Parity-SAT is the problem of determining whether a given CNF formula has an odd number of satisfying assignments. As a canonical ⊕P-complete problem, it represents a fundamental variant of the exact model counting problem (#SAT). Under the Strong Exponential Time Hypothesis (SETH), Parity-SAT admits no O^*((2-ε)ⁿ)-time or O^*((2-ε)^m)-time algorithm for any constant ε > 0, where n and m denote the numbers of variables and clauses, respectively. Thus, breaking the 2ⁿ or 2^m barrier appears impossible in full generality. In this work, we revisit this barrier through structural restrictions and a refined exploitation of parity. We study Parity-d-occ-SAT, where each variable appears in at most d clauses, and obtain three main results. First, we design {a randomized} O^*(2^{m(1-1/O(d))})-time algorithm, thereby breaking the 2^m barrier for every fixed d. Second, for the special case d = 2, we develop a significantly sharper branching algorithm running in O^*(1.1193ⁿ) time or O^*(1.3248^m) time. Third, leveraging the structural insights underlying the d = 2 case, we obtain an O^*(1.1052^L)-time algorithm for general Parity-SAT, where L denotes the formula length. All algorithms use only polynomial space. Notably, our running-time bounds are better than the best known bounds for the corresponding exact counting counterparts, highlighting a genuine algorithmic advantage of parity over counting. Conceptually, our results demonstrate that parity admits finer structural reductions and more efficient branching than exact model counting, and that bounded occurrence can be systematically leveraged to circumvent classical exponential barriers.

The automatic complexity of a finite word (string) is an analogue for finite automata of Sipser’s distinguishing complexity (1983) and was introduced by Shallit and Wang (2001). For a finite alphabet Σ of at least two elements, we consider the non-deterministic automatic complexity given by exactly - yet not necessarily uniquely - accepting automata: a word x ∈ Σ^* has exact non-deterministic automatic complexity k ∈ ℕ if there exists a non-deterministic automaton of k states which accepts x while rejecting every other word of the same length as x, and no automaton of fewer states has this property. Importantly, and in contrast to the classical notion, the witnessing automaton may have multiple paths of computation accepting x. We denote this measure of complexity by A_{Ne}, and study a class of languages of low A_{Ne}-complexity defined as L_q = {x ∈ Σ^* : A_{Ne}(x) < q|x|}, which is parameterised by rationals q ∈ (0,1/2) (generalising a class of sets first studied by Kjos-Hanssen). We show that for every q ∈ (0,1/2), this class is neither context-free nor recognisable by certain Boolean circuits. In the process, we answer an open question of Kjos-Hanssen quantifying the complexity of L_{1/3} in terms of Boolean circuits, and also prove the Shannon effect for A_{Ne}.

This paper studies the recursion-theoretic aspects of large-scale geometries of infinite strings, a subject initiated by Khoussainov and Takisaka (2017). We investigate several notions of quasi-isometric reductions between recursive infinite strings and prove various results on the equivalence classes of such reductions. The main result is the construction of two infinite recursive strings $α$ and $β$ such that $α$ is strictly quasi-isometrically reducible to $β$, but the reduction cannot be made recursive. This answers an open problem posed by Khoussainov and Takisaka.

This paper studies the complexity of operations on finite automata and the complexity of their decision problems when the alphabet is unary. Let $n$ denote the maximum of the number of states of the input finite automata considered in the corresponding results. The following main results are obtained: (1) Given two unary NFAs recognising $L$ and $H$, respectively, one can decide whether $L \subseteq H$ as well as whether $L = H$ in time $2^{O((n \log n)^{1/3})}$. The previous upper bound on time was $2^{O((n \log n)^{1/2})}$ as given by Chrobak (1986), and this bound was not significantly improved since then. (2) Given two unary UFAs (unambiguous finite automata) recognising $L$ and $H$, respectively, one can determine a UFA recognising $L \cup H$ and a UFA recognising complement of $L$, where these output UFAs have the number of states bounded by a quasipolynomial in $n$. However, in the worst case, a UFA for recognising concatenation of languages recognised by two $n$-state UFAs, uses $2^{Θ((n \log^2 n)^{1/3})}$ states. (3) Given a unary language $L$, if $L$ contains the word of length $k$, then let $L(k)=1$ else let $L(k)=0$. Let $ω_L$ be the $ω$-word $L(0)L(1)\ldots$ and let $\cal L$ be a fixed $ω$-regular language. The last section studies how difficult it is to decide, given an $n$-state UFA or NFA

Abstract The current work includes a result announced in the year 2012 which was unproven for 10 years. The result shows that there is a co-r.e. tree with uncountably many infinite branches such that the nonisolated infinite branches of the constructed tree are all nonrecursive, generalised low, and hyperimmune-free and form a perfect tree. This article is the journal version of two conference articles from 2012 and 2022; the article contains the main result proved in 2022 and also the other major results from 2012.

