We introduce a new technique for proving bounds on the Kolmogorov complexity of geometric objects in Euclidean space, such as points and lines. We apply this technique to prove two theorems on algorithmic information theory, both of which have consequences for well-known problems in geometric measure theory. First, we show that for any point $x$ in the plane and any other point $y$ sufficiently independent of $x$, the distance between $x$ and $y$ retains at least half the complexity of the original point $x$. By the point-to-set principle of J. Lutz and N. Lutz, this yields an improved lower bound on the Hausdorff dimension of pinned distance sets, a topic closely related to Falconer's distance set conjecture. Second, we prove an analogous result for orthogonal projections: for any point $x$ in the plane and any line through the origin which is sufficiently independent of $x$, the projection of $x$ onto that line retains at least half the complexity of $x$. As a consequence, we obtain a generalization of a theorem of Bourgain on exceptional sets for orthogonal projections.

Characterizing neural networks in terms of better-understood formal systems has the potential to yield new insights into the power and limitations of these networks. Doing so for transformers remains an active area of research. Bhattamishra and others have shown that transformer encoders are at least as expressive as a certain kind of counter machine, while Merrill and Sabharwal have shown that fixed-precision transformer encoders recognize only languages in uniform $TC^0$. We connect and strengthen these results by identifying a variant of first-order logic with counting quantifiers that is simultaneously an upper bound for fixed-precision transformer encoders and a lower bound for transformer encoders. This brings us much closer than before to an exact characterization of the languages that transformer encoders recognize.

Although transformers are remarkably effective for many tasks, there are some surprisingly easy-looking regular languages that they struggle with.Hahn shows that for languages where acceptance depends on a single input symbol, a transformer's classification decisions become less and less confident (that is, with crossentropy approaching 1 bit per string) as input strings get longer and longer.We examine this limitation using two languages: PAR-ITY, the language of bit strings with an odd number of 1s, and FIRST, the language of bit strings starting with a 1.We demonstrate three ways of overcoming the limitation suggested by Hahn's lemma.First, we settle an open question by constructing a transformer that recognizes PARITY with perfect accuracy, and similarly for FIRST.Second, we use layer normalization to bring the cross-entropy of both models arbitrarily close to zero.Third, when transformers need to focus on a single position, as for FIRST, we find that they can fail to generalize to longer strings; we offer a simple remedy to this problem that also improves length generalization in machine translation.

Abstract We consider the question “Is every nonzero generic degree a density-1-bounding generic degree?” By previous results [8] either resolution of this question would answer an open question concerning the structure of the generic degrees: A positive result would prove that there are no minimal generic degrees, and a negative result would prove that there exist minimal pairs in the generic degrees. We consider several techniques for showing that the answer might be positive, and use those techniques to prove that a wide class of assumptions is sufficient to prove density-1-bounding. We also consider a historic difficulty in constructing a potential counterexample: By previous results [7] any generic degree that is not density-1-bounding must be quasiminimal, so in particular, any construction of a non-density-1-bounding generic degree must use a method that is able to construct a quasiminimal generic degree. However, all previously known examples of quasiminimal sets are also density-1, and so trivially density-1-bounding. We provide several examples of non-density-1 sets that are quasiminimal. Using cofinite and mod-finite reducibility, we extend our results to the uniform coarse degrees, and to the nonuniform generic degrees. We define all of the above terms, and we provide independent motivation for the study of each of them. Combined with a concurrently written paper of Hirschfeldt, Jockusch, Kuyper, and Schupp [4], this paper provides a characterization of the level of randomness required to ensure quasiminimality in the uniform and nonuniform coarse and generic degrees.

Abstract Soare [20] proved that the maximal sets form an orbit in ${\cal E}$ . We consider here ${\cal D}$ -maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of ${\cal D}$ -maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the ${\cal D}$ -maximal sets. Although these invariants help us to better understand the ${\cal D}$ -maximal sets, we use them to show that several classes of ${\cal D}$ -maximal sets break into infinitely many orbits.

Abstract We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the proof-theoretic strength of the regularity of Lebesgue measure for Gδ sets. Our constructions essentially settle the reverse mathematical classification of this principle.

In reverse mathematics, one formalizes theorems of ordinary mathematics in second order arithmetic and attempts to discover which set theoretic axioms are required to prove these theorems. Often, this project involves making choices between classically equivalent definitions for the relevant mathematical concepts. In this paper, we consider a number of equivalent definitions for the notions of well quasi-order and better quasi-order and examine how difficult it is to prove the equivalences of these definitions. As usual in reverse mathematics, we work in the context of subsystems of second order arithmetic and take RCA0 as our base system. RCA0 is the subsystem formed by restricting the comprehension scheme in second order arithmetic to formulas and adding a formula induction scheme for formulas. For the purposes of this paper, we will be concerned with fairly weak extensions of RCA0 (indeed strictly weaker than the subsystem ACA0 which is formed by extending the comprehension scheme in RCA0 to cover all arithmetic formulas) obtained by adjoining certain combinatorial principles to RCA0. Among these, the most widely used in reverse mathematics is Weak König's Lemma; the resulting theory WKL0 is extensively documented in [11] and elsewhere. We give three other combinatorial principles which we use in this paper. In these principles, we use k to denote not only a natural number but also the finite set {0, …, k − 1}.

Abstract We show that if A and are automorphic via Φ then the structures (A) and ( ) are Δ30-isomorphic via an isomorphism Ψ induced by Φ. Then we use this result to classify completely the orbits of hhsimple sets.

Abstract A computably enumerable (c.e.) degree is a maximal contiguous degree if it is contiguous and no c.e. degree strictly above it is contiguous. We show that there are infinitely many maximal contiguous degrees. Since the contiguous degrees are definable, the class of maximal contiguous degrees provides the first example of a definable infinite anti-chain in the c.e. degrees. In addition, we show that the class of maximal contiguous degrees forms an automorphism base for the c.e. degrees and therefore for the Turing degrees in general. Finally we note that the construction of a maximal contiguous degree can be modified to answer a question of Walk about the array computable degrees and a question of Li about isolated formulas.

Abstract We show there is a non-recursive r.e. set A such that if W is any low r.e. set. then the join W ⊕ A is also low. That is. A is “almost deep”. This answers a question of Joekusch. The almost deep degrees form an definable ideal in the r.e. degrees (with jump.)

Abstract We study the proof–theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT k n denote Ramsey's theorem for k –colorings of n –element sets, and let RT <∞ n denote (∀ k )RT k n . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k –coloring of the n –element sets of natural numbers, there is an infinite homogeneous set X with X ″ ≤ T 0 ( n ) . Let I Σ n and B Σ n denote the Σ n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low 2 ) to models of arithmetic enables us to show that RCA 0 + I Σ 2 + RT 2 2 is conservative over RCA 0 + I Σ 2 for Π 1 1 statements and that RCA 0 + I Σ 3 + RT <∞ 2 is Π 1 1 -conservative over RCA 0 + I Σ 3 . It follows that RCA 0 + RT 2 2 does not imply B Σ 3 . In contrast, J. Hirst showed that RCA 0 + RT <∞ 2 does imply B Σ 3 , and we include a proof of a slightly strengthened version of this result. It follows that RT <∞ 2 is strictly stronger than RT 2 2 over RCA 0 .

Effective model theory is the subject that analyzes the typical notions and results of model theory to determine their effective content and counterparts. The subject has been developed both in the former Soviet Union and in the west with various names (recursive model theory, constructive model theory, etc.) and divergent terminology. (We use “effective model theory” as the most general and descriptive designation. Harizanov [6] is an excellent introduction to the subject as is Millar [13].) The basic subjects of model theory include languages, structures, theories, models and various types of maps between these objects. There are many ways to introduce considerations of effectiveness into the area. The two most prominent derive from starting, on the one hand, with the notion of a theory and its models or, on the other, with just structures. If one begins with theories, then a natural version of effectiveness is to consider decidable theories (i.e., ones with a decidable (equivalently, computable or recursive) set of theorems). When one moves to models and wants them to be effective, one might start with the requirement that the model (of any theory) have a decidable theory (i.e., Th ( ), the set of sentences true in , is decidable). Typically, however, one wants to be able to talk about the elements of the model as well as its theory in the given language. Thus one naturally considers the model as a structure for the language expanded by adding a constant ai, for each element ai of . Of course, one requires that the mapping from the constants to the corresponding elements of be effective (computable). We are thus lead to the following basic definition: A structure or model is decidable if there is a computable enumeration ai of A, the domain of , such that Th( , ai,) is decidable. (Of course, ai, is interpreted as ai, for each i Є ω.)

We prove conjectures of Herrmann and Stob by showing that the dense simple sets are orbit complete w.r.t. the simple sets.

The class of locally stratified logic programs is shown to be Pi-1-1 complete by the construction of a reducibility of the class of infinitely branching nondeterministic finite register machines.

Abstract The main result of this paper is to show that for every recursive ordinal α ≠ 0 and for every nonrecursive r.e. degree d there is a r.e. set of rank α and degree d.

Most theories of learning consider inferring a function f from either (1) observations about f or, (2) questions about f. We consider a scenario whereby the learner observes fand asks queries to some set A. EX[A] is the set of concept classes EX-learnable by an inductive inference machine with oracle A. A and F are EX-equivalent if EX[A] = EX[B]. The equivalence classes induced are the degrees of inferability. We prove several results about these degrees: (1) There are an uncountable number of degrees. (2) For A r.e., REC e BC[A] iff O'' ≤T A´, and there is evidence this holds for all sets A. (3) For A, B r.e., A ≡T B iff EX[A] = EX[B]. (4) There exists A, B low2 r.e., A|RB, EX[A] = EX[B]. (hence (3) is optimal).

An important program in the study of the structure the lattice of r.e. sets modulo finite sets, is the classification of the orbits under Aut( ), the automorphism group of , If Φ ∈ Aut( ) and Φ(We) =* Wh(e) for all e ∈ ω, then h is called a presentation of Φ. Define Autχ( ) to be the class of those elements of Aut( ) that have a presentation in the class of functions X, where for instance X might be the class of Δn functions for n ∈ ω. If Φ ∈ Autx( ) then we will say that Φ is an X-automorphism. Note that we only need to consider Δn- and Πn-orbits and automorphisms since if f is a Σn-presentation, then f is a total function and therefore Δn. Definition. If X is a class of functions and ⊆ {Wi: i < ω}, then is an X-orbit iff is an orbit under Aut( ) and for all A, B ∈ there is a Φ ∈ Autx( ) satisfying Φ(A) =* B. (Here Φ: → .) Harrington proved that the creative sets form a Δ0-orbit and Soare proved that the maximal sets form Δ3-orbit. We will show that there is no Boolean algebra such that {A: A is r.e. and ℒ*(A) ≈ ] forms a Δ2-orbit, where ℒ*(A) is the principal filter of A in ; ℒ*(A) = {B: B ⊆* A & B ∈ }. The idea behind the proof of this theorem is very similar to the proof by Soare that maximal sets do not form a Δ2-orbit (see Soare [1974] or Soare [1987]).