arXiv subjects
Publications and source records attributed to Laurent Bienvenu.
Many statements studied in reverse mathematics can be seen as mathematical problems, formulated in terms of instances and solutions. We develop a framework of typicality encompassing measure and genericity, and we classify the reverse mathematics zoo in terms of which problems admit typical solutions. It turns out that even very weak problems do not admit typical solutions.
For a fixed alphabet $A$, an infinite sequence $X$ is said to be normal if every word $w$ over $A$ appears in $X$ with the same frequency as any other word of the same length. A classical result of Agafonov (1966) relates normality to finite automata as follows: a sequence $X$ is normal if and only if any subsequence of $X$ selected by a finite automaton is itself normal. Another theorem of Schnorr and Stimm (1972) gives an alternative characterization: a sequence $X$ is normal if and only if no gambler can win large amounts of money by betting on the sequence $X$ using a strategy that can be described by a finite automaton. Both of these theorems are established in the setting of deterministic finite automata. This raises the question as to whether they can be extended to the setting of probabilistic finite automata. In the case of the Agafonov theorem, this question was positively answered by Léchine et al. (2024) in a restricted case of probabilistic automata with rational transition probabilities. In this paper, we settle the full conjecture by proving that both the Agafonov and the Schnorr-Stimm theorems hold true for arbitrary probabilistic automata. Specifically, we show that a sequence $X$ is normal if and only if any probabilistic automaton selects a normal subsequence of $X$ with probability $1$. We also show that a sequence $X$ is normal if and only if a probabilistic finite-state gambler fails to win on $X$ with probability $1$.
For a fixed alphabet A, an infinite sequence X is said to be normal if every word w over A appears in X with the same frequency as any other word of the same length. A classical result relates normality to finite automata as follows: a sequence X is normal if and only if all gambling strategies implementable with finite deterministic automata lose all their capital when trying to predict the next bit of X after seing the ones before. More precisely, Schnorr and Stimm (1972) proved that the capital goes exponentially fast to zero unless the automaton represents the gambler that never bets, in which case the capital remains constant. In this paper we show that an analogous result holds when considering probabilistic automata: a sequence X is normal if and only if for any gambling strategy implementable with probabilistic finite automaton it holds that the expected value of the capital of the gambler converges exponentially fast to a finite value when playing against X. To obtain this result, we show a more general statement related to the convergence of martingales given by finite sets of non-negative matrices {M a } a$\in$A . In particular, we show that X is normal if and only if ||vM X[1] . . . M X[n] || converges exponentially fast to a finite value for any non-negative starting vector v. Moreover, we distinguish three distinctive behaviours that this sequence can attain, and prove that the problem of recognizing, given a family of matrices, to which case it belongs, is decidable.
Finite-state dimension quantifies the asymptotic rate of information in an infinite sequence as perceived by finite automata. For a fixed alphabet, the infinite sequences that have maximal finite-state dimension are exactly those that are Borel normal, i.e., in which all words of any given length appear with the same frequency. A theorem of Schnorr and Stimm (1972) shows that a real number is Borel normal if and only if, for every finite-state irreducible Markov chain with fair transitions, when the chain is simulated using the binary expansion of the given number, the empirical distribution of states converges to its stationary distribution. In this paper we extend this correspondence beyond normal numbers. We show that the finite-state dimension of a sequence can be characterized in terms of the conditional Kullback-Leibler divergence between the limiting distributions arising from the simulation of Markov chains using the given sequence and their stationary distributions. This provides a new information-theoretic characterization of finite-state dimension which generalizes the Schnorr-Stimm result. As an application, we prove a generalization of Agafonov's theorem for normal numbers. Agafonov's theorem states that a sequence is normal if and only if every subsequence selected by a finite automaton is also normal. We extend this to arbitrary sequences by establishing a tight quantitative relationship between the finite-state dimension of a sequence and the finite-state dimensions of its automatic subsequences.
The notion of an individual random sequence goes back to von Mises. We describe the evolution of this notion, especially the use of martingales (suggested by Ville), and the development of algorithmic information theory in 1960s and 1970s (Solomonov, Kolmogorov, Martin-Lof, Levin, Chaitin, Schnorr and others). We conclude with some remarks about the use of the algorithmic information theory in the foundations of probability theory.