It is shown that the parity game can be solved in quasi-polynomial time. The parameterized parity game---with $n$ nodes and $m$ distinct values (a.k.a. colors or priorities)---is proven to be in the class of fixed parameter tractable problems when parameterized over $m$. Both results improve known bounds, from runtime $n^{O(\sqrt{n})}$ to $O(n^{\log(m)+6})$ and from an XP algorithm with runtime $O(n^{\Theta(m)})$ for fixed parameter $m$ to a fixed parameter tractable algorithm with runtime $O(n^5+2^{m\log(m)+6m})$. As an application, it is proven that colored Muller games with $n$ nodes and $m$ colors can be decided in time $O((m^m \cdot n)^5)$; it is also shown that this bound cannot be improved to $2^{o(m \cdot \log(m))} \cdot n^{O(1)}$ in the case that the exponential time hypothesis is true. Further investigations deal with memoryless Muller games and multidimensional parity games.

We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure μ on the space of infinite bit sequences is Martin-Löf absolutely continuous if the non-Martin-Löf random bit sequences form a null set with respect to μ. We think of this as a weak randomness notion for measures. We begin with examples, and a robustness property related to Solovay tests. Our main work connects our property to the growth of the initial segment complexity for measures μ; the latter is defined as a μ-average over the complexity of strings of the same length. We show that a maximal growth implies our weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We briefly discuss K-triviality for measures, which means that the growth of initial segment complexity is as slow as possible. We show that full Martin-Löf randomness of a measure implies Martin-Löf absolute continuity; the converse fails because only the latter property is compatible with having atoms. In a final section we consider weak randomness relative to a general ergodic computable measure. We seek appropriate effective versions of the Shannon-McMillan-Breiman theorem and the Brudno theorem where the bit sequences are replaced by measures.

Abstract We prove that the continuous function ${\rm{\hat \Omega }}:2^\omega \to $ that is defined via $X \mapsto \mathop \sum \limits_n 2^{ - K\left( {Xn} \right)} $ for all $X \in {2^\omega }$ is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that $\mathop \smallint \nolimits _0^1{\rm{\hat{\Omega }}}\left( X \right)\,{\rm{d}}X$ is a left-c.e. $wtt$ -complete real having effective Hausdorff dimension ${1 / 2}$ . We further investigate the algorithmic properties of ${\rm{\hat{\Omega }}}$ . For example, we show that the maximal value of ${\rm{\hat{\Omega }}}$ must be random, the minimal value must be Turing complete, and that ${\rm{\hat{\Omega }}}\left( X \right) \oplus X{ \ge _T}\emptyset \prime$ for every X. We also obtain some machine-dependent results, including that for every $\varepsilon > 0$ , there is a universal machine V such that ${{\rm{\hat{\Omega }}}_V}$ maps every real X having effective Hausdorff dimension greater than ε to a real of effective Hausdorff dimension 0 with the property that $X{ \le _{tt}}{{\rm{\hat{\Omega }}}_V}\left( X \right)$ ; and that there is a real X and a universal machine V such that ${{\rm{\Omega }}_V}\left( X \right)$ is rational.

X3SAT is the problem of whether one can satisfy a given set of clauses with up to three literals such that in every clause, exactly one literal is true and the others are false. A related question is to determine the maximal Hamming distance between two solutions of the instance. Dahllöf provided an algorithm for Maximum Hamming Distance XSAT, which is more complicated than the same problem for X3SAT, with a runtime of $O(1.8348^n)$; Fu, Zhou and Yin considered Maximum Hamming Distance for X3SAT and found for this problem an algorithm with runtime $O(1.6760^n)$. In this paper, we propose an algorithm in $O(1.3298^n)$ time to solve the Max Hamming Distance X3SAT problem; the algorithm actually counts for each $k$ the number of pairs of solutions which have Hamming Distance $k$.

