Questions tagged [mu-recursion]
11 questions
4
votes
2 answers
Can Turing machines simulate the unbounded minization operator applied to a partial function?
I am a little bit confused with the unbounded minimization ($\mu$ operator of the $\mu$ recursive functions).
The $\mu$ operator is $\mu(f)(x) = \min(n | f(x, n) > 0)$ and the operator is said safe if we only apply it to functions such that for…
mouton5000
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3
votes
3 answers
Are partial recursive functions analogous to recursive languages or r.e. languages?
From Ullman and Hopcroft's Introduction to Automata Theory, Language, and Computation 1ed 1979:
The assumption that the intuitive notion of "computable function"
can be identified with the class of partial recursive functions is
known as…
Tim
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3
votes
0 answers
Prove that variable projection is recursive
Let $\varphi:\mathbb{N}\to\mathbb{N}^*$ be an arbitrary recursive enumeration of finite strings and
$\mathcal{I}^n_i(x_1,...,x_n) = x_i $
be the $i$-th projection over $n$ variables.
I would like to show from the point of view of recursion theory…
Manlio
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3
votes
1 answer
What does the exact $\mu$-recursive program for minimization look like?
The minimization of a given primitive recursive function $f$ is computed by the following expression:
$
\newcommand{\pr}[2]{\text{pr}^{#1}_{#2}}
\newcommand{\gpr}{\text{Pr}}
\newcommand{\sig}{\text{sgn}}
\text{Mn}[f] = \gpr[g, h] \circ (\pr{1}{1},…
lo tolmencre
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3
votes
2 answers
Undefined behaviour when composing primitive-recursive with $\mu$-recursive functions?
It is quite easy to show that the following two functions are primitive recursive and thus also $\mu$-recursive:
$$ifthen(n,a,b) = \begin{cases}a & n > 0 \\ b & else\end{cases} $$
$$ eq(x,y) = \begin{cases} 1 & x = y \\ 0 & else\end{cases} $$
Now,…
Corristo
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3
votes
2 answers
Intuition for Church-Turing thesis for Turing machines
I can very clearly see "why" mu-recursion is a universal model of computation, i.e. why the Church-Turing thesis -- that any physically computable algorithm can be executed with mu-recursion -- holds for mu-recursion. It reflects exactly the type of…
Abhimanyu Pallavi Sudhir
- 151
- 3
2
votes
3 answers
Does the normal form theorem imply that every partially computabe function is primitive recursive?
This is Normal Form Theorem (Second Edition of Computability, Complexity, and Languages written by Martin Davis page 75):
Let $f(x_1,...,x_n)$ be a partially computable function. Then there is a primitive recursive predicate $R(x_1,...,x_n,y)$ such…
M a m a D
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1
vote
0 answers
Predecessor function with recursive types
I am defining the type Nat of natural numbers a recursive sum type:
$$ Nat = \mu X. Unit \oplus X$$
Now, I have defined zero as the term:
zero : Nat
zero = fold Nat (Inl tmUnit tyNat tyUnit)
and successor as
succ : Nat -> Nat
succ n = fold Nat…
Noel Arteche
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0
votes
1 answer
prove that $h(x)$ is partial recursive
If $f:A^*\rightarrow A*$ and $g:A^* \rightarrow A^*$ are partial recursive we want to prove $h:A^* \rightarrow A^*$ with the following definition is partial recursive
$$ h(x) =
\begin{cases}
\lambda & f(x) = \lambda\\
g(x) &…
Karo
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0
votes
1 answer
Does it matter for this function if the set we check membership of is finite?
I have the following problem.
Let $\Phi$ be an admissible numbering of the single-parameter partially-recursive functions. That is, $\Phi(i, x) = f_i(x)$ with $f_i$ the $i$th partially-recusive function with one parameter. Denote with…
David Hamide
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0
votes
2 answers
Is the function that computes the minimum of a countable set computable?
Given $A$ a countable set of numbers and $\min$ the function returning the minimum of a set (if exists).
Is $\min(A)$ computable? My first try is thinking $A$ as infinite list $A = [a_0, a_1, a_2,...]$ such that $a_i < a_j$ for every $i < j$ and…
Saber98
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