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I was creating an example for a casual talk on mutual information. I considered a system of two coins, which with probability 1/2 are copies of each other, and with probability 1/2 are independent.
Then $P(A=B)=1/2+1/2\times 1/2=3/4$, so $P(HH)=P(TT)=3/8$ and $P(HT)=P(TH)=1/8$.
Total information is then $E(-\log_2 p) =... |
I am trying to generate a graph (the structure with edges and nodes), that as a structure like an Order-7 triangular tiling of specified diameter around a central node.
http://en.wikipedia.org/wiki/Order-7_triangular_tiling
!http://en.wikipedia.org/wiki/Order-7_triangular_tiling#mediaviewer/File:H2_tiling_237-4.png
In ... |
I'm just starting my second year in computer science and one of my classes briefly touched upon deterministic vs. non-deterministic algorithms. This got me thinking - is there any use for algorithms which return deterministic output for certain inputs, but non-deterministic output for other inputs? Say, something along... |
There's a class of data structures where you preprocess some other data structure in order to answer queries about it more efficiently.
That includes trivial things like sorting/hashing, less trivial things like quad trees, and includes far less trivial things such as dominator trees, interval decompositions of DAGs or... |
We define the language
$$
L = \{a^nb^n : n\geq0 \}
$$
and we want to prove the following
$$
L = \mathrm{DSPACE}(\log n)\,.
$$
So we have to prove that by using $\log n$ space on the work tape of a Turing machine we can decide this language.
In this case, do we have to describe the delta function in detail, or can we j... |
Is there any efficient algorithm to convert a given truth table of a Boolean function to its equivalent algebraic normal form (ANF)?
I have seen that Sage has one implementation (official documentation):
sage: from sage.crypto.boolean_function import BooleanFunction
sage: B = BooleanFunction("12fe342a")
sage: B.algeb... |
Languages such as $\text{HALT}_{TM}$ are $\textsf{RE-complete}$ under many-one reductions. It is trivial to see that $\text{co-RE}$ has complete problems, too. S. Schmitz [1] considers some classes inbetween $\text{ELEM}$ and $\text{REC}$. They present complete problems for these classes under specifically crafted red... |
How to I show that any language in NTIME(T(n)) can be accepted by a non-deterministic 2-tape O(T(n)) time-bounded Turing machine, with modifiable input tape?
I've seen the k to 2 tape reduction for deterministic Turing machines, which increase the running time by a logarithmic factor, but I'm not sure if I can modify t... |
I've been having trouble proving the next statement:
Let $L_n=\{ww, |w|=n\}$ (the set of equal-length palindromes (EPAL) restricted to length $2n$). Prove that $L^c_n$ can be accepted by an NFA with at most $O(n^2)$ states.
What I've been trying to do is that the complement of $L_n$ is
$\qquad\displaystyle L^c_n = \... |
The Turing model of computing is widely accepted, as a Tape, Head, State register, and a finite state Table to manage transitions. The Bulk Synchronous Parallel Machine model has for its part similarity to the PRAM model, but with additional features in state management, by communication, and synchronization. So, in a ... |
The word choice of "complexity" in analysis of algorithms to describe temporal and spatial resource requirements has always struct me as an odd one. These are certainly useful and meaningful concepts. This question is concerned with the language used to describe them.
Colloquially, one would probably not say that inser... |
Or in other words, find all $v \in V$ such that there exists a path $\forall w \in V$ $v \rightarrow w$ or $w \rightarrow v$. This is for a directed acyclic graph. I need to find an $O(|E| + |V|)$ algorithm for this.
I can see how to identify if a given vertex meets these traits (perform a BFS starting at that vertex,... |
The book I'm currently reading specifies that Dynamic Power = (1/2) * Capacitive Load * (V^2) * frequency However, that is assuming it's a single-core system. I wonder if we're calculating the Dynamic Power of a dual-core system, would that be twice the formula of Dynamic Power above ?
|
Suppose we work with some particular programming language (like C++) on some particular computer. Furthermore, we want to know which values are minimum and maximum for some particular numeric data type of this language. We can't use any specific tools of our language (like numeric_limits), and have no access to languag... |
I am reading the book Structured Computer Organization (5th edition) by Tanenbaum and at a certain point, in the second chapter, he talks about data path cycle and fetch-decode-execute cycle, and I am not understanding if they are really the same thing.