XSAT is defined as the following: Given a propositional formula in conjunctive normal form, can one find an assignment to variables such that there is exactly only 1 literal that is true in every clause, while the other literals are false. The decision problem XSAT is known to be NP-complete. Crescenzi and Rossi [Pierluigi Crescenzi and Gianluca Rossi, 2002] introduced the variant where one searches for a pair of two solutions of an X3SAT instance with maximal Hamming Distance among them, that is, one wants to identify the largest number k such that there are two solutions of the instance with Hamming Distance k. Dahllöf [Vilhelm Dahllöf, 2005; Vilhelm Dahllöf, 2006] provided an algorithm using branch and bound method for Max Hamming Distance XSAT in O(1.8348^n); Fu, Zhou and Yin [Linlu Fu and Minghao Yin, 2012] worked on a more specific problem, the Max Hamming Distance X3SAT, and found for this problem an algorithm with runtime O(1.6760^n). In this paper, we propose an exact exponential algorithm to solve the Max Hamming Distance XSAT problem in O(1.4983^n) time. Like all of them, we will use the branch and bound technique alongside a newly defined measure to improve the analysis of the algorithm.

This paper introduces and studies a notion of \emph{algorithmic randomness} for subgroups of rationals. Given a randomly generated additive subgroup $(G,+)$ of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of $(G,+)$; second, what learnability properties can one extract from $G$ and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of $(G,+)$ coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for $G$ with respect to any generating sequence for $G$ is not even semi-decidable, one can build a generating sequence $β$ such that the word problem for $G$ with respect to $β$ is co-recursively enumerable (assuming that the set of generators of $G$ is limit-recursive). In regard to the second question, it is proven that there is a generating sequence $β$ for $G$ such that every non-trivial finitely generated subgroup of $G$ is recursively enumerable and the class of all such subgroups of $G$ is behaviourally correctly learnable, that is, every non-trivial finitely generated subgroup can be semantically identified in the limit (again assuming that the set of generators of $G$ is limit-recursive). On the other hand, the class of non-trivial finitely generated subgroups of $G$ cannot be syntactically identified in the limit with respect to any generating sequence for $G$. The present work thus contributes to a recent line of research studying algorithmically random infinite structures and uncovers an interesting connection between the arithmetical complexity of the set of generators of a randomly generated subgroup of rationals and the learnability of its finitely generated subgroups.

The present paper deals with the inductive inference of recursively enumerable languages from positive data (also called text). It introduces the learning models of \emph{verifiability} and \emph{comparability}. The input to a verifier is an index $e$ and a text of the target language $L$, and the learner has to \emph{verify} whether or not the index $e$ input is correct for the target language $L$. A comparator receives two indices of languages from the target class $\cL$ as input and has to decide in the limit whether or not these indices generate the same language. Furthermore, \emph{standardisability} is studied, where a \emph{standardiser} receives an index $j$ of some target language $L$ from the class $\cL$, and for every $L∈\cL$ there must be an index $e$ such that $e$ generates $L$ and the standardiser has to map every index $j$ for $L$ to $e$. Additionally, the common learning models of \emph{explanatory learning}, \emph{conservative explanatory learning}, and \emph{behaviourally correct learning} are considered. For almost all learning models mentioned above it is also appropriate to consider the number of times a learner changes its mind. In particular, if no mind change occurs then we obtain the \emph{finite} variant of the models considered. Occasionally, also learning with the help of an oracle is taken into consideration. The main goal of this paper is to figure out to what extent verifiability, comparability, and standardisability are helpful for the inductive inference of classes of recursively enumerable languages. Here we also distinguish between \emph{indexed families}, \emph{one-one enumerable classes}, and \emph{recursively enumerable classes}. Our results are manyfold, and an almost complete picture is obtained. In particular, for indexed families and recursively enumerable classes finite comparability, finite standardisability, and finite verifiability always imply finite learnability. If at least one mind change is allowed, then there are differences, i.e., for indexed families, comparability or verifiability imply conservative explanatory learning, but standardisability does not; still explanatory learning can be achieved.

We introduce the notion of limit-depth, as a notion similar to Bennett depth, but well behaved on Turing degrees, as opposed to truth-table degrees for Bennett depth. We show limit-depth satisfies similar properties to Bennett depth, namely both recursive and sufficiently random sequences are not limit-deep, and limit-depth is preserved over Turing degrees. We show both the halting problem and Chaitin's omega are limit-deep. We show every limit-deep set has DNR wtt-degree, and some limit-cuppable set does not have a limit-deep wtt degree.

10.1093/logcom/exx030

We study the connections between the learnability of automatic families of languages and the types of text used to present them to a learner. More precisely, we study how restrictions on the number of times that a correct datum appears in a text influence what classes of languages are automatically learnable. We show that an automatic family of languages is automatically learnable from fat text iff it is automatically learnable from thick text iff it is verifiable from balanced text iff it satisfies Angluin's tell-tale condition. Furthermore, many automatic families are automatically learnable from exponential text. We also study the relationship between automatic learnability and verifiability and show that all automatic families are automatically partially verifiable from exponential text and automatically learnable from thick text.

