Unary Predicates Research Articles

ORDERED STABILITY AND EXPANSIONS OF A PURE LINEAR ORDER BY A UNARY FUNCTION

A linearly ordered structure is said to be o-stable if each of its Dedekind cut has a “small” number of extensions to complete 1-types. This concept, which was introduced by B.S. Baizhanov and V.V. Verbovsky, generalizes such widely known concepts among specialists in model theory as weak o-minimality, (weak) quasi-o-minimality and dp-minimality of ordered structures. It is based on a combination of the concepts of o-minimality and stability. As we know, the elementary theory of any pure linear order is o-superstable. Indeed, this follows from the fact, which Rubin proved in the late 70s of the 20th century, that any type of one variable is determined by its cut and definable subsets, distinguished by unary predicates or formulas with one free variable. In this paper, we explore the question of what happens if a pure linear order is expanded with a unary function. Two examples were constructed when o-stability is violated; in addition, sufficient conditions for preserving o-stability with such language expansion were found. Research work on this topic is not yet finished, ideally, it would be good to find a criterion for preserving ordered stability when enriching a structure with pure linear order with a new function of one variable.

Convex geometries representable with colors, by ellipses on the plane, and impossible by circles

A convex geometry is a closure system satisfying the anti-exchange property. This paper, following the work of Adaricheva and Bolat (Discrete Math 342(N3):726–746, 2019) and the Polymath REU 2020 team (Convex geometries representable by at most 5 circles on the plane. arXiv:2008.13077), continues to investigate representations of convex geometries on a 5-element base set. It introduces several properties: the opposite property, nested triangle property, area Q property, and separation property, of convex geometries of circles on a plane, preventing this representation for numerous convex geometries on a 5-element base set. It also demonstrates that all 672 convex geometries on a 5-element base set have a representation by ellipses, as given in the appendix for those without a known representation by circles, and introduces a method of expanding representation with circles by defining unary predicates, shown as colors.

Enriching a predicate and tame expansions of the integers

Given a structure [Formula: see text] and a stably embedded [Formula: see text]-definable set [Formula: see text], we prove tameness preservation results when enriching the induced structure on [Formula: see text] by some further structure [Formula: see text]. In particular, we show that if [Formula: see text] and [Formula: see text] are stable (respectively, superstable, [Formula: see text]-stable), then so is the theory [Formula: see text] of the enrichment of [Formula: see text] by [Formula: see text]. Assuming simplicity of [Formula: see text], elimination of hyperimaginaries and a further condition on [Formula: see text] related to the behavior of algebraic closure, we also show that simplicity and NSOP1 pass from [Formula: see text] to [Formula: see text]. We then prove several applications for tame expansions of weakly minimal structures and, in particular, the group of integers. For example, we construct the first known examples of strictly stable expansions of [Formula: see text]. More generally, we show that any stable (respectively, superstable, simple, NIP, NTP2, NSOP1) countable graph can be defined in a stable (respectively, superstable, simple, NIP, NTP2, NSOP1) expansion of [Formula: see text] by some unary predicate [Formula: see text].

AN ALGEBRAIC PROOF OF COMPLETENESS FOR MONADIC FUZZY PREDICATE LOGIC

Abstract Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$ -valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$ , which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$ . The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$ , we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$ , giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$ , and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras.

CLASSIFICATION OF -CATEGORICAL MONADICALLY STABLE STRUCTURES

AbstractA first-order structure $\mathfrak {A}$ is called monadically stable iff every expansion of $\mathfrak {A}$ by unary predicates is stable. In this paper we give a classification of the class $\mathcal {M}$ of $\omega $ -categorical monadically stable structure in terms of their automorphism groups. We prove in turn that $\mathcal {M}$ is the smallest class of structures which contains the one-element pure set, is closed under isomorphisms, and is closed under taking finite disjoint unions, infinite copies, and finite index first-order reducts. Using our classification we show that every structure in $\mathcal {M}$ is first-order interdefinable with a finitely bounded homogeneous structure. We also prove that every structure in $\mathcal {M}$ has finitely many reducts up to interdefinability, thereby confirming Thomas’ conjecture for the class $\mathcal {M}$ .