In this article, we study the relationship between notions of depth for sequences, namely, Bennett's notions of strong and weak depth, and deep $Π^0_1$ classes, introduced by the authors and motivated by previous work of Levin. For the first main result of the study, we show that every member of a $Π^0_1$ class is order-deep, a property that implies strong depth. From this result, we obtain new examples of strongly deep sequences based on properties studied in computability theory and algorithmic randomness. We further show that not every strongly deep sequence is a member of a deep $Π^0_1$ class. For the second main result, we show that the collection of strongly deep sequences is negligible, which is equivalent to the statement that the probability of computing a strongly deep sequence with some random oracle is 0, a property also shared by every deep $Π^0_1$ class. Finally, we show that variants of strong depth, given in terms of a priori complexity and monotone complexity, are equivalent to weak depth.
Bennett's notion of depth is usually considered to describe the usefulness and internal organization of the information encoded into an object such as an infinite binary sequence. We consider a natural way to relativize the notion of depth for such sets, and we investigate for various kinds of oracles whether and how the unrelativized and the relativized version of depth differ. Intuitively speaking, access to an oracle increases computation power. Accordingly, for most notions for sets considered in computability theory, for the corresponding classes trivially for all oracles the unrelativized class is contained in the relativized class or for all oracles the relativized class is contained in the unrelativized class. Examples for these two cases are given by the classes of computable and of Martin-Löf random sets, respectively. However, in the case for depth the situation is different. It turns out that the classes of deep sets and of sets that are deep relative to the halting set $\emptyset '$ are incomparable with respect to set-theoretical inclusion. On the other hand, the class of deep sets is strictly contained in the class of sets that are deep relative to any given Martin-Löf-random oracle. The set built in the proof of the latter result can also be used to give a short proof of the known fact that every PA-complete degree is Turing-equivalent to the join of two Martin-Löf-random sets. In fact, we slightly strengthen this result by showing that every DNC$_2$ function is truth-table-equivalent to the join of two Martin-Löf random sets. Furthermore, we observe that the class of deep sets relative to any given K-trivial oracle either is the same as or is strictly contained in the class of deep sets. Obviously, the former case applies to computable oracles. We leave it as an open problem which of the two possibilities can occur for noncomputable K-trivial oracles.
Can a probabilistic gambler get arbitrarily rich when all deterministic gamblers fail? We study this problem in the context of algorithmic randomness, introducing a new notion -- almost everywhere computable randomness. A binary sequence $X$ is a.e.\ computably random if there is no probabilistic computable strategy which is total and succeeds on $X$ for positive measure of oracles. Using the fireworks technique we construct a sequence which is partial computably random but not a.e.\ computably random. We also prove the separation between a.e.\ computable randomness and partial computable randomness, which happens exactly in the uniformly almost everywhere dominating Turing degrees.
The $Ω$ numbers-the halting probabilities of universal prefix-free machines-are known to be exactly the Martin-L{ö}f random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-L{ö}f random left-c.e. real $α$, a universal prefix-free machine U whose halting probability is $α$. We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real $α$, one cannot uniformly produce a left-c.e. real $β$ such that $α$ -- $β$ is neither left-c.e. nor right-c.e.
Many constructions in computability theory rely on "time tricks". In the higher setting, relativising to some oracles shows the necessity of these. We construct an oracle~$A$ and a set~$X$, higher Turing reducible to~$X$, but for which $Ψ(A)\ne X$ for any higher functional~$Ψ$ which is consistent on all oracles. We construct an oracle~$A$ relative to which there is no universal higher ML-test. On the other hand, we show that badness has its limits: there are no higher self-PA oracles, and for no~$A$ can we construct a higher $A$-c.e.\ set which is also higher $A$-ML-random. We study various classes of bad oracles and differentiate between them using other familiar classes. For example, bad oracles for consistent reductions can be higher ML-random, whereas bad oracles for universal tests cannot.
The rate of randomness (or dimension) of a string $σ$ is the ratio $C(σ)/|σ|$ where $C(σ)$ is the Kolmogorov complexity of $σ$. While it is known that a single computable transformation cannot increase the rate of randomness of all sequences, Fortnow, Hitchcock, Pavan, Vinodchandran, and Wang showed that for any $0<α<β<1$, there are a finite number of computable transformations such that any string of rate at least $α$ is turned into a string of rate at least $β$ by one of these transformations. However, their proof only gives very loose bounds on the correspondence between the number of transformations and the increase of rate of randomness one can achieve. By translating this problem to combinatorics on (hyper)graphs, we provide a tight bound, namely: Using $k$ transformations, one can get an increase from rate $α$ to any rate $β< kα/(1+(k-1)α)$, and this is optimal.
Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better approximations of $p$ as we read the sequence. We study in this paper a similar question, but from the viewpoint of inductive inference. We suppose now that $p$ is a computable real, but one asks for more: as we are reading more and more bits of our random sequence, we have to eventually guess the exact parameter $p$ (in the form of a Turing code). Can one do such a thing uniformly on all sequences that are random for computable Bernoulli measures, or even on a `large enough' fraction of them? In this paper, we give a negative answer to this question. In fact, we prove a very general negative result which extends far beyond the class of Bernoulli measures.
In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence (as shown by Nies, Stephan, and Terwijn). We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence (which answers an open question posed by Barmpalias, Day, and Lewis) and that every Demuth random sequence forms a minimal pair with every pb-generic sequence (where pb-genericity is an effective notion of genericity that is strictly between 1-genericity and 2-genericity). Moreover, we prove that for every comeager $\mathcal{G}\subseteq 2^ω$, there is some weakly 2-random sequence $X$ that computes some $Y\in\mathcal{G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity.
Relativizing computations of Turing machines to an oracle is a central concept in the theory of computation, both in complexity theory and in computability theory(!). Inspired by lowness notions from computability theory, Allender introduced the concept of "low for speed" oracles. An oracle A is low for speed if relativizing to A has essentially no effect on computational complexity, meaning that if a decidable language can be decided in time $f(n)$ with access to oracle A, then it can be decided in time poly(f(n)) without any oracle. The existence of non-computable such A's was later proven by Bayer and Slaman, who even constructed a computably enumerable one, and exhibited a number of properties of these oracles as well as interesting connections with computability theory. In this paper, we pursue this line of research, answering the questions left by Bayer and Slaman and give further evidence that the structure of the class of low for speed oracles is a very rich one.
A set of infinite binary sequences $\mathcal{C}\subseteq2^ω$ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of $Π^0_1$ classes. In this paper, we introduce the notion of depth for $Π^0_1$ classes, which is a stronger form of negligibility. Whereas a negligible $Π^0_1$ class $\mathcal{C}$ has the property that one cannot probabilistically compute a member of $\mathcal{C}$ with positive probability, a deep $Π^0_1$ class $\mathcal{C}$ has the property that one cannot probabilistically compute an initial segment of a member of $\mathcal{C}$ with high probability. That is, the probability of computing a length $n$ initial segment of a deep $Π^0_1$ class converges to 0 effectively in $n$. We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call $\mathit{tt}$-negligibility. We also provide a number of examples of deep $Π^0_1$ classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, we examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin's $Ω$.
A set C of reals is said to be negligible if there is no probabilistic algorithm which generates a member of C with positive probability. Various classes have been proven to be negligible, for example the Turing upper-cone of a non-computable real, the class of coherent completions of Peano Arithmetic or the class of reals of minimal degrees. One class of particular interest in the study of negligibility is the class of diagonally non-computable (DNC) functions, proven by Kucera to be non-negligible in a strong sense: every Martin-Löf random real computes a DNC function. Ambos-Spies et al. showed that the converse does not hold: there are DNC functions which compute no Martin-Löf random real. In this paper, we show that such the set of such DNC functions is in fact non-negligible. More precisely, we prove that for every sufficiently fast-growing computable~$h$, every 2-random real computes an $h$-bounded DNC function which computes no Martin-Löf random real. Further, we show that the same holds for the set of reals which compute a DNC function but no bounded DNC function. The proofs of these results use a combination of a technique due to Kautz (which, following a metaphor of Shen, we like to call a `fireworks argument') and bushy tree forcing, which is the canonical forcing notion used in the study of DNC functions.
We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood, we say that a Ramsey-type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey-type variants of problems related to König's lemma, such as restrictions of König's lemma, Boolean satisfiability problems, and graph coloring problems. We find that sometimes the Ramsey-type variant of a problem is strictly easier than the original problem (as Flood showed with weak König's lemma) and that sometimes the Ramsey-type variant of a problem is equivalent to the original problem. We show that the Ramsey-type variant of weak König's lemma is robust in the sense of Montalban: it is equivalent to several perturbations of itself. We also clarify the relationship between Ramsey-type weak König's lemma and algorithmic randomness by showing that Ramsey-type weak weak König's lemma is equivalent to the problem of finding diagonally non-recursive functions and that these problems are strictly easier than Ramsey-type weak König's lemma. This answers a question of Flood.
Algorithmic randomness theory starts with a notion of an individual random object. To be reasonable, this notion should have some natural properties; in particular, an object should be random with respect to image distribution if and only if it has a random preimage. This result (for computable distributions and mappings, and Martin-Löf randomness) was known for a long time (folklore); in this paper we prove its natural generalization for layerwise computable mappings, and discuss the related quantitative results.