Parity games play an important role in model checking and synthesis. In their paper, Calude et al. have recently shown that these games can be solved in quasi-polynomial time. We show that their algorithm can be implemented efficiently: we use their data structure as a progress measure, allowing for a backward implementation instead of a complete unravelling of the game. To achieve this, a number of changes have to be made to their techniques, where the main one is to add power to the antagonistic player that allows for determining her rational move without changing the outcome of the game. We provide a first implementation for a quasi-polynomial algorithm, test it on small examples, and provide a number of side results, including minor algorithmic improvements, a quasi bi-linear complexity in the number of states and edges for a fixed number of colours, and matching lower bounds for the algorithm of Calude et al.

It is shown that the parity game can be solved in quasipolynomial time. The parameterised parity game - with n nodes and m distinct values (aka colours or priorities) - is proven to be in the class of fixed parameter tractable (FPT) problems when parameterised over m. Both results improve known bounds, from runtime nO(√n) to O(nlog(m)+6) and from an XP-algorithm with runtime O(nΘ(m)) for fixed parameter m to an FPT-algorithm with runtime O(n5)+g(m), for some function g depending on m only. As an application it is proven that coloured Muller games with n nodes and m colours can be decided in time O((mm · n)5); it is also shown that this bound cannot be improved to O((2m · n)c), for any c, unless FPT = W[1].

The present work investigates inductive inference from the perspective of reverse mathematics. Reverse mathematics is a framework which relates the proof strength of theorems and axioms throughout many areas of mathematics in an interdisciplinary way. The present work looks at basic notions of learnability including Angluin's tell-tale condition and its variants for learning in the limit and for conservative learning. Furthermore, the more general criterion of partial learning is investigated. These notions are studied in the reverse mathematics context for uniformly and weakly represented families of languages. The results are stated in terms of axioms referring to domination and induction strength.

Abstract In contrast with the notion of complexity, a setAis called anti-complex if the Kolmogorov complexity of the initial segments ofAchosen by a recursive function is always bounded by the identity function. We show that, as for complexity, the natural arena for examining anti-complexity is the weak-truth table degrees. In this context, we show the equivalence of anti-complexity and other lowness notions such as r.e. traceability or being weak truth-table reducible to a Schnorr trivial set. A setAis anti-complex if and only if it is reducible to another setBwithtiny use, whereby we mean that the use function for reducingAtoBcan be made to grow arbitrarily slowly, as gauged by unbounded nondecreasing recursive functions. This notion of reducibility is then studied in its own right, and we also investigate its range and the range of its uniform counterpart.

This paper investigates the complexity of verbal languages and pattern languages of Thurston automatic groups in terms of the Chomsky hierarchy. Here the language generated by a pattern is taken as the set of representatives of all strings obtained when chosing values for the various variables. For noncommutative free groups, it is shown that the complexity of the verbal and pattern languages (in terms of level on the Chomsky hierarchy) does not depend on the Thurston automatic representation and that verbal languages cannot be context-free (unless they are either the empty word or the full group). They can however be indexed languages. Furthermore, it is shown that in the general case, it might depend on the exactly chosen Thurston automatic representation which level a verbal language takes in the Chomsky hierarchy. There are examples of groups where, in an appropriate representation, all pattern languages are regular or context-free, respectively.

Abstract The truth-table degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domain-random strings, and show that the truth-table degrees of these sets depend on the underlying acceptable numbering. We achieve some additional properties for the truth-table incomplete versions of these sets, namely retraceability and approximability. We give priority-free constructions of bounded truth-table chains and bounded truth-table antichains inside the truth-table complete degree by identifying an acceptable set of domain-random strings within each degree.

We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-Löf, computably, Schnorr, and Kurtz random sets, weakly 1-generics and their complementary classes, we find that there exist characterizations of the third and fourth levels of the arithmetic hierarchy purely in terms of these notions. More generally, there exists an equivalence between arithmetic complexity and existence of numberings for classes of left-r.e. sets with shift-persistent elements. While some classes (such as Martin-Löf randoms and Kurtz nonrandoms) have left-r.e. numberings, there is no canonical, or acceptable , left-r.e. numbering for any class of left-r.e. randoms. Finally, we note some fundamental differences between left-r.e. numberings for sets and reals.

Abstract We give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations allow us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth-table degrees. This answers a question of Downey. Griffiths and LaForte.

