David A. Tanzer, March 26, 2021

For languages in general, consonants are either voiced — involving vibration of the vocal cords — or unvoiced. A voiced consonant may have as its “partner” an unvoiced consonant produced by suppressing its vocal action. For example, the ‘b’ consonant in English becomes ‘p’ when devoiced. The articulation of the two consonants is the same — e.g. the motion of the lips — except … Read more

David A. Tanzer, February 11, 2021

In Part 6, we saw a program with quadratic complexity. The collection of all languages that can be decided in $O(n^k)$ time for some $k$ is called P, for polynomial time complexity.

Now let’s consider languages that appear to require time that is exponential in the size of the input, and hence lie outside of P.

Here is a decision problem that is believed … Read more

David A. Tanzer, February 10, 2021

Quadratic complexity

In Part 5 we introduced big O notation for describing linear complexity. Now let’s look at a function with greater than linear complexity.

def square_length(text): 
    # compute the square of the length of text
    # FIXME: not the most elegant or efficient approach
    n = length(text)  
    counter = 0    
    for i = 1 to n:        
        for j = 1 to n:             
            counter 

David A. Tanzer, February 9, 2021

Big O analysis

Recall our function $f(n)$ from Part 4, which gives the values 2, 13, 14, 25, 26, 37, …

Using ‘big O’ notation, we write $f(n) = O(n)$ to say that $f$ is linearly bounded.

This means that $f(n)$ will eventually become less than some linear function.

As we said, $f(n)$ has a “rough slope” of 6. So f could never … Read more

David A. Tanzer, February 8, 2021

Summarizing computational complexity

In Part 3 we defined, for each program P, a detailed function P'(n) that gives the worst case number of steps that P must perform when given some input of size n. Now we want to summarize P into general classes, such as linear, quadratic, etc.

What’s in a step?

But we should clarify something before proceeding: what is meant by … Read more

David A. Tanzer, February 7, 2021

A detailed measure of computational complexity

In Part 2, we identified language complexity by how “difficult” it is for a program to decide if some given text is a sentence of the language — in other words, by the complexity of the decision problem for the language.  

Note there may be many decision procedures — which are implementations of functions – that solve … Read more

David A. Tanzer, January 30, 2021

Decision problems, decision procedures and complexity

In Part 1, we introduced the formal idea of a language, as being just a set of sentences. Now let’s approach the topic of language complexity.

For any given language, there is an associated decision problem:   given a candidate string of characters, determine whether or not it belongs to the language.   A decision procedure, or … Read more

David A. Tanzer, December 31, 2020. This article was cross-posted to Azimuth, click here to discuss it.

A simple view of languages

How complex is the English language? The question has many dimensions and nuances. What does complexity mean, and how could it be measured? This a tough nut to crack, so in this post we’ll make a retrenchment and reconsider the question in a more formal setting — … Read more