We extend an independence result proved in [arana2]. We show that for all n, there is a special set of Π_n sentences {φ_a}_{a∈ H} corresponding to elements of a linear ordering (H,<_H) of order type ω₁^CK(1+η). These sentences allow us to build completions {T_a}_{a∈ H} of PA such that for a<_H b, T_a ∩ Σ_n ⊂ T_b ∩ Σ_n, with φ_a∈ T_a, ¬φ_a ∈ T_b. Our method uses the Barwise—Kreisel Compactness Theorem.

The notion of quasi-minimal structures was defined by B. Zil’ber as a natural generalization of minimal structures. Inspired by his work, we study here basic model theoretic properties of quasi-minimal structures. Main result is the construction of ω-saturated quasi-minimal models under ω-stability assumption.

A Carter subgroup is a self-normalizing locally nilpotent subgroup. For studying these subgroups in groups of finite Morley rank, we introduce the new notion of a locally closed subgroup. We show that every solvable group of finite Morley rank has a unique conjugacy class of Carter subgroups.

A type analysable in one-based types in a simple theory is itself one-based.

We show that there is a structure of countably infinite signature with P=N₂P and a structure of finite signature with P=N₁P and N₁P ≠ N₂P. We give a further example of a structure of finite signature with P ≠ N₁P and N₁P ≠ N₂P. Together with a result from [Koiran] this implies that for each possibility of P versus NP over structures there is an example of countably infinite signature. Then we show that for some finite ℒ the class of ℒ-structures with P=N₁P is not closed under ultraproducts and obtain as corollaries that this class is not Δ-elementary and that the class of ℒ-structures with P ≠ N₁P is not elementary. Finally we prove that for all f dominating all polynomials there is a structure of finite signature with the following properties: P ≠ N₁P, N₁P ≠ N₂P, the levels N₂TIME(nⁱ) of N₂P and the levels N₁TIME(nⁱ) of N₁P are different for different i, indeed DTIME(n^i’) ⊈ N₂TIME(nⁱ) if i’>i, DTIME(f) ⊈ N₂P, and N₂P ⊈ DEC. DEC is the class of recognizable sets with recognizable complements. So this is an example where the internal structure of N₂P is analyzed in a more detailed way. In our proofs we use methods in the style of classical computability theory to construct structures except for one use of ultraproducts.

Assuming 𝖢𝖧, Hindman [ht1] showed that the existence of certain ultrafilters on the power set of the natural numbers is equivalent to Hindman’s Theorem. Adapting this work to a countable setting formalized in RCA₀, this article proves the equivalence of the existence of certain ultrafilters on countable Boolean algebras and an iterated form of Hindman’s Theorem, which is closely related to Milliken’s Theorem. A computable restriction of Hindman’s Theorem follows as a corollary.

We use a κ⁺-Mahlo cardinal to give a forcing construction of a model in which there is no sequence 〈 A_γ:γ<ω₂ 〉 of sets of cardinality ω₁ such that {λ<ω₂:∃ c⊂λ (⋃ c=λ & otp(c)=ω₁&∀γ<λ (c∩γ∈ A_γ))} is stationary.

A family A⊆𝒫(ω) is called countably splitting if for every countable F⊆[ω]^ω, some element of A splits every member of F. We define a notion of a splitting tree, by means of which we prove that every analytic countably splitting family contains a closed countably splitting family. An application of this notion solves a problem of Blass. On the other hand we show that there exists an F_σ splitting family that does not contain a closed splitting family.

This paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].

We introduce an operation exists_n(S) on properties S that says "there are n components having S". We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As a corollary, if the first order quantifiers are not already absorbed in V, then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure of V are strict.

We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of properties S, S cannot belong to the positive first order closure of a monadic prefix class W unless it already belongs to W.

We introduce another operation alt(S) on properties which has the same relationship with the Circuit Value Problem as reach(S) (defined in [JM01]) has with the Directed Reachability Problem. We use alt(S) to show that Π_n⊈ FO(Σ_n), Σ_n⊈ FO(Δ_n), and Δ_n+1⊈ FOB(Σ_n), solving some open problems raised in [Mat98].

We prove that a computably enumerable set A is effectively speedable (effectively levelable) if and only if there exists a splitting (A₀,A₁) of A such that both A₀ and A₁ are effectively speedable (effectively levelable). These results answer two questions raised by J. B. Remmel.

We prove, in ZFC, the existence of a definable, countably saturated elementary extension of the reals.

