Problem Detail: Someone told me that the $log$ function was introduced to make the calculation easier. If we have to calculate $xy$, we can calculate instead $log x+log y$ since $log xy=log x+log y$. How this can make the calculation easier? Read More …
Category: Uncategorized
[Solved]: How to evaluate all derivatives of a polynomial at a point with FFT?
Problem Detail: I found this problem: Evaluating all derivatives of a polynomial at a point Given a polynomial A(x) of degree-bound n, its tth derivative is defined by From the coefficient representation $(a_0, a_1, . . . , a_{n-1})$ of A(x) Read More …
[Solved]: Combinatory interpretation of lambda calculus
Problem Detail: According to Peter Selinger, The Lambda Calculus is Algebraic (PDF). Early in this article he says: The combinatory interpretation of the lambda calculus is known to be imperfect, because it does not satisfy the $ξ$-rule: under the interpretation, $M Read More …
[Solved]: Why is the PageRank vector also the eigenvector of the web adjacency matrix?
Problem Detail: From wikipedia: The PageRank values are the entries of the dominant eigenvector of the modified adjacency matrix. This makes PageRank a particularly elegant metric Can anyone please elaborate on the connection between the eigenvector and the PR vector? Why Read More …
[Solved]: Non-termination of types in Martin-Löf’s Type:Type?
Problem Detail: In the pre-history of dependent type theory, Per Martin Löf introduced a calculus that is in some sense the simplest dependent type theory and the most general form of impredicative polymorphism. It is often referred to as Type:Type because Read More …
[Solved]: Binary rooted tree isomorphism
Problem Detail: My trees are rooted and have at most two children at every vertex. I need references that help me solve any or all of the questions below: How many isomorphism classes of trees with n vertices are there? What Read More …
[Solved]: Quantifiers in lambda calculus
Problem Detail: I’m learning lambda calculus, however I’m quite confused about the quantifiers in lambda calculus. As far as I know, quantifiers such as “∃” are concepts of first order logic (FOL), which are not needed by lambda calculus. Moreover, I Read More …
[Solved]: Why are pushdown automata countable?
Problem Detail: I began a chapter in a textbook on computational theory where they begin to talk about decidable languages. The problems in this section are pretty confusing and I honestly don’t know how to begin them because I’m not 100% Read More …
[Solved]: Halting problem
Problem Detail: I have some concerns about the Halting problem. This is the proof I know: Let $h(M, i)$ be a function, $M$ being Turing machine and $i$ input for the Turing machine. Let $h(M, i)$ output true whenever $M(i)$ halts. Read More …
[Solved]: Proof of correctness of A star search algorithm
Problem Detail: I’ve been looking for the proof of correctness for the A star (A*) algorithm but none of the texts and websites offer it. Mostly they are talking about the proof of optimality of the A* algorithm. I’m looking for Read More …
