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Questions tagged [computability]

Computability theory a.k.a. recursion theory.

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17
votes
2answers
985 views

Smallest possible universal combinator

I am looking for the smallest possible universal combinator, measured by the number of abstractions and applications required to specify such a combinator in the lambda calculus. Examples of universal ...
1
vote
0answers
42 views

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
votes
0answers
75 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 ...
-1
votes
1answer
120 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 ...
10
votes
4answers
425 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 those ...
11
votes
1answer
449 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 ...
6
votes
1answer
219 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 $M$ ...
2
votes
1answer
449 views

Langton's ant questions

I'm a mathematician currently working on the Langton's ant conjecture, just for fun. I have some result but I don't know if they are meaningless. So that is why I'm asking. 1) Is there a mathematical ...
37
votes
7answers
6k views

What do we know about provably correct programs?

The ever increasing complexity of computer programs and the increasingly crucial position computers have in our society leaves me wondering why we still don't collectively use programming languages in ...
3
votes
2answers
152 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, ...
9
votes
3answers
427 views

Defining primitive recursive functions over general data types

The primitive recursive functions are defined over the natural numbers. However, it seems as if the concept should generalise to other data types, allowing one to talk about primitive recursive ...
6
votes
1answer
358 views

Is there any research on Turing machines with transition relation homomorphic to given algebraic structure?

A Turing machine is defined as a structure $ TM(L,Q,T) $, where $L,Q$ are sets of symbols and internal states of TM respectively, and T is a transition relation: $T: L \times Q \to L \times Q $ for ...
2
votes
0answers
70 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 ...
-1
votes
1answer
2k views

Decidability of the halting problem on finite computers [closed]

I've seen two competing and contrary arguments for this problem. One states that real computers are linear-bounded automata, and therefore the halting problem is decidable. The other states that ...
-1
votes
1answer
127 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
votes
1answer
261 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 ...
-7
votes
2answers
578 views

Validity implies NP=#P? [closed]

Valid progams for NP imply every solution is a valid answer. NP not equals #P implies not all solutions are answers. Therefore, Validity implies NP=#P. NP is the problem class for ...
5
votes
1answer
167 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 ...
15
votes
2answers
3k views

A total language that only a Turing complete language can interpret

Any language which is not Turing complete can not write an interpreter for it self. I have no clue where I read that but I have seen it used a number of times. It seems like this gives rise to a kind ...
1
vote
0answers
131 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 ...
5
votes
3answers
628 views

Isolation in Turing-complete reversible cellular automata

I don't know much about the terminology and the results on cellular automata, but I would like to ask a question about an conjecture I thought. Consider Turing-complete reversible cellular automata. ...
11
votes
0answers
136 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, ...
83
votes
10answers
14k views

What would it mean to disprove Church-Turing thesis?

Sorry for the catchy title. I want to understand, what should one have to do to disprove the Church-Turing thesis? Somewhere I read it's mathematically impossible to do it! Why? Turing, Rosser etc ...
11
votes
1answer
1k views

Given a PDA M such that L(M) is in DCFL construct a DPDA N such that L(N) = L(M)

Is it possible to construct an algorithm which takes as input a pushdown automaton $M$ along with the promise that the language accepted by this automaton $L(M)$ is a deterministic context-free ...
1
vote
1answer
188 views

Is there any research on approximation of reals with computable numbers

I was wondering if there is any research in the field of Diophantine Approximation using the computable numbers. It seems to be a good fit, a dense countable set with a variety of different potential ...
9
votes
1answer
304 views

Is the Kolmogorov complexity of the truth tables of the halting problem known asymptotically?

Let $HALT_n$ denote the string of length $2^n$ corresponding to the truth table of the halting problem for inputs of length $n$. If the sequence of Kolmogorov complexities $K(HALT_n)$ were $O(1)$, ...
6
votes
2answers
262 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
votes
1answer
168 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
votes
3answers
209 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
votes
1answer
277 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\...
4
votes
1answer
122 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 ...
6
votes
3answers
12k views

What is the best text of computation theory/theory of computation?

In University we used the Sipser text and while at the time I understood most of it, I forgot most of it as well, so it of course didn't leave all to great of an impression. I borrowed that book and ...
2
votes
3answers
114 views

Reference request: Arithmetic circuit complexity

I am completely new to this field. I want to read the prelims and unfortunately, I don't see any book written for a beginner in this area. Can anyone give me some basic starting points/references(text/...
-1
votes
2answers
485 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}) \...
48
votes
3answers
2k views

Is there a sensible notion of an approximation algorithm for an undecidable problem?

Certain problems are known to be undecidable, but it is nevertheless possible to make some progress on solving them. For example, the halting problem is undecidable, but practical progress can be ...
6
votes
2answers
587 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) ...
6
votes
2answers
3k views

Was Babbage's Analytical Engine really turing-complete?

According to literature, Babbage's Analytical Engine is turing-complete because it supports conditional branching: it can perform different operations depending on the sign of the result last ...
3
votes
2answers
326 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 ...
1
vote
2answers
201 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 ...
2
votes
2answers
254 views

How to translate general recursion into a set of $\mu$-recursive operator applications?

I'm trying to find a scheme to translate a functional language with let rec into a set of primitives called "generalized arrows", i.e. $\kappa$-calculus with ...
-5
votes
1answer
123 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
votes
2answers
659 views

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 ...
3
votes
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
votes
4answers
457 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-...
4
votes
1answer
466 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 ...
12
votes
0answers
242 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 ...
10
votes
1answer
216 views

Equilibrium in a Halting Game

Consider the following 2-player game: Nature randomly picks a program Each player plays a number in [0, infinity] inclusive in response to nature's move Take the minimum of the players’ numbers, and ...
-1
votes
1answer
88 views

What is the relation between P-immune languages and NP-complete languages? [closed]

Can a NP-complete language be P-immune? Why can't existence of P-immune languages separate NP from P?
1
vote
1answer
109 views

Set of languages that can represent every c. e. languange

Could we find any set of languages $S$, such that it can represent every c. e. languange as it's union, intersection, complement, production(times of element ), and $S\subset X$, where $X\subseteq c.e....
3
votes
1answer
67 views

Can any c.e. language with infinite words be decomposed into infinite CFLs with infinite words?

Suppose $L$ is a computably enumerable language, can it be decompose into infinite CFLs with infinite words ? $$L=\bigcup_{L_i\in CFL }^{\infty}L_i$$ Second question: if it is possible that every $L$...