# Questions tagged [primitive-recursion]

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### Are there situations where we can decrease the time complexity of a program by increasing its ordinal complexity?

Are there (interesting) situations where we can decrease the time complexity of a program by increasing its ordinal complexity? For example, is it possible to find a primitive recursive function such ...
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### Why is this function primitive recursive?

Let $f:\mathbb{N}^{p+1} \to \mathbb{N}$ a primitive recursive function and $g:\mathbb{N}^{p+1} \to \mathbb{N}$ the bounded sum defined by : $g(\bar{a},x)=\sum\limits_{i=0}^x f(\bar a , i)$. To show ...
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### How does primitive recursion handle mutual recursion?

My intuition is that you can't call a function that has not yet been defined, although I have yet to find a source confirming this. Is this true? Thanks, friends :)
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### What can be proven regarding the differences in power between unary ECMAScript regex functions and primitive recursive functions?

In 2014, inspired by Regex Golf, I started exploring, along with a mathematician going by the name teukon, what could be done in the unary domain in ECMAScript regex that went significantly beyond ...
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### Showing that the quotient function is primitive recursive

I'm asked to show that the quotient function is primitive recursive. I know that the operation of integer division $div$ is not total, as it is not defined when the denominator is zero, and a ...
1 vote
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### What is known about the sets enumerated by primitive recursive functions?

Let's say that a set of natural numbers $S \subseteq \mathbb{N}$ is primitive recursively enumerable if there exists some primitive recursive function $f$ such that $S$ is the range of $f$. That is, ...
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### Computability: Proving a predicate is not recursively enumerable

Let P(p) <=> for each x, comp(p,x) is defined. Can anyone explain to me how to prove that P is not RE (recursively enumerable) ?
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### Determining whether Turing machine halts on input: primitive recursive?

In Elements of the Theory of Computation by Lewis and Papadimitriou, the authors use a specific function for proving that application of unbounded minimization on a primitive recursive function need ...
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### Quotient in LOOP program [closed]

I want to construct a LOOP-computable program for the integer division (quotient): x = a DIV b The LOOP specification can be seen here: https://en.wikipedia.org/wiki/LOOP_(programming_language) I ...
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### Primitive recursive plus Ackermann

Let us consider the class $\cal F$ of functions that contains all constant functions all projections the successor function the Ackermann function as basic functions, and that is closed under ...
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### What is the definition of computable partial function?

Let $f:\mathbb{N} \to \mathbb{N}$ be a computable partial functions and $T$ a Turing Machine which computes it. By this I understand that $T$ writes $f(n)$ on its tape and halts when $n$ is an input ...
1 vote
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### How to prove that if the set and its complement are recursively enumerable, then both are recursive?

How to prove that if the set and its complement are recursively enumerable, then this set and its complement are recursive? My idea is that we can make the characteristic function of recursively ...
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### The evolution of the term "recursive" from Goedel to Church to present day

I'm currently studying some of the history of computation / computability, in the early days known as recursion theory. I see Goedel's definition of recursive functions seems significant in his paper,...
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### Is there a name for the class of functions whose totality can be proved using "Ackermann-like" reasoning?

Primitive recursion is recursion where totality can be proved because there is a single natural number parameter that strictly decreases in every recursive call. Put another way, the recursion ...
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### Is there a broader class of total functions than $PR$? [duplicate]

In total functional programming programs are restricted to total computable functions. A well-known class of total functions are the primitive recursive functions ($PR$). However the Ackermann ...
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### Are all functions with constant space complexity in $REG$?

The Wikipedia article about regular languages mentions that $DSPACE(O(1))$ is equal to $REG$. Can I conclude from this that every function in $R$ with constant space complexity is in $REG$?
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### Primitive Recursion equipped with an evaluator function

The wikipedia article for primitive recursion mentions a limitation that primitive recursive function can't compute the function $ev(i,j)$ which computes the $i$th primitive recursive function on ...
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### Showing that the number of primitive-recursion programs for each function is countably-infinite

Problem Statement Prove that if a function $f$ is primitive recursive, then there are countably infinite number of primitive recursive definitions of $f$ Yes, this is a homework question. My ...
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### How do I prove that all primitive recursive functions are computable?

I stumbled across an exam question, and I am not sure how to prove that that all primitive recursive functions are computable. Is there a formal definition of this?
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### Primitive Recursive Function for Division and Hailstone function

Are division and Hailstone primitive recursion function? $$\text{Div}(x,y) = \begin{cases} x/y, & \text{if y divides x } \\ 0, & \text{otherwise} \end{cases}$$ \text{Hailstone}(n) =\...