Questions tagged [computability]
Computability theory a.k.a. recursion theory.
354
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1answer
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Where, if any, is there currently any research being done on the subject of ternary computers? [closed]
I had the experience several years ago of working with a team that had developed a ternary computing system. It ran out of funding and was abandoned but I felt it was ahead of its time. Currently, ...
3
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1answer
233 views
Proof and computational complexity
I couldn't find documents elaborating on this: if the Curry Howard correspondence is to be interpreted as establishing a strong relation between proofs and programs, should there not be a strong ...
1
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0answers
44 views
A proved computationally-irreductible function
In arXiv:1111.4121 and arXiv:1304.5247, Hervé Zwirn and Jean-Paul Delahaye propose a formal definition of computational irreducibility. Is there a (possibly artificial) function that can be proved to ...
0
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0answers
123 views
What is the difference between a model of computation and a programming language?
https://en.wikipedia.org/wiki/Model_of_computation includes sequential models, functional models and concurrency models.
Sequential models include finite state machine, Turing machines, random access ...
4
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1answer
135 views
Is this variant of bitwise cyclic tag Turing-complete?
Cross-posted to MathOverflow.
CT is an extremely minimalist programming language that can simulate arbitrary tag systems, and is therefore Turing-complete. A program consists simply of a string of 3 ...
3
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1answer
105 views
What is the meaning of 'uniform' in Blum's paper “On the Size of Machines”
In Blum's "On the Size of Machines" he uses the word 'uniform' in several places.
For instance, in Theorem 1, he writes
Let $f$ be any recursive function. Then there exists $i,j\in \mathbb{...
15
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1answer
635 views
Computability and continuity
Say $L_1$ and $L_2$ are computable languages. Let $f$ be a function $L_1 \rightarrow L_2$. Let $C$ be the statement, "if $l \subseteq L_2$ and $l$ is a computable language, the preimage $f^{-1}(...
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0answers
104 views
Functional abbreviation for Inst expression in Turing's 1936 paper
In Turing's 1936 paper "On Computable Numbers",
For a Turing Machine $M$, $Inst(q_i S_j S_k L q_l ) $ means that if $M$ scans symbol $S_j $ under $m-configuration$ $q_i$, then the symbol ...
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1answer
110 views
Are all computable functions monoidal from a category theory POV? [closed]
I have been studying some category theory only to understand the "why" behind the various categorical constructs we use ("borrow") in the world of functional programming.
One of ...
11
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1answer
202 views
Is there a language of first-order logic such that every r.e. set is Turing-equivalent to some finitely axiomatizable theory in that language?
I hope that mathematical logic / recursion theory type questions are welcome here. I am sorry this question is so long and technical, but I believe that if you read it you will find that it is well-...
1
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1answer
154 views
How come Wikipedia says that Random Turing Machines can provide uncomputable output?
Wikipedia article mentioned : Hypercomputation
The third paragraph starts off with:
Technically, the output of a random Turing machine is uncomputable; however, most hypercomputing literature focuses ...
1
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1answer
127 views
What kind of computational model is the brain? [duplicate]
I was wondering what kind of computational model is the human brain (as it seems superior to a Turing machine).
Another thing that should be a separate question, What would be a perfect computer model ...
12
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2answers
402 views
Complexity relative to the graph of the Busy-Beaver function
This question is inspired by the comments made on this other question that I asked, and by an attempt to provide an explicit example of a complexity question beyond the Turing degree $\mathbf{0}$. (...
5
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0answers
194 views
Has there been any development of complexity theory for other Turing degrees than 0?
(I'm not sure if this question is better suited for MathOverflow or here. I'll try here first, and move over to MO later if it appears to be more appropriate.)
Complexity theory can be very broadly ...
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0answers
66 views
Given a program specification, S, what can be said about the size and efficiency of programs that exactly satsify S, with respect to the size of S?
Suppose we are given a program specification, $S$, and we want to reason about programs $P$ that satisfy $S$. One might like to think that if the specification is 'simple', the the program should be '...
3
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0answers
95 views
Logic of learning
Does Robust logic (Leslie Valiant), Default logic (Raymond Reiter) and Circumscription logic (John McCarthy) have any relation?
I was Mathematician and Computer Science (dual degree undergraduate) ...