We prove that Standardization fails in every nontrivial universe definable in the nonstandard set theory BST, and that a natural characterization of the standard universe is both consistent with and independent of BST. As a consequence we obtain a formulation of nonstandard class theory in the ∈-language.

The paper discusses the notion of finite model truth definitions (or FM-truth definitions), introduced by M. Mostowski as a finite model analogue of Tarski’s classical notion of truth definition.

We compare FM-truth definitions with Vardi’s concept of the combined complexity of logics, noting an important difference: the difficulty of defining FM-truth for a logic ℒ does not depend on the syntax of ℒ, as long as it is decidable. It follows that for a natural ℒ there exist FM-truth definitions whose evaluation is much easier than the combined complexiy of ℒ would suggest.

We apply the general theory to give a complexity-theoretical characterization of the logics for which the Σ^d_m classes (prenex classes of higher order logics) define FM-truth. For any d≥ 2, m≥ 1 we construct a family {[Σ^d_m]^{≤ k}}_k∈ω of syntactically defined fragments of Σ^d_m which satisfy this characterization. We also use the [Σ^d_m]^{≤ k} classes to give a refinement of known results on the complexity classes captured by Σ^d_m.

We close with a few simple corollaries, one of which gives a sufficient condition for the existence, given a vocabulary σ, of a fixed number k such that model checking for all first order sentences over σ can be done in deterministic time n^k.

We define the notion of approximate Euler characteristic of definable sets of a first order structure. We show that a structure admits a non-trivial approximate Euler characteristic if it satisfies weak pigeonhole principle WPHP²ⁿ_n: two disjoint copies of a non-empty definable set A cannot be definably embedded into A, and principle CC of comparing cardinalities: for any two definable sets A, B either A definably embeds in B or vice versa. Also, a structure admitting a non-trivial approximate Euler characteristic must satisfy WPHP²ⁿ_n.

Further we show that a structure admits a non-trivial dimension function on definable sets if and only if it satisfies weak pigeonhole principle WPHPⁿ²_n: for no definable set A with more than one element can A² definably embed into A.

A compact complex space X is viewed as a 1-st order structure by taking predicates for analytic subsets of X, X \times X, … as basic relations. Let f: X→ Y be a proper surjective holomorphic map between complex spaces and set X_y:=f^-1(y). We show that the set A_k,d:={y∈ Y: the number of d-dimensional components of X_y is <k} is analytically constructible, i.e., is a definable set when X and Y are compact complex spaces and f: X→ Y is a holomorphic map. The analogous result in the context of algebraic geometry gives rise to the definability of Morley degree.

We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal μ, of cofinality ω, such that every μ⁺-chromatic graph X on μ⁺ has an edge colouring c of X into μ colours for which every vertex colouring g of X into at most μ many colours has a g-colour class on which c takes every value.

The paper also contains some generalisations of the above statement in which μ⁺ is replaced by other cardinals >μ.

This article is a continuation of our search for tautologies that are hard even for strong propositional proof systems like EF, cf. [Kra-wphp,Kra-tau]. The particular tautologies we study, the τ-formulas, are obtained from any 𝒫/poly map g; they express that a string is outside of the range of g. Maps g considered here are particular pseudorandom generators. The ultimate goal is to deduce the hardness of the τ-formulas for at least EF from some general, plausible computational hardness hypothesis.

In this paper we introduce the notions of pseudo-surjective and iterable functions (related to free functions of [Kra-tau]). These two properties imply the hardness of the τ-formulas from the function but unlike the hardness they are preserved under composition and iteration. We link the existence of maps with these two properties to the provability of circuit lower bounds, and we characterize maps g yielding hard τ-formulas in terms of a hitting set type property (all relative to a propositional proof system). We show that a proof system containing EF admits a pseudo-surjective function unless it simulates a proof system WF introduced by Jeřábek [Jer], an extension of EF. We propose a concrete map g as a candidate function possibly pseudo-surjective or free for strong proof systems. The map is defined as a Nisan-Wigderson generator based on a random function and on a random sparse matrix. We prove that it is iterable in a particular way in resolution, yielding the output/input ratio n^3-ε (that improves upon a direct construction of Alekhnovich et al. [ABRW]).

Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and “nearly” minimal size, i.e., within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. A lim-computable function is (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be “not-so-nearly” minimal size, e.g., to be within a lim-computable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are lim-computable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of lim-computability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the lim-computable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight.