Problem Detail: In Algorithmic Randomness and Complexity from Downey and Hirschfeldt, it is stated on page 129 that $qquad displaystyle sum_{K(sigma)downarrow} 2^{-K(sigma)} leq 1$, where $K(sigma)downarrow$ means that $K$ halts on $sigma$, $sigma$ being a binary string. $K$ denotes the prefix-free Read More …
Category: Uncategorized
[Solved]: Determining Number of States in a Turing Machine
Problem Detail: I am looking at an example Turing machine in my textbook, Automata and Computability by Dexter C. Kozen, and I’m confused as to how they determine the number of states this particular machine has. Here is the example: “Here Read More …
[Solved]: Why are comparisons so expensive on a GPU?
Problem Detail: While trying to improve the performance of my collision detection class, I found that ~80% of the time spent at the gpu, it spent on if/else conditions just trying to figure out the bounds for the buckets it should Read More …
[Solved]: Proving that English is not a regular language
Problem Detail: I am wanting to try and prove that the English language is not regular. The alphabet is the set of all words in the English dictionary. Looking at sentences, I was able to use this pattern of sentences { Read More …
[Solved]: Dijkstra’s algorithm to compute shortest paths using k edges?
Problem Detail: I am aware of using Bellman-Ford on a graph $G=(V,E)$ with no negative cycles to find the single-source single-destination shortest paths from source $s$ to target $t$ (both in $V$) using at most $k$ edges. Assuming we have no Read More …
[Solved]: Can one consider living (biological) cell to be Turing Complete?
Problem Detail: Universal Turing Machine can be boiled down to two components. Infinite tape of input and an action table, a finite state machine that moves read/write head along the tape and writes to it depending on input provided by the Read More …
[Solved]: Efficient algorithm for detecting overlaps in time intervals
Problem Detail: Let’s say I am building a time clock app. The employee can log into the interface and then enter time for multiple projects so that we can generate reports for billing as well as payroll. A table in the Read More …
[Solved]: Binary code with constraint
Problem Detail: Suppose I have an alphabet of n symbols. I can efficiently encode them with $lceil log_2nrceil$-bits strings. For instance if n=8: A: 0 0 0 B: 0 0 1 C: 0 1 0 D: 0 1 1 E: 1 Read More …
[Solved]: Rewriting gates such as XOR into three basic gates?
Problem Detail: How would I rewrite an XOR gate into the three basic logic gates (AND, OR, NOT). To be more specific, I have to write it in such a way with 2 NOT gates, 2 OR gates, and 1 AND Read More …
[Solved]: Can nodes in red-black trees have one nil child and one non-nil child?
Problem Detail: I don’t recall hearing that nodes in red-black trees can’t have one nil child and one non-nil child. However, I did hear that red-black trees have a worst-case height of $2log_2(n + 1)$, where n is the number of Read More …