The data path cycle should be the number of small and minimal step... |
I understand the notion of crossing sequences when talking about time, however how are they used to actually prove strict lower bounds for some decision/search problems?
For example, suppose that you have an array of size N and you want to find the maximum number in it.
Speaking informally, we need at least a pointer t... |
I'm still fighting with the aforementioned paper "Improved upper bounds for vertex cover" by Chen, Kanj, Xia (PDF kindly provided by Yuval Filmus).
My current problem is that it's specified that the main vertex cover algorithm VC (page 7/21) has the following header VC(G, T, k), where G is the graph to process, T is a ... |
Assume that the nodes know that the topology G is either a hypercube or a
tree. Assuming a unique initiator, design an algorithm to discover the topology. In other words, you would like a node to reach terminal state TREE if G is a tree, terminal state HYPERCUBE if instead it is a hypercube. Let n be the number of node... |
Prove that any binary tree with $n$ nodes has at least $1+\log_2 n$ levels.
I tried setting $n=8$ and plugging in $8\geq\log_2 8 = 8\geq 3$. But I'm not sure how I can prove this by induction.
|
Is the cardinality of the set of partially recursive functions greater than the cardinality of the set of recursive functions ?
|
I recently came across the concept of "forwarding pointer" and have not been able to find sources of a precise definition. In my case this was mentioned and briefly (but poorly) explained during a discussion on distributed systems.
Could you elaborate on the concept, its usual context(s) and possible other names by whi... |
Could anyone tell me from where could I get the c compiler source code.In particular the lexical analyzer for c.I searched the web but no links show how a compiler is defined.
|
My big question is the following: What is the meaning of a computation being "effectively computable" (EC) in theory and in practice?
In trying to understand these concepts further, I have a couple of sub-questions as well:
Is there a relationship between a computation being EC in theory and being EC in practice? (It... |
Nick's Class (NC) is the class of problems that can be decided in poly-log time using a polynomial number of processors.
I want to know about the exponential analogue, which would cover problems that can be decided in polynomial time using an exponential number of processors.
What I'm looking for is a name for this cla... |
We have a string $s$ containing $n \leq 100$ bits. The move we can make on it is erasing from $s$ some substring $x$, but only if $x$ is directly preceded by $x^R$, where $x^R$ means string $x$ reversed. In other words, we're choosing some even-length palindrome and erasing it's "right half".
The goal is to minimize th... |
show whether $n2^n = 2^{O(n)}$ is true or not.
In my opinion, it's false because O(n) can be n and thus the equality will be wrong, because $n2^n$ grows much faster than $2^n$
|
I am trying to understand Tarjan's strongly connected component algorithm and I have a few questions (the line numbers I am referring to are from Algoritmy.net):
On line 33 why is node.lowlink = min(node.lowlink, n.index) — shouldn't it be same as line 31: node.lowlink = min(node.lowlink, **n.lowlink**)?
Do we have to... |
When talking about Constraint Satisfaction Problems (CSP), what does the term extensional constraint mean?
|
This may be a silly question. It seem clear that an FSA, since it is finite, can only count the number of symbols in its input string up to a number bounded by the number of its states. But now suppose we equip the FSA with output (e.g. printing) capabilities. It would then be very easy to construct a machine capable o... |
In writing a decider for a machine to see if it has made a left move or not on an input of w, it is said that if we continue the computation for $|w|+N+1$ ($N$ : number of states) number of steps, we can make a decision for this decide.
I didn't get why $|w|+N+1$ number of steps. Could someone explain this in more deta... |
This may be basic to some of you, but excuse my inexperience with comp. geometry:
Given a set of $n$ circles with centers $(x_i, y_i)$ for $1 \leq i \leq n$ and each having radii $r$. Also given a rectangle. All objects are on a plane. How to verify that every point inside the rectangle (including its edges) is fully... |
Considering this FSM:
Can someone explain me why the grammar is not:
$S \rightarrow aA$
$A \rightarrow aA \mid bB \mid \varepsilon$
$B \rightarrow bB \mid \varepsilon$
Why does A not have a transition to $\varepsilon$?
|
When Savitch's famous theorem is stated, one often sees the requirement that $S(n)$ be space constructible (interestingly, it is omitted in Wikipedia). My simple question is: Why do we need this? I understand the requirement for $S(n)$ being in $\Omega(\log n)$, which is clear from the proof. But no proof I have seen s... |
I just learned PDAs in class today, but am having problems understanding the syntax of the transition function. Could someone please explain to me the meaning of this syntax:
$\delta(q, \lambda, S) = \{(q, aaB), (q, bbA)\}$
This is one of the rules for my language. I am unsure of what the meanings of this syntax exact... |
According to my understanding, a Turing machine that's valid has to have finite steps to finish a certain step. If this is right, what else determines the validity of a turing machine?
|
We are given an (unsorted) list $L=(a_1,\dots,a_n)$ of numbers of size $n$, where $a_i\in \{ 1,\dots,B\}$.
