Infinite Binary Strings Research Articles

Languages with membership determined by single letter factors

The full scan condition on a language L introduced in [1] ensures that a word w must be completely inspected in order to decide whether or not w lies in L. We strengthen the condition by replacing the factor words in that definition by single letters. After examining the general case for both arbitrary and regular languages, we investigate the two-letter alphabet to find a complete description of the corresponding languages, which may be coded as infinite binary strings. Regularity of these languages corresponds to the associated numbers being rational and we find the minimal automata in all cases, which may be pictured as a cylinder of tape with a protruding end, although this cylinder is replaced by a Möbius strip for a special class of rational languages.

Radix sort trees in the large

Author(s): Evans, SN; Wakolbinger, A | Abstract: The trie-based radix sort algorithm stores pairwise different infinite binary strings in the leaves of a binary tree in a way that the Ulam-Harris coding of each leaf equals a prefix (that is, an initial segment) of the corresponding string, with the prefixes being of minimal length so that they are pairwise different. We investigate the radix sort tree chains – the tree-valued Markov chains that arise when successively storing the finite collections of random infinite binary strings Z1,…,Zn, n = 1, 2,… according to the trie-based radix sort algorithm, where the source strings Z1, Z2,… are independent and identically distributed. We establish a bijective correspondence between the full Doob–Martin boundary of the radix sort tree chain with a symmetric Bernoulli source (that is, each Zk is a fair coin-tossing sequence) and the family of radix sort tree chains for which the common distribution of the Zk is a diffuse probability measure on {0, 1}∞. In essence, our result characterizes all the ways that it is possible to condition such a chain of radix sort trees consistently on its behavior “in the large”.

Profiles of PATRICIA Tries

A PATRICIA trie is a trie in which non-branching paths are compressed. The external profile $$B_{n,k}$$ , defined to be the number of leaves at level k of a PATRICIA trie on n nodes, is an important “summarizing” parameter, in terms of which several other parameters of interest can be formulated. Here we derive precise asymptotics for the expected value and variance of $$B_{n,k}$$ , as well as a central limit theorem with error bound on the characteristic function, for PATRICIA tries on n infinite binary strings generated by a memoryless source with bias $$p > 1/2$$ for $$k\sim \alpha \log n$$ with $$\alpha \in (1/\log (1/q) + \varepsilon , 1/\log (1/p) - \varepsilon )$$ for any fixed $$\varepsilon > 0$$ . In this range, $$ {\mathbb {E}}[B_{n,k}] = \varTheta ( {\mathrm {Var}}[B_{n,k}])$$ , and both are of the form $$\varTheta (n^{\beta (\alpha )}/\sqrt{\log n})$$ , where the $$\varTheta $$ hides bounded, periodic functions in $$\log n$$ whose Fourier series we explicitly determine. The compression property leads to extra terms in the Poisson functional equations for the profile which are not seen in tries or digital search trees, resulting in Mellin transforms which are only implicitly given in terms of the moments of $$B_{m,j}$$ for various m and j. Thus, the proofs require information about the profile outside the main range of interest. Our derivations rely on analytic techniques, including Mellin transforms, analytic de-Poissonization, the saddle point method, and careful bounding of complex functions.

SAFE RECURSIVE SET FUNCTIONS

AbstractWe introduce the safe recursive set functions based on a Bellantoni–Cook style subclass of the primitive recursive set functions. We show that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets are exactly the polynomial growth rate functions computed by alternating exponential time Turing machines with polynomially many alternations. We also show that the functions computed by safe recursive set functions under a more efficient binary tree encoding of finite strings by hereditarily finite sets are exactly the quasipolynomial growth rate functions computed by alternating quasipolynomial time Turing machines with polylogarithmic many alternations.We characterize the safe recursive set functions on arbitrary sets in definability-theoretic terms. In its strongest form, we show that a function on arbitrary sets is safe recursive if and only if it is uniformly definable in some polynomial level of a refinement of Jensen's J-hierarchy, relativized to the transitive closure of the function's arguments.We observe that safe recursive set functions on infinite binary strings are equivalent to functions computed by infinite-time Turing machines in time less than ωω. We also give a machine model for safe recursive set functions which is based on set-indexed parallel processors and the natural bound on running times.

The descriptive complexity of stochastic integration

For most functionals for which pathwise stochastic integration with respect to Brownian motion is defined, sample Brownian paths for which the integral exists are very hard to construct. There exist on the unit interval, functions ω that can be uniformly approximated by sequences of continuous piece-linear functions (ωn) such that each ωn is encoded by a finite binary string of high Kolmogorov–Chaitin complexity. Such functions ω are called complex oscillations. Their set has Wiener measure 1 and they are fully characterised by infinite binary strings of high complexity. In this paper we study stochastic integration from the point of view of complex oscillations. We prove that, under some computability properties on integrands, pathwise stochastic integrals exist for any complex oscillation. We prove also that Itô’s lemma holds for each complex oscillation. Thus constructing a continuous function satisfying Itô’s lemma is reduced to constructing an infinite binary string of high Kolmogorov–Chaitin complexity.

