On maximal subgroups of the group of recursive permutations
The book contains the necessary information from the algorithm theory, graph theory, combinatorics. It is considered partially recursive functions, Turing machines, some versions of the algorithms (associative calculus, the system of substitutions, grammars, Post's productions, Marcov's normal algorithms, operator algorithms). The main types of graphs are described (multigraphs, pseudographs, Eulerian graphs, Hamiltonian graphs, trees, bipartite graphs, matchings, Petri nets, planar graphs, transport nets). Some algorithms often used in practice on graphs are given. It is considered classical combinatorial configurations and their generating functions, recurrent sequences. It is put in a basis of the book long-term experience of teaching by authors the discipline «Discrete mathematics» at the business informatics faculty, at the computer science faculty of National Research University Higher School of Economics, and at the automatics and computer technique faculty of National research university Moscow power engineering institute. The book is intended for the students of a bachelor degree, trained at the computer science faculties in the directions 09.03.01 Informatics and computational technique, 09.03.02 Informational systems and technologies, 09.03.03 Applied informatics, 09.03.04 Software Engineering, and also for IT experts and developers of software products.
It is known that the normalized algorithmic information distance is not computable and not semicomputable. We show that for all 𝜀<1/2, there exist no semicomputable functions that differ from N by at most 𝜀. Moreover, for any computable function f such that |lim𝑡𝑓(𝑥,𝑦,𝑡)−N(𝑥,𝑦)|≤𝜀 and for all n, there exist strings x, y of length n such that
∑_𝑡 |𝑓(𝑥,𝑦,𝑡+1)−𝑓(𝑥,𝑦,𝑡)| ≥ 𝛺(log 𝑛)
This is optimal up to constant factors.
We also show that the maximal number of oscillations of a limit approximation of N is 𝛺(𝑛/log𝑛). This strengthens the 𝜔(1) lower bound from [K. Ambos-Spies, W. Merkle, and S.A. Terwijn, 2019, Normalized information distance and the oscillation hierarchy].
We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations R, S, a componentwise reducibility is defined by
R ≤ S ⇔ ∃f ∀x, y [x R y ↔ f (x) S f (y)].
Here, f is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and f must be computable. We show that there is a -complete equivalence relation, but no -complete for k ≥ 2. We show that preorders arising naturally in the above-mentioned areas are -complete. This includes polynomial time m-reducibility on exponential time sets, which is , almost inclusion on r.e. sets, which is , and Turing reducibility on r.e. sets, which is .