Algorithmic randomness and splitting of supermartingales

Randomness in the sense of Martin-L"of can be defined in terms of lower semicomputable supermartingales. We show that such a supermartingale cannot be replaced by a pair of supermartingales that bet only on the even bits (the first one) and on the odd bits (the second one) knowing all preceding bits.

💡 Research Summary

The paper investigates whether the definition of Martin‑Löf randomness based on lower‑semicomputable supermartingales can be reduced to a pair of supermartingales that bet only on even positions and only on odd positions, respectively. While for computable martingales such a reduction is well‑known – any computable martingale can be split into an “even‑step” martingale and an “odd‑step” martingale whose product reproduces the original capital – the authors show that this property fails dramatically for lower‑semicomputable supermartingales, and consequently for Martin‑Löf random sequences.

Background. A supermartingale m is a non‑negative real‑valued function on binary strings satisfying
 m(x) ≥ (m(x0)+m(x1))/2 for all x.
If m’s capital becomes unbounded along the prefixes of an infinite binary sequence ω, we say that m “wins” on ω. The class of lower‑semicomputable supermartingales (those whose values can be approximated from below by a computable, monotone sequence of rationals) has a maximal element up to a multiplicative constant; its winning set coincides with the set of Martin‑Löf random sequences.

Even/odd decomposition for computable martingales. For any computable martingale t one can construct computable martingales t₀ (betting only on even steps) and t₁ (betting only on odd steps) such that whenever t wins on ω, at least one of t₀ or t₁ also wins on ω. The construction simply lets t₀ follow t on even moves while keeping its capital unchanged on odd moves, and vice‑versa for t₁. Consequently, the winning set of t is contained in the union of the winning sets of t₀ and t₁.

Main theorem. The authors prove that there exists a Martin‑Löf random sequence ω for which no pair of lower‑semicomputable supermartingales that are restricted to even or odd steps respectively can win. In other words, the even/odd splitting property that holds for computable martingales breaks down for the more powerful lower‑semicomputable supermartingales that characterize Martin‑Löf randomness.

Proof technique – an infinite game. The core of the proof is a two‑player infinite game played on the full binary tree. Player A (the adversary) enumerates two lower‑semicomputable supermartingales t₀ and t₁, constrained to bet only on even and odd positions respectively. Player M (the mathematician) simultaneously builds a lower‑semicomputable supermartingale t with no such restriction. Moves consist of providing finite approximations (non‑decreasing rational values) for the three supermartingales; all three start with value 1 at the root. The game proceeds forever, and after it ends a referee looks for an infinite branch ω such that t’s capital along ω is unbounded while both t₀ and t₁ stay bounded.

The authors first define a finite‑tree version of the game: on a binary tree of fixed height h, M tries to force a leaf where t’s value exceeds a constant C > 1 while t₀ and t₁ never exceed 1 on the path to that leaf. They exhibit a computable winning strategy for M on this finite game. The intuition is that M can “invest” a little extra capital C in a leaf; A must raise either t₀ or t₁ on the path to that leaf to prevent M’s gain, but doing so inevitably forces A to raise the same supermartingale on sibling branches, which would prematurely disqualify those branches. By carefully choosing the order of leaves and the constant C (e.g., C = N/(N‑1) where N is the number of leaves), M can force A into a deterministic pattern that eventually makes one of t₀, t₁ exceed 1 on the chosen leaf, while t’s value at that leaf stays above C.

From finite to infinite. M repeats the finite‑tree strategy on a sequence of nested sub‑trees. After winning on the first finite tree, M starts a second finite game on the subtree rooted at the winning leaf, but now with all of M’s capital doubled. Repeating this construction yields a sequence of leaves where t’s value grows as 2, 4, 8, … while t₀ and t₁ remain ≤ 1 along the entire concatenated path. Because each finite game is computable and the nesting is effective, the limit supermartingale t is lower‑semicomputable. Consequently, there exists an infinite branch ω on which t wins and both t₀, t₁ lose, establishing the theorem.

Relation to van Lambalgen’s theorem. Van Lambalgen’s theorem states that a binary sequence α is Martin‑Löf random iff its even‑indexed subsequence α₀ is random relative to α₁ and vice‑versa. This suggests that randomness of the whole sequence is equivalent to a kind of “oracle‑enhanced” randomness of the two halves. The present result shows that a weaker, “online” version—where each half must be random without any look‑ahead—does not suffice. In game‑theoretic terms, even if each referee (even‑step player and odd‑step player) can keep his own capital bounded while the other referee’s future bits are unknown, the whole sequence can still be non‑random for a supermartingale that uses the full information.

Significance. The paper’s contributions are threefold:

1. It clarifies the boundary between computable martingales (where even/odd splitting works) and lower‑semicomputable supermartingales (where it fails).
2. It introduces a novel infinite‑game framework that translates the existence of a counterexample into a concrete, constructive strategy for one player.
3. It deepens our understanding of the structural requirements of algorithmic randomness, showing that the ability to bet on every step is essential for the supermartingale characterization of Martin‑Löf randomness.

Overall, the work demonstrates that the intuitive idea “if each half of a sequence looks random, the whole should be random” is false when randomness is defined via lower‑semicomputable supermartingales, highlighting a subtle but fundamental distinction between different notions of algorithmic randomness.