For the natural notion of splitting classes into two disjoint subclasses via a recursive classifier working on texts, the question of how these splittings can look in the case of learnable classes is addressed. Here the strength of the classes is compared using the strong and weak reducibility from intrinsic complexity. It is shown that, for explanatorily learnable classes, the complete classes are also mitotic with respect to weak and strong reducibility, respectively. But there is a weakly complete class that cannot be split into two classes which are of the same complexity with respect to strong reducibility. It is shown that, for complete classes for behaviorally correct learning, one-half of each splitting is complete for this learning notion as well. Furthermore, it is shown that explanatorily learnable and recursively enumerable classes always have a splitting into two incomparable classes; this gives an inductive inference counterpart of the Sacks splitting theorem from recursion theory.

Following a line of research that aims at relating the computation power and the initial segment complexity of a set, the work presented here investigates into the relations between Turing reducibility, defined in terms of computation power, and C-reducibility and H-reducibility, defined in terms of the complexity of initial segments. The global structures of all C-degrees and of all H-degrees are rich and allows to embed the lattice of the powerset of the natural numbers under inclusion. In particular, there are C-degrees, as well as H-degrees, that are different from the least degree and are the meet of two other degrees, whereas on the other hand there are pairs of sets that have a meet neither in the C-degrees nor in the H-degrees; these results answer questions in a survey by Nies and Miller. There are r.e. sets that form a minimal pair for C-reducibility and Sigma20sets that form a minimal pair for H-reducibility, which answers questions by Downey and Hirschfeldt. Furthermore, the following facts on the relation between C-degrees, H-degrees and Turing degrees hold. Every C-degree contains at most one Turing degree and this bound is sharp since there are C-degrees that do contain a Turing degree. For the comprising class of complex sets, neither the C-degree nor the H-degree of such a set can contain a Turing degree, in fact, the Turing degree of any complex set contains infinitely many C-degrees. Similarly the Turing degree of any set that computes the halting problem contains infinitely many H-degrees, while the H-degree of any 2-random set R is never contained in the Turing degree of R. By the latter, H-equivalence of Martin-Lof random sets does not imply their Turing equivalence. The structure of the Cdegrees contained in the Turing degree of a complex sets is rich and allows to embed any countable distributive lattice; a corresponding statement is true for the structure of H-degrees that are contained in the Turing degree of a set that computes the halting problem.

Abstract In this paper we answer the following well-known open question in computable model theory. Does there exist a computable not ℵ0-categorical saturated structure with a unique computable isomor-phism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an ℵ1-categorical but not ℵ0-categorical saturated -structure with a unique computable isomorphism type. In addition, using the construction we give an example of an ℵ1-categorical but not ℵ0-categorical theory whose only non-computable model is the prime one.

We study the existence of automatic presentations for various algebraic structures. An automatic presentation of a structure is a description of the universe of the structure by a regular set of words, and the interpretation of the relations by synchronised automata. Our first topic concerns characterising classes of automatic structures. We supply a characterisation of the automatic Boolean algebras, and it is proven that the free Abelian group of infinite rank, as well as certain Fraisse limits, do not have automatic presentations. In particular, the countably infinite random graph and the random partial order do not have automatic presentations. Furthermore, no infinite integral domain is automatic. Our second topic is the isomorphism problem. We prove that the complexity of the isomorphism problem for the class of all automatic structures is \Sigma_1^1-complete.

This article extends Post's; programme to finite levels of the Ershov hierarchy of Δ2 sets. Our initial characterization, in the spirit of Post (1994, Bulletin of the American Mathematical Society, 50, 284–316), of the degrees of the immune and hyperimmune n-enumerable sets leads to a number of results setting other immunity properties in the context of the Turing and wtt-degrees derived from the Ershov hierarchy. For instance, we show that any n-enumerable hyperhyperimmune set must be co-enumerable, for each n ≥ 2. The situation with regard to the wtt-degrees is particularly interesting, as demonstrated by a range of results concerning the wtt-predecessors of hypersimple sets. Finally, we give a number of results directed at characterizing basic classes of n-enumerable degrees in terms of natural information content. For example, a 2-enumerable degree contains a 2-enumerable dense immune set iff it contains a 2-enumerable r-cohesive set iff it bounds a high enumerable set. This result is extended to a characterization of n-enumerable degrees which bound high enumerable degrees. Furthermore, a characterization for n-enumerable degrees bounding only low2 enumerable degrees is given.

Abstract A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x). f is a k(n)-enumerator if for every input x of length n. h(x) is among the first k(n) elements enumerated by f. If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A.

