The axiomatic power of Kolmogorov complexity

Abstract

A preliminary version of this paper was presented on a special session of the Computability in Europe conference (CiE), June 2012.International audienceThe famous Gödel incompleteness theorem states that for every consistent sufficiently rich formal theory T there exist true state- ments that are unprovable in T. Such statements would be natural can- didates for being added as axioms, but how can we obtain them? One classical (and well studied) approach is to add to some theory T an axiom that claims the consistency of T. In this paper we discuss another approach motivated by Chaitin’s version of G ̈odel’s theorem where ax- ioms claiming the randomness (or incompressibility) of some strings are probabilistically added, and show that it is not really useful, in the sense that this does not help us to prove new interesting theorems. This result (cf. [She06]) answers a question recently asked by Lipton [LR11]. The situation changes if we take into account the size of the proofs: randomly chosen axioms may help making proofs much shorter (unless NP=PSPACE). This result partially answers the question asked in [She06].We then study the axiomatic power of the statements of type “the Kolmogorov complexity of x exceeds n” (where x is some string, and n is some integer) in general. They are Π1 (universally quantified) statements of Peano arithmetic. We show (Theorem 5) that by adding all true statements of this type, we obtain a theory that proves all true Π1-statements, and also provide a more detailed classification. In particular, as Theorem 7 shows, to derive all true Π1-statements it is enough to add one statement of this type for each n (or even for infinitely many n) if strings are chosen in a special way. On the other hand, one may add statements of this type for most x of length n (for every n) and still obtain a weak theory (Theorem 10). We also study other logical questions related to “random axioms” (hierarchy with respect to n, Theorem 8 in Section 3.3, independence in Section 3.6, etc.).Finally, we consider a theory that claims Martin-Löf randomness of a given infinite binary sequence. This claim can be formalized in different ways. We show that different formalizations are closely related but not equivalent, and study their properties

Computability and Randomness

“…Theorem 4 (Solovay (1975)). ( ) = ( )+ (2) ( )± ( (3) ( )), and this result is tight in that we cannot extend it to (4)…”

“…3 Other recent work has explored the effect of adding axioms asserting the incompressibility of certain strings in a probabilistic way. Bienvenu, Romashchenko, Shen, Taveneaux, and Vermeeren [4] have shown that this kind of procedure does not help to prove new interesting theorems, but that the situation changes if we take into account the sizes of the proofs: randomly chosen axioms (in a sense made precise in their paper) can help to make proofs much shorter under the reasonable complexity-theoretic assumption that NP ≠ PSPACE.…”

“…Otherwise, the probability that the answer is "yes" is very high. It is conjectured that PIT has a polynomial-time deterministic algorithm, 4 but no such algorithm is known.…”

27 Open Problems in Kolmogorov Complexity

“…Given a string x of complexity n, and some m < n, we can study the axiomatic power of the statement C(x) > m, in particular, the complexity of the universal statements that follow from it. (Universal statements are statements about non-termination of some program in an optimal programming language, and their complexity is the length of this program, see [12] for details.) If x and x ′ have small information distance, then we can positively check this, and therefore the high complexity of x provably implies (almost as) high complexity of x ′ and vice versa.…”