{-# OPTIONS --without-K --exact-split --safe #-} module Base.Complexity.Basic where
Let 𝑇ₙ be a totally ordered set of size 𝑛. Let 𝐴 be a set (the alphabet). We can model the set 𝑊ₙ, of words (strings of letters from 𝐴) of length 𝑛 by the type 𝑇ₙ → 𝐴 of functions from 𝑇ₙ to 𝐴.
The set of all (finite length) words is then
[ W = ⋃[n ∈ ℕ] Wₙ ]
The length of a word 𝑥 is given by the function size x, which will be defined below.
An algorithm is a computer program with infinite memory (i.e., a Turing machine).
A function 𝑓 : 𝑊 → 𝑊 is computable in polynomial time if there exist an algorithm and numbers 𝑐, 𝑑 ∈ ℕ such that for each word 𝑥 ∈ 𝑊 the algorithm stops in at most (size 𝑥) 𝑐 + 𝑑 steps and computes 𝑓 𝑥.
At first we will simplify by assuming 𝑇ₙ is Fin n.