A set A is autoreducible if one can compute, for all x, the value $A(x)$ by querying A only at places $y \neq x$. Furthermore, A is infinitely‐often autoreducible if, for infinitely many x, the value $A(x)$ can be computed by querying A only at places $y \neq x$. For all other x, the computation outputs a special symbol to signal that the reduction is undefined. It is shown that for polynomial time Turing and truth‐table autoreducibility there are A, B, C in the class EXP of all exponential‐time computable sets such that A is not infinitely‐often Turing autoreducible, B is Turing autoreducible but not infinitely‐often truth‐table autoreducible and C is truth‐table autoreducible with $g(n)+1$ queries but not infinitely‐often Turing autoreducible with $g(n)$ queries. Here n is the length of the input, g is nondecreasing, and there exists a polynomial p such that $p(n)$ bounds both the computation time and the value of g at input of length n. Furthermore, connections between notions of infinitely‐often autoreducibility and notions of approximability are investigated. The Hausdorff‐dimension of the class of sets which are not infinitely‐often autoreducible is shown to be 1.

Abstract We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Löf random relative to ∅(n − 1). We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C(x) ≥ ∣x∣ − c. The ‘only if’ direction was obtained independently by Joseph Miller. This characterization can be extended to the case of time-bounded C-complexity. Next we prove some results on lowness. Among other things, we characterize the 2-random sets as those l-random sets that are low for Chaitin's Ω. Also, 2-random sets form minimal pairs with 2-generic sets. The r.e. low for Ω. sets coincide with the r.e. K-trivial ones. Finally we show that the notions of Martin-Löf randomness, recursive randomness, and Schnorr randomness can be separated in every high degree while the same notions coincide in every non-high degree. We make some remarks about hyperimmune-free and PA-complete degrees.

We answer a question of Ambos-Spies and Kucera in the affirmative. They asked whether, when a real is low for Schnorr randomness, it is already low for Schnorr tests.

We investigate partial orders that are computable, in a precise sense, by finite automata. Our emphasis is on trees and linear orders. We study the relationship between automatic linear orders and trees in terms of rank functions that are related to Cantor--Bendixson rank. We prove that automatic linear orders and automatic trees have finite rank. As an application we provide a procedure for deciding the isomorphism problem for automatic ordinals. We also investigate the complexity and definability of infinite paths in automatic trees. In particular, we show that every infinite path in an automatic tree with countably many infinite paths is a regular language.

This paper studies the existence of automatic presentations for various algebraic structures. The automatic Boolean algebras are characterised, and it is proven that the free Abelian group of infinite rank and many Fraisse limits do not have automatic presentations. In particular, the countably infinite random graph and the universal partial order do not have automatic presentations. Furthermore, no infinite integral domain is automatic. The second topic of the paper is the isomorphism problem. We prove that the complexity of the isomorphism problem for the class of all automatic structures is /spl Sigma//sub 1//sup 1/-complete.

We investigate partial orders that are computable, in a precise sense, by finite automata. Our emphasis is on trees and linear orders. We study the relationship between automatic linear orders and trees in terms of rank functions that are versions of Cantor-Bendixson rank. We prove that automatic linear orders and automatic trees have finite rank. As an application we provide a procedure for deciding the isomorphism problem for automatic ordinals. We also investigate the complexity and definability of infinite paths in automatic trees. In particular, we show that every infinite path in an automatic tree with countably many infinite paths is a regular language.

In this paper we investigate effective versions of Hausdorff dimension which have been recently introduced by Lutz. We focus on dimension in the class E of sets computable in linear exponential time. We determine the dimension of various classes related to fundamental structural properties including different types of autoreducibility and immunity. By a new general invariance theorem for resource-bounded dimension we show that the class of p-m-complete sets for E has dimension 1 in E. Moreover, we show that there are p-m-lower spans in E of dimension /spl Hscr/(/spl beta/) for any rational /spl beta/ between 0 and 1, where /spl Hscr/(/spl beta/) is the binary entropy function. This leads to a new general completeness notion for E that properly extends Lutz's concept of weak completeness. Finally we characterize resource-bounded dimension in terms of martingales with restricted betting ratios and in terms of prediction functions.

