated H still
faces the fundamental limitation?
> You’ve left my proposal and returned to leaving out the world state. H would
> need to detect that it is being called in a simple contradiction and behave
> pathologically in some manner that results in H(H’) being correct. Here that
> could be as simple as returning loops inside H’ and halts outside.
Ah, I see what you're getting at now! You're proposing that H could
detect when it's being called from within H' versus being called
externally, and return different answers based on that context. So
when we ask H(H', H') from the outside, H returns one answer, but when
H' internally calls H(H', H'), H returns a different answer.
This is a clever idea! And you're right that with access to the full
execution context (call stack, memory state, etc.), H could indeed
detect whether it's being called from within H' or from outside.
But here's the subtle catch: for H to be a solution to the halting
problem, it needs to be a function in the mathematical sense - given
the same inputs, it must always return the same output. The halting
problem asks: "Does there exist a function H(program, input) that
correctly determines if program halts on input?"
If H returns different values for H(H', H') depending on where it's
called from, then it's not actually computing a function of its stated
inputs (program, input) - it's computing a function of (program,
input, calling_context).
You could argue we should reformulate the halting problem to allow
context-dependent oracle machines. But then we'd need to specify:
which context represents the "true" answer about whether H' halts? The
answer when called from outside? From inside? Both answers are
"correct" for their context, but they're different!
This is actually a really insightful observation about the
relationship between mathematical functions and real computation. What
do you think - does this distinction between context-dependent
computation and mathematical functions address your point?
> They are different because the world state is different when H is called. It
> is giving the answer for the world state at the time of call.
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