We want to find the minimum number $x$ from $\{ 1,\dots,B\} \backslash L$.
What is the space complexity of this problem ? (The space to store the input, $L$, does not count.) What if the input $L$ is in a strea... |
I have to solve the following problem:
Consider the problem Connected:
Input: An unweighted, undirected graph $G$.
Output: True if and only if $G$ is connected.
Show that Connected can be decided in polynomial time.
I have been at this for hours, and I can't seem to find a way to prove this.
Any hints?
|
I know that there exists a Turing Machine, if a function is computable. Then how to show that the function is not computable or there aren't any Turing Machine for that. Is there anything like a Pumping lemma?
Similarly, how can we show a language is not recursively enumerable?
|
Is there any set of rules or methods to convert any context free grammar to a push down automata?
I already found some slides online but I wasn't able to understand them.
In slide 10 he speaks about some rules could anyone explain that?
|
Please help me understand, and if possible, tips, to determine a pumping length $p$.
Suppose I have the example :
Let $G$ be a Context-Free-Grammar with a set of variables $\{S,A,B,C\}$, set of terminals $\{0,1\}$, start variable $S$, and rules
$S \to ABA \mid SS$
$A \to S0 \mid 1C1$
$B \to S1 \mid 0$
$C \to 0$
Now g... |
I am a little bit confused by the definition of dominant strategy and winning strategy, what's the difference between them.
Following are the definitions taken from wikipedia, below every definition I added some thoughts. If you fell that my thoughts are not correct or ambigues please let me know.
Dominant strategy.In... |
PIC16F887 Block Diagram
According to the block diagram above, since we already have Program Memory, which may be used to store our program, why should we still need EEPROM? What is it for?
|
Is there a formal definition for a size of a context-free grammar? The only definition I have seen so far is on this wiki page:
The size of a grammar is the sum of the sizes of its production rules, where the size of a rule is one plus the length of its right-hand side.
So, how does one define the size of a producti... |
If there is a Turing Machine $M_e$ (computing some partially computable function $f$), is there an algorithm to decide if $f$ diverges for all possible inputs?
|
Why did the sets of languages accepted/decided by a TM get the name Recursively Enumerable and Recursive, respectively?
|
How would I go about using Cartesian construction to find the intersection between a PDA and a DFA?
Is there another term for Cartesian construction? Would it be similar to the procedure for finding the intersection/union of 2 DFAs?
|
I've been reading about hypercube connection template for parallel algorithms. The general scheme is explained in Designing and Building Parallel Programs by Ian Foster and it's pretty clear.
What I don't understand is how it's applied on the merge sort in §11.4 The point I'm most interested is the parallel_merge funct... |
If I am given a graph which forms a tree, I am interested in finding a vertex which maximizes the minimum distance to any leaf.
I am sure this problem has been studied before.
Does anybody know the name of this problem or an algorithm for solving it?
|
In computability and complexity theory (and maybe other fields), reductions are ubiquitous. There are many kinds, but the principle remains the same: show that one problem $L_1$ is at least as hard as some other problem $L_2$ by mapping instances from $L_2$ to solution-equivalent ones in $L_1$. Essentially, we show tha... |
The Kolmogorov complexity of a string $x$ is the size of the smallest Turing machine $M$ that started on empty tape produces $x$. To make it computable, we can add a bound on the time used by $M$ to produce $x$:
$C^{t}(x) = \min \{|M| : U(M) = x$ in less than $t(n)$ steps $ n = |x| \}$
And for a nice function $f(n) <... |
Why is the laplace transform not popular for image processing convolution? Most textbooks only conver the Fourier transforms.
|
From that question about differences between Quantum annealing and simulated annealing, we found (in commets to answer) that physical implementation of quantum annealing is exists (D-Wave quantum computers).