Abstract The following theorems on the structure inside nonrecursive truth-table degrees are established: Dëgtev's result that the number of bounded truth-table degrees inside a truth-table degree is at least two is improved by showing that this number is infinite. There are even infinite chains and antichains of bounded truth-table degrees inside every truth-table degree. The latter implies an affirmative answer to the following question of Jockusch: does every truth-table degree contain an infinite antichain of many-one degrees? Some but not all truth-table degrees have a least bounded truth-table degree. The technique to construct such a degree is used to solve an open problem of Beigel, Gasarch and Owings: there are Turing degrees (constructed as hyperimmune-free truth-table degrees) which consist only of 2-subjective sets and therefore do not contain any objective set. Furthermore, a truth-table degree consisting of three positive degrees is constructed where one positive degree consists of enumerable semirecursive sets, one of coenumerable semirecursive sets and one of sets, which are neither enumerable nor coenumerable nor semirecursive. So Jockusch's result that there are at least three positive degrees inside a truth-table degree is optimal. The number of positive degrees inside a truth-table degree can also be some other odd integer as for example nineteen, but it is never an even finite number.

Abstract For a fixed set A. the number of queries to A needed in order to decide a set S is a measure of S's complexity. We consider the complexity of certain sets defined in terms of A: and, for m > 2, where #nA. (x1….. xn) = A(x1) + A(xn)(We identify with , where χA is the characteristic function of A.) If A is a nonrecursive semirecursive set or if A is a jump, we give tight bounds on the number of queries needed in order to decide ODDnA and MODmnA: • ODDnA can be decided with n parallel queries to A, but not with n − 1. • ODDnA can be decided with ⌈log(n + 1)⌉ sequential queries to A but not with ⌈log(n + 1)⌉ − 1. • MODmnA can be decided with ⌈n/m⌉ + ⌊n/m⌋ parallel queries to A but not with ⌈n/m⌉ + ⌊n/m⌋ − 1. • MODmnA can be decided with ⌈log(⌈n/m⌉ + ⌊n/m⌋ + 1)⌉ sequential queries to A but not with ⌈log(⌈n/m⌉ + ⌊n/m⌋ + 1)⌉ − 1. The lower bounds above hold for nonrecursive recursively enumerable sets A as well. (Interestingly, the lower bounds for recursively enumerable sets follow by a general result from the lower bounds for semirecursive sets.) In particular, every nonzero truth-table degree contains a set A such that ODDnA cannot be decided with n − 1 parallel queries to A. Since every truth-table degree also contains a set B such that ODDnB can be decided with one query to B, a set's query complexity depends more on its structure than on its degree. For a fixed set A, Q(n, A) = {S: S can be decided with n sequential queries to A}. Q∥ (n, A) = {S : S can be decided with n parallel queries to A}. We show that if A is semirecursive or recursively enumerable, but is not recursive, then these classes form non-collapsing hierarchies: • Q(0,A) ⊂ Q (1, A) ⊂ Q(2, A) ⊂ … Q∥ (0, A) ⊂ Q∥ (1, A) ⊂ Q∥ (2, A) ⊂ … The same is true if A is a jump.

Article Free Access Share on Robust learning aided by context Authors: John Case Department of CIS, University of Delaware, Newark, DE Department of CIS, University of Delaware, Newark, DEView Profile , Sanjay Jain Department of Information Systems and Computer Science, National University of Singapore, Singapore 119260, Republic of Singapore Department of Information Systems and Computer Science, National University of Singapore, Singapore 119260, Republic of SingaporeView Profile , Matthias Ott Institut für Logik, Komplexität und Deduktionssysteme Universität Karlsruhe, 76128 Karlsruhe, Germany Institut für Logik, Komplexität und Deduktionssysteme Universität Karlsruhe, 76128 Karlsruhe, GermanyView Profile , Arun Sharma School of Computer Science and Engineering, University of New South Wales, Sydney 2052, Australia School of Computer Science and Engineering, University of New South Wales, Sydney 2052, AustraliaView Profile , Frank Stephan Mathematisches Institut, Universität Heidelberg, 69120 Heidelberg, Germany Mathematisches Institut, Universität Heidelberg, 69120 Heidelberg, GermanyView Profile Authors Info & Claims COLT' 98: Proceedings of the eleventh annual conference on Computational learning theoryJuly 1998 Pages 44–55https://doi.org/10.1145/279943.279952Published:24 July 1998Publication History 1citation230DownloadsMetricsTotal Citations1Total Downloads230Last 12 Months8Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Under the assumption that the Polynomial-Time Hierarchy does not collapse we show for a regular language L: the unbalanced polynomial-time leaf language class determined by L equals iff L is existentially but not quantifierfree definable in FO[<, min, max, +1, −1]. Furthermore, no such class lies properly between NP and co-1-NP or NP⊕co-NP. The proofs rely on a result of Pin and Weil characterizing the automata of existentially first-order definable languages.