An Ehrenfeucht‐Fraïssé game for

In this paper we develop an Ehrenfeucht‐Fraïssé game for . Unlike the standard Ehrenfeucht‐Fraïssé games which are modeled solely after the behavior of quantifiers, this new game also takes into account the behavior of connectives in logic. We prove the adequacy theorem for this game. We also apply the new game to prove complexity results about infinite binary strings.

Weakly Integrally Closed Numerical Monoids and Forbidden Patterns

An integral domain D is weakly integrally closed if whenever x is in the quotient field of D, and J is a nonzero finitely generated ideal of D such that xJ ⊆ J 2, then x is in D. We define weakly integrally closed (WIC) numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. The characteristic function of a numerical monoid M can be thought of as an infinite binary string s(M). A pattern of finitely many 0's and 1's is called forbidden if whenever s(M) contains it, then M is not weakly integrally closed. The pattern 11011 is forbidden. We show that a numerical monoid M is WIC if and only if s(M) contains no forbidden patterns. We also show that for every finite set S of forbidden patterns, there exists a numerical monoid M that is not WIC and for which s(M) contains no stretch (in a natural sense) of a pattern in S.

Dynamics of a generic Brownian motion: Recursive aspects

We study the local fluctuations of Brownian motions which are represented by infinite binary strings which are random in the sense of Kolmogorov–Chaitin. We show how the dynamical properties of such a Brownian motion at a point depend on its recursive properties.

Two-pattern strings II—frequency of occurrence and substring complexity

The previous paper in this series introduced a class of infinite binary strings, called two-pattern strings, that constitute a significant generalization of, and include, the much-studied Sturmian strings. The class of two-pattern strings is a union of a sequence of increasing (with respect to inclusion) subclasses T λ of two-pattern strings of scope λ, λ = 1 , 2 , … . Prefixes of two-pattern strings are interesting from the algorithmic point of view (their recognition, generation, and computation of repetitions and near-repetitions) and since they include prefixes of the Fibonacci and the Sturmian strings, they merit investigation of how many finite two-pattern strings of a given size there are among all binary strings of the same length. In this paper we first consider the frequency f λ ( n ) of occurrence of two-pattern strings of length n and scope λ among all strings of length n on { a , b } : we show that lim n → ∞ f λ ( n ) = 0 , but that for strings of lengths n ⩽ 2 λ , two-pattern strings of scope λ constitute more than one-quarter of all strings. Since the class of Sturmian strings is a subset of two-pattern strings of scope 1, it was natural to focus the study of the substring complexity of two-pattern strings to those of scope 1. Though preserving the aperiodicity of the Sturmian strings, the generalization to two-pattern strings greatly relaxes the constrained substring complexity (the number of distinct substrings of the same length) of the Sturmian strings. We derive upper and lower bounds on C 1 ( k ) (the number of distinct substring of length k) of two-pattern strings of scope 1, and we show that it can be considerably greater than that of a Sturmian string. In fact, we describe circumstances in which lim k → ∞ ( C 1 ( k ) − k ) = ∞ .

Compressibility and Resource Bounded Measure

We give a new definition of resource bounded measure based on compressibility of infinite binary strings. We prove that the new definition is equivalent to the one commonly used. This new characterization offers us a different way to look at resource bounded measure, shedding more light on the meaning of measure zero results and providing one more tool to prove such results. The main contribution of the paper is the new definition and the proofs leading to the equivalence result. We then show how this new characterization can be used to prove that the class of linear autoreducible sets has p- measure 0. We also prove that the class of sets that are truth-table reducible to a p-selective set has p-measure 0 and that the class of sets that Turing reduce to a subpolynomial dense set has p-measure 0. This strengthens various results.

The Descriptive Complexity of Brownian Motion

A continuous function x on the unit interval is a generic Brownian motion when every probabilistic event which holds almost surely with respect to the Wiener measure is reflected in x, provided that the event has a suitably effective description. We show that a generic one-dimensional Brownian motion can be computed from an infinite binary string which is complex in the sense of Kolmogorov–Chaitin. Conversely, one can construct a Kolmogorov–Chaitin random string from the values at the rational numbers of a generic Brownian motion. In this way, we construct a recursive isomorphism between encoded versions of generic Brownian motions and Kolmogorov–Chaitin random reals.