Can anyone explain that algorithm in terms of quantum gates and quantum algorithms, or in physical terms (a part... |
I have laid out the various cases that would make this not a context free language already and proved all but one for this set:
\begin{equation}
A = \{a^f b^g \mid f = g^2\}
\end{equation}
I am having trouble trying to explain how if v was a's and x was b's for this problem. Exactly how would I prove that us... |
In one of the text book its mentioned that 'running time of this algorithm is 200 computer years'. Can somebody please explain what is the meaning of a computer year?
|
I'm having a hard time understanding the actual proof of this proposition:
$\qquad \mathsf{NP} \subseteq \mathsf{P}/\log \implies \mathsf{P} = \mathsf{NP}$
The sketch of the proof is on slides 6-8 of this PDF.
So I let $L \in \mathsf{NP}$. That means that there's a deterministic polynomial-time TM $M^1_L$ s.t. $x \in L... |
My teacher describes true quantified boolean formula (TQBF) as an interactive game between two players $\exists$ and $\forall$, and asks us to show a winning strategy for the existential player $\exists$. Can someone describe what this game is? I feel like this is background knowledge I am missing.
|
A bridge (critical edge) in an undirected graph is an edge whose removal increases the number of connected components.
I need to determine all critical edges in an undirected graph, in $O(V+E)$ time. From what I found out, I need to use a modified DF search, but all pseudo-code algorithms I found have low[v] and d[v] w... |
I am reading Operating Systems book by Galvin. Galvin explains, what are kernel & user modes, instruction privileges given for both modes & also about mode-bit. But I am interested to know how mode changes from one to another. Basically I want to solve following question :
A CPU has 2 modes, privileged and non-privile... |
I have been reading up on algorithm for finding the strongly connected components in a directed graph $G=(V,E)$. It considers two DFS search and the second step is transposing the original graph $G^T$.
The algorithm is the following :
Execute DFS on $G$ (starting at an arbitrary starting vertex), keeping track of the... |
I'm looking for something i would call rating functions.
I'm searching for some literature about this concept.
I'm not really sure about the terminology, but what I mean should be pretty obvious.
A type of function that returns a rating of some input.
Lets have a function that gets some input and returns a number betwe... |
Is the language $ L = \{0^n 1^m \mid n \text{ and } m \text{ are co-prime}\}$ context-free ?
I guess that it's not context free because it seems too complicated for a PDA to decided whether 2 numbers are co-prime or not.
I tried using the pumping lemma to no avail.
Any help would be gladly appreciated.
Edit:
Here is on... |
I have this problem:
Let $L_1$ and $L_2$ be two regular languages. Show that $L_3 = \{xx^r : x \in L_1, x^r \in L_2 \}$ is a context-free language.
I am unsure how to prove that some language is context-free. Could someone please provide the steps?
|
Traditionally, linear programming is used to find the one optimal solution to a set of constraints, variables and a goal (all described as linear relationships). Sometimes, when the objective is parallel to a constraint, there are infinite or many equally good optimal solutions. I am not asking about this latter case.
... |
I have a problem which I suspect is NP-complete. It is easy to prove that it is NP. My current train of thought revolves around using a reduction from knapsack but it would result in instances of 0-1-Knapsack with the value of every item being equal to its weight.
Is this still NP-complete? Or am I missing something?
|
In an exercise I have to show that minimizing a multivariate polynomial with $n$ variables over the hyper-cube $H = \{ (x_1, \ldots, x_n) : 0 \leq x_i \leq 1 \}$ is NP-Hard. Formally, given $p(x_1, \ldots, x_n)$ and $\alpha$, does $\min_{0 \leq x_i \leq 1} p(x_1, \ldots, x_n) \leq \alpha$?
My idea is to reduce it to MA... |
I am looking for some sort of temporal logic for expressing invariants in interfaces. Since interfaces do not specify data representation, the invariants must rely solely on the publicly available functions of the interface.
For example, suppose we have a simple interface (Java style) Sum:
interface Sum {
public vo... |
It seems, from a quick (and incomprehensive) survey of the literature (e.g. N. Ailon, B. Chazelle, SIAM J. Comput. 39 (2009), 302-322.) that there are many fast algorithms (and constructions) for the Johnson-Lindenstrauss transform: $f \in \mathbb{R}^{d} \to \mathbb{R}^{k}$, for $n$ points in $\mathbb{R}^d$ and $k \in ... |
A program takes as input a balanced binary search tree with $n$ leaf nodes and computes the value of a function $g(x)$ for each node $x$. If the cost of computing $g(x)$ is
$\qquad \min(\#\text{leaves in } L(x), \#\text{leaves in } R(x))$
for $L(x), R(x)$ the left resp. right subtree of $x$, then the worst-case time c... |
I'm reading about MinHash technique to estimate the similarity between 2 sets: Given set A and B, h is the hash function and $h_\min(S)$ is the minimum hash of set S, i.e. $h_\min(S) = \min(h(s))$ for s in S. We have the equation:
$$
p(h_\min(A) = h_\min(B)) = \frac{|A \cap B|}{|A \cup B|}
$$
Which means the probabilit... |
The paper Eventually-Serializable Data Services (PODC'96) presented an eventually-serializable data service.