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1answer
84 views
Definition of Omega Jump [closed]
In the book Computability theory (Rebecca Weber) I stumbled about Exercise 7.1.24 with the definition of the omega jump.
The book says:
The omega jump of $A$, $A^{(\omega)}$ is the join of all $A^{(n)}...
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0answers
32 views
Non Turing-Complete Models, Conditional-Complete function?
I know well the distinction between the class of Partial Recursive Functions, and $\mu$-Recursive, i.e. the latter is Turing Complete and the former is equivalent to the LOOP-Program model of ...
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1answer
105 views
Is this a good definition of computability? [closed]
I still haven't found a good definition of computability. All the definitions are either too vague, or they delegate the definition to another loaded term like "anything that uses math to solve a ...
1
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1answer
151 views
Extended Church's thesis and internal parametricity
I am wondering if there is any known relationship between these 2 concepts in intensional MLTT as formulated here.
Does $Internal\ parametricity \implies ECT$ hold?
For forumlation of ECT see https://...
0
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2answers
171 views
Is the decidability of a language decidable? [closed]
Is there a Turing machine that takes a language as input and decides/semi-decides if it is a decidable language?
Comments + answer say trivially the answer is yes; however, I'm wondering here would ...
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2answers
153 views
Turing meta-oracle
Let H(P) be some program that given P('s source code) computes whether or not P terminates, i.e. solves the halting problem. H only needs to terminate if P terminates. (This disallows solutions like ...
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0answers
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Is there an alternative technique to define computable functions other than their definition on natural numbers? [closed]
I am not completely sure that why computable functions is defined as a subset of all functions from/to Natural Numbers, but as far as I know, computation comes from real (physical) world so ...
5
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0answers
81 views
Are there languages require many variables to achieve $\Sigma_n^0$ completeness?
The proof of Post's Theorem that I am familiar with assumes you have access to as many variables as you wish in your language. Matiyasevich's Theorem by contrast gives a $\Sigma_n^0$-complete formula ...
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1answer
128 views
Gödel-Numbering of the Context-Sensitive Languages
I would like to have a Gödel-numbering of the context-sensitive languages. Because there is no obvious syntactic distinction between LBAs and TMs, I cannot number the former in an immediate way. So I ...
7
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1answer
267 views
Uniform mortality problem for Turing Machines
Consider the following generalisation of the mortality problem for Turing Machines.
Given a Turing Machine $M$. Is there a bound $k_M$ such that starting
from any configuration $c$ machine $M$ ...
3
votes
2answers
185 views
Compactness of domino tilings
I've read in Lemma 2 of the paper 1 that if every square region of the plane admits a tiling, then the whole plain admits a tiling, but the proof is omitted. This sounds like a compactness property, ...
2
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0answers
83 views
how is time complexity defined in computational learning theory
In general, when we say an algorithm $A$ PAC learns $C$ in time $t$, we say $A$ takes time $t$ before outputting a hypothesis $h$, and the hypothesis can be evaluated (on every $x$) in time $t$.
Now ...
0
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1answer
165 views
Why does the Placid Platypus function grow faster than any computable function?
I came across the Placid Platypus function $PP(n)$ today, defined as the minimal number of states needed for a turing machine that prints a string of $n$ ones and halts. This function is claimed to (...
6
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1answer
319 views
Is a binary sequence computable iff the Kolmogorov complexity of its initial segments is bounded?
Disclaimer: I am mostly unfamiliar with theoretical computer science, making it hard for me to navigate literature in the field. I ask the following out of curiosity.
Background/Motivation: Coming ...
5
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1answer
183 views
Minimal information needed for determine some function
From calculus, we know that if someone has a continuous function $f$, it is enough to know $f$'s values on the rationals in order to know $f$ on the entire line. In some sense, a "countable amount of ...
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0answers
159 views
Solving the Halting problem for most inputs [closed]
Is it possible to solve the following version of the Halting problem : given any Turing machine and some input tape, the program should answer if this pair halts or not except possibly for one Turing ...
11
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0answers
177 views
Can two-tape read-only Turing machines recognize any recursive language?
Suppose that a $k$-tape read-only Turing machine receives its input on each $k$ tapes.