Kolmogorov complexity and symmetric relational structures

AbstractWe study partitions of Fraïssé limits of classes of finite relational structures where the partitions are encoded by infinite binary strings which are random in the sense of Kolmogorov-Chaitin.

Descriptive Complexity and Reflective Properties of Combinatorial Configurations

We study the Ramsey theoretic properties of combinatorial configurations which are generated by infinite binary strings which are random in the sense of Kolmogorov-Chaitin. Introduction In this paper we investigate the properties of combinatorial configurations which are generated or encoded by infinite binary strings which are random in the sense of Kolmogorov-Chaitin [3,9,4]. Using ideas from Ramsey theory, we introduce various means of assessing the degree of reflectiveness (self-similarity) of classes of combinatorial configurations and show how Kolmogorov-Chaitin strings (or KC strings) bring these reflective properties into realisation. We show for many categories C of combinatorial configurations (including graphs, posets, lattices and trees) that one can find a countable object U with the following properties. The object U is universal in the sense that every finite object in C can be embedded into U. Moreover, if /? is any finite object in C, we can use an arbitrary KC string e (i) for serving as a 2-colouring of the copies (images under embeddings) of fi in U and (ii) for generating a copy U' of U in U such that all the copies of fi which are contained in U' are of the same colour with respect to the colouring e. Since an infinite binary string is, with probability 1, a KC string, one can therefore conclude that for a random and uniform 2-colouring x of the copies of ft in U there is, with probability 1, a copy U' of U in U such that all the copies of fi in U' are monochromatic with respect to xIn a sense, KC strings are themselves highly reflective objects. Indeed, an important factor for the wjde acceptance of Kolmogorov's definition [9] of random strings has been that strings which are random according to his definition also satisfy the Church-Von Mises-Loveland criteria for randomness (see [18] for a historical survey). In particular, the sequence of outcomes brought about by a recursive gambling strategy against a KC string will itself be a KC string. Another way of depicting the reflective properties of a KC string e: N -*• {0,1} is by considering its graph-theoretic representation. Let Km be the complete graph on the set of natural numbers and let ex,e2, ...be a recursive enumeration without repetition of the edges of K^. Let G = (N, Ve) be the subgraph of Kx such that, for every ie N, we have ete Ve if and only if e(i) = 1. It is shown in Section 4 that Ge is isomorphic to the random graph of Rado [16]. In particular, Ge has every countable graph as an induced subgraph. Received 9 December 1992. 1991 Mathematics Subject Classification 68Q30. J. London Math. Soc. (2) 54 (1996) 199-208

Unpredictable strings are collectives

In [Z] we defined the notion of an “inference method” which given a finite binary string guesses its continuation. We showed that a special class of these inference methods could be used to construct a very general theory of confirmation. In this paper we investigate the class of strings which are unpredictable with respect to these inference method-that is, infinite strings which are such that given any initial segment a given inference method will guess its continuation incorrectly. We show that such unpredictable strings are also what Von Mises [6] called “collectives” (where the ambiguity in Von Mises’ definition is appropriately resolved). Thus strings which are unpredictable from initial segments are also statistically random. This can be regarded as a combinatorial analogue of a result of MartinLof [S] on computable inference methods. Martin-Liif showed that if the shortest algorithmic descriptions of longer and longer initial segments of an infinite string are of length rougly equal that of the initial segments they describe, then that infinite string is statistically random. We discuss the application of minimal algorithmic descriptions to inductive inference in [4]. We will use the following notation. S is a (finite or infinite) binary string. (S)j is the jth bit of S. [Sli is the j-length initial segment of S. ISI is the length of S. ISI 1 is the number of l’s in the string S. For a finite S, the density of S, D(S)= ISI,/ISI. For an infinite S, D(S)=lim,,, D([S],).

Information-theoretic characterizations of recursive infinite strings

Loveland and Meyer have studied necessary and sufficient conditions for an infinite binary string x to be recursive in terms of the program-size complexity relative to n of its n-bit prefixes x n . Meyer has shown that x is recursive iff ∃ c, ∀ n, K( x n ⧸ n) ⩽ c, and Loveland has shown that this is false if one merely stipulates that K( x n ⧸ n) ⩽ c for infinitely many n. We strengthen Meyer's theorem. From the fact that there are few minimal-size programs for calculating n given result, we obtain a necessary and sufficient condition for x to be recursive in terms of the absolute program-size complexity of its prefixes: x is recursive iff ∃ c, ∀ n, K( x n ) ⩽ K( n) + c. Again Loveland's method shows that this is no longer a sufficient condition for x to be recursive if one merely stipulates that K( x n ) ⩽ K( n) + c for infinitely many n.