In the abstract, it says：
… and generalizes their algorithm (in the related work Lazy replication) by allowing general operations …
In section 1.1 (Background of our Work), it says:
… Also, their algorithm (... |
There is this standard algorithm for finding longest path in undirected trees using two depth-first searches:
Start DFS from a random vertex $v$ and find the farthest vertex from it; say it is $v'$.
Now start a DFS from $v'$ to find the vertex farthest from it. This path is the longest path in the graph.
The questio... |
Given a set of n jobs with [start time, end time, cost] find a subset so that no 2 jobs overlap and the cost is maximum.
Now I'm not sure if a greedy algorithm will do the trick. That is, sort by cost and always take the next job that doesn't intersect and with max cost between the two.
Is this equivalent to a knapsa... |
I want an algorithm that calculates which element, among two, appears more often than the other in a sorted array. The array will have only two types of elements.
Example : $aaaaaabbb$
Here $a>b$.
I have to find an constant time algorithm. Is it possible? The only thing I could come up with was using stack. Push all... |
Suppose you have some loop and and integer k:
int k = 5;
for (int i = 0 ; i < N; i++)
{
//(*)
//do something
}
The loop invariant at (*) is:
$\{ K=k\}$
Does that guarantee that $k$ doesn't change in between iterations? if not, is there any otherway to guarantee that?
|
So in what complexity class would all decision problems in that class can be transformed into a problem (by many-one reduction) that its solution only involves basic arithmetic operation of multiplication, division, addition and subtraction?
So, like this: for every problem, there is input and output (we consider only... |
Given a partial ordered set (poset) $S$, is there a known procedure or algorithm to find the set of chains (i.e. subsets of $S$ where every two elements are comparable)?
Note: I am asking here instead of math.SE because i'm looking for an algorithm for the problem.
|
I have a little problem to understand the proof of the Time Hierarchy Theorem (Hennie and Stearns, 1966) that ensures the existence of a language acceptable in $U(n)$ but not acceptable in $T(n)$ for any functions $T(n),U(n)$, such that $U(n)$ is time-constructible and
$$n \leq T(n) = o\left(\frac{U(n)}{\log T(n)}\righ... |
Given a graph $G = (V,E)$.
Is there any algorithm which finds the minimum number of vertices to be removed from $G$ so that every vertex in the graph becomes disjoint, i.e., every vertex is disconnected from every other vertex?
|
Note that I had asked this question in GIS forum, although it
has gotten many up-votes, still has not received any answer. Hope you can
break the silence, some collaboration :)
Consider a region (2D) filled with lines randomly (following Figure). We are interested in filling the empty spaces between lines includi... |
A TM for a recursive language corresponds to our informal notion of an algorithm.
as per Automata Theory, Languages and Computation by Ullman et al. Then there are languages called RE and $L_d$, where there exist TM that halts for accepting strings in the case of RE languages.
This means that we cannot construct a al... |
Let's say we a have flow network with $m$ edges and integer capacities.
Prove that there exists a sequence of at most $m$ augmenting paths that yield the maximum flow.
A good way to start thinking about this is to imagine that we know the maximum flow already. How can we figure the sequence of $m$ paths?
|
Basically, is the following statement true?
$A \leq_p B$ $\rightarrow$ $B \leq_p A$
|
Given an unweighted DAG (directed acyclic graph) $D = (V,A)$ and two vertices $s$ and $t$, is it possible to find the shortest and longest path from $s$ to $t$ in polynomial time? Path lengths are measured by the number of edges.
I am interested in finding the range of possible path lengths in polynomial time.
