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Questions tagged [mu-calculus]

14 questions
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Why use $\mu$-calculus and not LTL,CTL,CTL*?

It is known that the temporal logics LTL,CTL,CTL* can be translated/embedded into the $\mu$-calculus. In other words, the (modal) $\mu$-calculus subsumes these logics, (i.e. it is more expressive.) Could you please explain/point me to papers/books…
Dimiter
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Why do we have free variables in mu-calculus?

I'm reading the Model Chekcing book of E. M. Clarke and in the chapter about µ-calculus there is an example formula $$ f = \mu Z. ((q \mathrel{\mathrm{AND}} Y) \mathrel{\mathrm{OR}} \langle a \rangle Z) $$ As the author has mentioned, here $Y$ is a…
Y. Tang
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CTL and the modal mu-calculus

Is there any literature / results to which fragment of the modal mu-calculus the logic CTL corresponds? I know that in order to tranlsate the until operators of CTL, it suffices to use a single mu-operator. Does this imply that CTL coincides with…
Riddle
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Have I got the right understanding of the mu operator?

I have a homework problem that says: For $g(x,y)=xy-5$ compute $h(x) = \mu y(g(x,y))$ and determine its domain. I was under the impression that this means the least y such that $g(x,y)=0$, so then $y = \frac{5}{x}, D=\{x \in \mathbb{N}, x \neq…
Stephen
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Intuitive meaning of modal $\mu$-calculus formula

I am solving one of the past exams and I am not certain with my solution to one of the exercises. The exercise is asking to give intuitive meaning to modal $\mu$-calculus formula: $$ \phi = \mu Z. \langle - \rangle tt \wedge [-a]Z $$ According to an…
Martin Jonáš
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What does least fix point and greatest fix point mean in Safety games

In safety games there are these mathematical notation about greatest fix points and least fix points but I don't get it. How would we describe them plain English without mathematical symbols.
kaitlynrutledge
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Using the μ (mu) operator

Problem I've got this function: $f(x,y)=(6-3\cdot x)\cdot(y+2)$, with $(x,y)\in\mathbb{N}^2$ Now I have to find $g=\mu f$. Proposed solution My solution was to find the smallest $n\in\mathbb{N}$ to find $f(n,y)=0$ and show that $\forall 0\leq m\leq…
polym
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Using diagonal argument to prove that $H(x) = \mu y T(x,x,y)$ has no total computable extension

Hello everyone just like the title says I want to prove that $H(x) = \mu y T(x,x,y)$ has no total computable extension such that if we had a function $BIG(x)$ that is both total and agrees with $H(x)$ whenever $H(x)$ is defined, then $BIG(x)$ is not…
InsigMath
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The minimization operator is an effective operator

Assume $\{f_i^{(n)}\}_{i=0}^\infty$ is a Gödel enumeration of the $\mu$-recursive functions of $n$ arguments, such that the $S^m_n$ theorem and the universal function theorem hold. Denote the set of (total and partial) functions $\{f:\mathbb…
superAnnoyingUser
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Techniques (tools) to convert temporal logic (CTL,CTL* or LTL) to μ-calculus formulae

Suppose one wants to use a μ-calculus model checker, but specify things in temporal logics, which is easier (more intuitive). Is there a technique (even better, a tool) that automatically translates formulae in any of these logics (CTL,CTL* or LTL)…
Dimiter
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Generalization of computability to continuous for loops?

A computable function, formulated in the sense of mu recursion, can compute a for or do loop over some (possibly infinite) integer range. I was wondering if a suitable generalization exists that allows the computation of a real-domain function $f$…
Abhimanyu Pallavi Sudhir
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Why is $\mu$ finite looping?

It is said that the intuitive meaning of $\mu$ is finite looping where as the intuitive meaning of $\nu$ is infinite looping in $\mu$ calculus. I understand this for finite systems, but why is this true in general? Is there any theorem which proves…
e_noether
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Is there a CTL* formula that translates into mu-calculus formula Y.νZ.(...) with alternation depths 2,1 for Y,Z?

A CTL* formula EFG p is equivalent to mu-calculus formula Y.(<>Y | νZ.(<>Z & p)). In this formula, the alternation depths are ad(Y)=ad(Z)=1. Is there a CTL* formula that translates into YνZ(...) with alternation depths 2,1 for Y,Z? And a similar…
Ayrat
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Understanding the reason behind the μ (mu) operator

What is the purpose of the $\mu$ operator? Is there a real world example? Is it correct that it can create partial functions out of total functions and it makes a function $g$ with k parameters out of a function $f$ with $k+1$ parameters?
polym
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