Speed of convergence in Gold's identification in the limit model can be measured by deriving bounds on the number of mind changes made by a learner before the onset of convergence. Two approaches to date are bounds given by constants (referred here as Type 1) and bounds expressed as constructive ordinals (referred as Type 2). The use of ordinals has recently been successfully employed to measure the mind change complexity of learning rich concept classes such as unions of pattern languages, elementary formal systems and logic programs. Motivated by these applications, the present work introduces two more general approaches to bounding mind changes. These are based on counting by going down in a linearly ordered set (Type 3) and on counting by going down in a partially ordered set (Type 4). In both cases the set must not contain infinite, descending, and computable sequences. These four types of mind changes yield a hierarchy and there are identifiable classes that cannot be learned wit...

We characterize FIN-, EX-and BC-learning, as well as the corresponding notions of team learning, in terms of isolated branches on uniformly strongly recursive sequences of trees.Further, the more restrictive models of FIN-learning and strong-monotonic BC-learning can be characterized in terms of isolated branches on a single tree.We discuss learning with additional information where the learner receives an index for a strongly recursive tree such that the function to be learned is isolated on this tree.We show that EX-learning with this type of additional information is strictly more powerful than EX-learning.

Inductive inference considers two types of queries: Queries to a teacher about the function to be learned and queries to a non-recursive oracle.This paper combines these two types it considers three basic models of queries to a teacher, namely QEX[SUCC], QEX[<] and QEX[+], together with membership queries to some oracle.The results for these three models of queryinference are very similar: If an oracle is already omniscient for query-inference, then it is already omniscient for EX.There is an oracle of trivial EX-degree, which allows nontrivial query-inference.Furthermore, queries to a teacher can not overcome differences between oracles and the query-inference degrees are a proper refinement of the EX-degrees.

We study the fine structure of the classification of sets of natural numbers A according to the number of queries which are needed to compute the n-fold characteristic function of A. A complete characterization is obtained, relating the question to finite combinatorics. In order to obtain an explicit description we consider several interesting combinatorial problems.

Much structural work on NP-complete sets has exploited SAT′s d-self-reducibility. In this paper, we exploit the additional fact that SAT is a d-cylinder to show that NP-complete sets are p-superterse unless P = NP. In fact, every set that is NP-hard under polynomial-time no(1)-tt reductions is p-superterse unless P = NP. In particular, no p-selective set is NP-hard under polynomial-time no(1)-tt reductions unless P = NP. In addition, no easily countable set is NP-hard under Turing reductions unless P = NP. Self-reducibility does not seem to suffice far our main result: in a relativized world, we construct a d-self-reducible set in NP − P that is polynomial-time 2-tt reducible to a p-selective set.

Article Free Access Share on Inclusion problems in parallel learning and games (extended abstract) Authors: Martin Kummer Institut für Logik, Komplexität, und Deduktionssysteme, D-76128 Universität Karlsruhe, Germany Institut für Logik, Komplexität, und Deduktionssysteme, D-76128 Universität Karlsruhe, GermanyView Profile , Frank Stephan Institut für Logik, Komplexität, und Deduktionssysteme, D-76128 Universität Karlsruhe, Germany Institut für Logik, Komplexität, und Deduktionssysteme, D-76128 Universität Karlsruhe, GermanyView Profile Authors Info & Claims COLT '94: Proceedings of the seventh annual conference on Computational learning theoryJuly 1994Pages 287–298https://doi.org/10.1145/180139.181156Published:16 July 1994Publication History 3citation13DownloadsMetricsTotal Citations3Total Downloads13Last 12 Months5Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Degrees of inferability have been introduced to measure the learning power of inductive inference machines which have access to an oracle. The classical concept of degrees of unsolvability measures the computing power of oracles. In this paper we determine the relationship between both notions. 1 Introduction We consider learning of classes of recursive functions within the framework of inductive inference [21]. A recent theme is the study of inductive inference machines with oracles ([8, 10, 11, 17, 24] and tangentially [12]; cf. [10] for a comprehensive introduction and a collection of all previous results.) The basic question is how the information content of the oracle (technically: its Turing degree) relates with its learning power (technically: its inference degree---depending on the underlying inference criterion). In this paper a definitive answer is obtained for the case of recursively enumerable oracles and the case when only finitely many queries to the oracle are allo...

A generic approach to constructing the topology of fixed degree directed networks for a given node degree and size is proposed. The topology has minimum diameter and makes it possible to answer Maekawa's problem partially. In addition, an expansion operator Ex(x,n) is provided to scale the network size. Fast routing algorithms for these networks that have logarithmic time complexity while using constant buffer size are presented.>