It cannot write on the tapes, but it can move on them in both ways, even move off from the input.
So for example, ...
6
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2answers
287 views
Algorithm for identifying unprovable statements
I understand that this may depend on the specific set of axioms, but is there a general way (algorithm) for automatically detecting unprovable statements within a set of axioms?
For example: If there ...
2
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1answer
179 views
Is there a difference between incompleteness and unknowable? [closed]
Godel proved that there are statements that are true but not provable. Unproven conjectures such "Twin Primes" might fall into such a class, as I understand. If so, does it mean that we will never ...
6
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3answers
257 views
Why/when do we ever need transfinite loop variants?
I do not understand why one would ever need a transfinite loop variant.
Why do natural-number-valued variants not suffice? My simple (but perhaps too naive) argument is: if a loop $L$ terminates ...
5
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1answer
303 views
Is there a useful notion of being “approximately computable”
It seems that we can define a notion of being “approximately computable” where a set, $S$, is approximately computable if there is a family of computable functions $f_n(x)$ such that $$\lim_{n\to\...
11
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1answer
563 views
Non-comparable natural numbers
The "name the biggest number game" asks two players to write down a number secretly, and the winner is the person who wrote down the larger number. The game commonly allows players to write down ...
5
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1answer
213 views
Complexity of type-checking in relation to complexity of normalization
In order to verify that a terminating program terminates, one thing that can be done is to actually run the program. That may take a lot of time. If the program is typed in a total type-system, we can ...
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2answers
968 views
Are these problems in NP class?
${\bf New\ version}$ [Version 1.2]
Let $f: \mathbb{N} \to \{0,1\}$ be a computable function, ${\bf Fin}(\mathbb{Z})$ be the set of all finite subsets of $\mathbb{Z}$, and $W: {\bf Fin}(\mathbb{Z}) \...
6
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2answers
837 views
a polynomial representation of boolean functions
I came up with this linear transformation to map boolean functions to polynomials and it seems to have some nice properties. I was wondering if there is any reference describing this (and/or similar) ...
1
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2answers
256 views
Is true randomness and the physical Church-Turing thesis incompatible?
As follow up to Does the physical Church-Turing thesis imply that all physical constants are computable?, I ask if true randomness (as predicted by QM) and the physical Church-Turing thesis are ...
3
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2answers
368 views
Does the physical Church-Turing thesis imply that all physical constants are computable?
The physical Church Turing thesis is a conjecture that any physically computable algorithm can be computed by a Turing machine.
Let us create a machine that, for example, outputs the digits of the ...
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votes
1answer
142 views
Are there proofs which show that AIs are bound to be “worse” than the human brain? [closed]
I was talking with two PhDs (both teach IT related subjects) about artificial intelligence the other day. They were in agreement that an AI can never reach the level of the human brain, but failed to ...
4
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2answers
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Enumerating decidable languages
[The assumption in this question is wrong. It is possible to enumerate exactly the decidable languages with semideciders.]
Lets say we have a TM $M_E$ enumerator that writes out codes of TM's on a ...
10
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4answers
539 views
Is there a notion of computability on sets other than the natural numbers?
Is there a notion of computability on sets other than the natural numbers? For the sake of argument, let's say on sets $S$ that biject with $\mathbb{N}$.
It's tempting to say "yes, they are ...
3
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0answers
32 views
references for optimal computation under memory constraint?
Can someone help me find some references for finding good execution schedule given memory constraint? Assuming computation graph is simple in some sense (ie, small tree-width)
There is this reference ...
8
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4answers
492 views
Impossibility in Computability and Complexity: always ultimately due to diagonal arguments?
In Computability, if we want to prove that a problem is not recursive or not recursively enumerable, we can use e.g. reductions from other non-recursive or non-r.e. problems, Rice's theorem, Rice-...
6
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1answer
1k views
Show that minimal CFG is undecidable via mapping reduction
Actually the problem below is an exercise in Sipser's textbook (Problem 5.36). However, from my perspective, it is not so trivial so that I post this question on this site instead of CS.SE.
The ...
14
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0answers
283 views
Computability of a “weird” set
The starting point of this question is the observation that the smallest positive integers $a,b,c$ satisfying
$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$$
are absurdly high. This leads to ...