Ps., th... |
You have an array of $n$ distinct elements. You have access to a comparator (a black box function taking two elements $a$ and $b$ and returning true iff $a < b$) and a truly random source of bits (a black box function taking no arguments and returning an independently uniformly random bit). Consider the following two t... |
If there is an interrupt from every keypress (or one for keydown, one for keyup, one for keypress?) how does the OS handle so many interrupts at the same time. A person typing at 120WPM would be sending more than 500 interrupts per minute. How does a process ever finish if there is a context switch for the CPU to handl... |
In database query processing, the approximate time for selection operation using primary index when equality is on key is $2(b_s + b_t)$ where $b_s$ is disk seek time and $b_t$ is disk transfer time (assuming one level of indexing), because one seek and transfer time will be needed for finding the index and another one... |
There is the complexity class ELEMENTARY that captures all problems that can be solved by using elementary recursive function only. So if algorithms for solving problems in some complexity class (e.g. NP or P) are converted to elementary recursive function form, would they retain time complexity of the complexity class... |
I am studying some old past test questions. Is this search tree correctly pruned?
|
Following question was asked in one of entrance exams for a graduation programme. Please help me try to solve it :
A computer system has an L1 cache, an L2 cache, and a main memory
unity connected as shown below. The block size in L1 cache is 4 words.
The block size in L2 cache is 16 words. The memory access times are... |
Specifying formal languages by giving formal grammars is a frequent task: we need grammars not only to describe languages, but also to parse them, or even do proper science. In all cases, it is important that the grammar at hand is correct, that is generates exactly the desired words.
We can often argue on a high-level... |
$L = \{a^i b^j a^k \ | \ k > i + j\}$
Use the pumping lemma to show that this language cannot be accepted by an FA.
Proof:
Suppose $L$ can be accepted by an FA.
Suppose a string $s = xyz \in L$, where
$$\begin{align}
&x=a^n \\
&y=b \\
&z=a^{n+2}
\end{align}
$$.
Then a string $t = a^n b^i a^{n+2}$ should also be in ... |
I understand that any CFL can be accepted by a PDA by final state or empty store but I have been rather stumped by this question.
The question states that the PDA has at most 2 states. Clearly 1 will be the start state while the other will be the final state (they cannot be the same since otherwise the empty string wil... |
As I understand the convoy effect, in the context of vehicular traffic in a road system. A slow moving group of vehicles passes through the system, slowing traffic even in areas which were not directly affected by the convoy.
How does this apply in the context of CPU scheduling? It does not seem to be an analogous situ... |
In the proof of Trakhtenbrot's theorem (as given in "Elements of Finite Model Theory" by Leonid Libkin), for every Turing machine $M$, author constructs a FO sentence $\Phi_M$ of vocabulary $\sigma$ such that $\Phi_M$ is finitely satisfiable iff $M$ halts on the empty input. Then he says that as the latter is known to ... |
Given a context-free language $L$, define the language $p(L)$ as containing all permutations of strings in $L$ (i.e. all strings in $L$ such that the order of symbols is not important). Is $p(L)$ context-free?
I found two papers dealing with similar, but not identical, questions:
Generating all permutations by context... |
A Hidoku is a $n \times n$ grid with some pre-filled integers from 1 to $n^2$. The goal is to find a path of successive integers (from 1 to $n^2$) in the grid. More concrete, each cell of the grid must contain a different integer from 1 to $n^2$ and each cell with value $z ≠ n^{2}$ must have a neighbor cell with value ... |
GeoQuery is a dataset used for benchmarking semantic parsers. It contains 880 queries about USA geography. The queries are in Prolog format, for example:
answer(A,longest(A,(river(A),traverse(A,B),const(B,countryid(usa)))))
(this represents a question for the longest river that passes through the us).
I have read sev... |
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 $y = \frac{5}{x}, D=\{x \in \mathbb{N}, x \neq 0\}$
So $h(x)=\frac{5}{x}$?
|
The Situation
I'm trying to read the book 'Digital Design Computer Architecture'.
In the part of Performance Analysis(7.3.4 in the book), Author refers to clock cycle for MIPS single cycle processor. But I think there's something wrong with author's evaluating clock cycle.
Author says,
T(Clock cycle) = T(pcq_pc) + T(... |
Let $m$ be the number of states (without the halting-state) of a deterministic turing-machine with $\Sigma$ = {} and $\Gamma$ = {$\square$, 0, 1}. Let $G(m)$ be the maximum number of ones, which can be generated by a deterministic turing-machine, before going into the halting-state. Prove that $G(m)$ $\in$ $\Omega(2^m)... |
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