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I looked into this last year while teaching. The other answers, including Prof. Erickson's excellent book, feel incomplete, because they handwave a step along the lines of "there is an optimal edit sequence that proceeds left-to-right" or "we start by lining up the two words in columns vertically...." (Even if that feels obvious, can you ...

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This problem can be solved with dynamic programming in pseudo-polynomial time (proof below). Therefore, it is not possible to show that this problem is strongly NP-hard (unless P=NP). First, let's restate the problem: Given: values $N$ and $T$ and positive integer intervals $R_1$, $R_2$, $\ldots$, and $R_n$ Output: the largest possible value of $\sum_{i=1}^... 3 The first problem is what does is even mean that a propositional proof system can prove its own properties: there is a serious discrepancy of the languages, because the propositional proof system can only express propositional formulas, whereas properties of the proof system are first-order statements in a language that can reason about finite strings, i.e., ... 3 There is no loop. The purpose of a formal system is to make reasoning principles explicit and to explain more precisely how reasoning works. The word "foundation" in "foundations of mathematics" does not mean "create a secure base for mathematics out of nothing" – that would be an indifensible position. There is absolutely ... 2 In general, proof systems can sometimes prove some of their properties within themselves. A nice example of this is the fact that NL=Co-NL can be proved "within NL". This video might also be useful: https://www.youtube.com/watch?v=TLjRGm8ZfyQ 2 I'm not sure whether it would work in your case, but in order to show succinctness results in modal/temporal logic (e.g. the fact the two-variable logic over words in exponentially more succinct than unary temporal logic) one can employ formula size games or Adler-Immerman games. Probably the most recent paper to read is by Lauri Hella and Miikka Vilander. ... 1 Your question is formally the same as this one: how many symbols$X$are needed to write the polynomial$XX + XXXX + XXXXXX = XX(1 + XX(1 + XX))$? Well, you obtained$6$as an upper bound. It is also a lower bound because any expression involving less than$6$symbols$X$would define a polynomial of degree$< 6$. For the general case, I would not be ... 1 A paper of mine with Gupta and Kumar titled On a bidirected relaxation for the MULTIWAY CUT problem was also based on running experiments. In fact we were trying to prove the converse of what we ended up proving. Vazirani, in the first edition of his book on approximation algorithms, suggested that the bidirected relaxation was at least as good as the ... 1 My recent paper with Karthik Chandrasekharan titled Hypergraph$k$-cut in deterministic polynomial time was based on extensive computational experiments. We explored different conjectures and submodular functions and found counter examples to several different approaches. The truth of the main structural theorem was suggested by experiments and we then ... 1 Some recent results in state complexity were found with the help of systematic brute-force search for worst-case examples. This is doable because there are not too many deterministic finite automata with a small number of states, for example if we concentrate on binary or ternary alphabets. Also, in many cases there are families of worst-case examples for$1,...

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In 2018, Aubrey de Grey found a 1581-vertex, non-4-colourable unit-distance graph. This gives a lower bound of five for the famous Hadwiger-Nelson problem. He used a computer to verify that the graph indeed has chromatic number at least five. Gil Kalai's blogpost covers some facts and further developments. An article in the quanta magazine reports that he ...

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