Companionability characterization for the expansion of an o-minimal theory by a dense subgroup

This paper provides a full characterization for when the expansion of a complete o-minimal theory, one that extends the theory of ordered divisible abelian groups, by a unary predicate that picks out a divisible, dense and codense group has a model companion. This result is motivated by criteria and questions introduced in the recent works [14] and [10] concerning the existence of model companions, as well as preservation results for some neostability properties when passing to the model companion. Examples are included both in which the predicate is an additive subgroup of a real ordered vector space, and where it is a multiplicative subgroup of the nonzero elements of an o-minimal expansion of a real closed field. The paper concludes with a brief discussion of neostability properties and examples that illustrate the lack of preservation (from the base o-minimal theory to the model companion of the expansion we define) for properties such as strong, NIP, and NTP2, though there are also examples for which some or all three of those properties are preserved.

Asymptotic conditional probabilities for binary probability functions

The paper investigates the class of probability functions defined on sentences of a predicate language containing binary and possibly also unary predicates which satisfy the Principle of Binary Exchangeability (a strengthening of the Principle of Exchangeability). Following a survey of relevant properties of such functions we prove the main theorems of this paper showing that under some mild conditions any such probability function can, by conditioning on suitable sentences, closely approximate any other probability function from this class. These results on conditioning extend earlier results obtained for purely unary languages.

ALMOST 1-TRANSITIVITY IN LINEARLY ORDERED STRUCTURES

The present paper concerns the notion of weak o-minimality introduced by M. Dickmann and originally deeply studied by D. Macpherson, D. Marker, and C. Steinhorn. Weak o-minimality is a generalization of the notion of o-minimality introduced by A. Pillay and C. Steinhorn in series of joint papers. As is known, the ordered field of real numbers is an example of an o-minimal structure. We continue studying model-theoretic properties of o-minimal and weakly o-minimal structures. In particular, we introduce the notion of almost 1-transitivity in linearly ordered structures and study tits properties. Almost 1-transitive o-minimal and weakly o-minimal linear orderings have been described. It has been established that an almost 1-transitive weakly o-minimal linear ordering is isomorphic to a finite number of concatenations of almost 1-transitive o-minimal linear orderings. Properties of expansions of families of almost 1-transitive linearly ordered theories are studied. Rank values for families of almost 1-transitive o-minimal and weakly o-minimal linear orderings have been found. A criterion for preserving both the almost 1-transitivity and weak o-minimality has been found at expanding an almost 1-transitive weak o-minimal theory by an arbitrary unary predicate. Dense ordering of an almost 1-transitive weakly o-minimal theory that is almost omega-categorical has been established.

Deciding FO-rewritability of Regular Languages and Ontology-Mediated Queries in Linear Temporal Logic

Our concern is the problem of determining the data complexity of answering an ontology-mediated query (OMQ) formulated in linear temporal logic LTL over (Z,<) and deciding whether it is rewritable to an FO(<)-query, possibly with some extra predicates. First, we observe that, in line with the circuit complexity and FO-definability of regular languages, OMQ answering in AC0, ACC0 and NC1 coincides with FO(<,≡)-rewritability using unary predicates x ≡ 0 (mod n), FO(<,MOD)-rewritability, and FO(RPR)-rewritability using relational primitive recursion, respectively. We prove that, similarly to known PSᴘᴀᴄᴇ-completeness of recognising FO(<)-definability of regular languages, deciding FO(<,≡)- and FO(<,MOD)-definability is also PSᴘᴀᴄᴇ-complete (unless ACC0 = NC1). We then use this result to show that deciding FO(<)-, FO(<,≡)- and FO(<,MOD)-rewritability of LTL OMQs is ExᴘSᴘᴀᴄᴇ-complete, and that these problems become PSᴘᴀᴄᴇ-complete for OMQs with a linear Horn ontology and an atomic query, and also a positive query in the cases of FO(<)- and FO(<,≡)-rewritability. Further, we consider FO(<)-rewritability of OMQs with a binary-clause ontology and identify OMQ classes, for which deciding it is PSᴘᴀᴄᴇ-, Π2p- and coNP-complete.

The subset relation and 2‐stratified sentences in set theory and class theory

AbstractHamkins and Kikuchi (2016, 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. This paper identifies the minimum subsystem of , , that ensures that the definable subset ordering of the universe interprets a complete theory, and classifies the structures that can be realised as the subset relation in a model of this set theory. Extending and refining Hamkins and Kikuchi's result for class theory, a complete extension, , of the theory of infinite atomic boolean algebras and a minimum subsystem, , of are identified with the property that if is a model of , then is a model of , where M is the underlying set of , is the unary predicate that distinguishes sets from classes and is the definable subset relation. These results are used to show that that decides every 2‐stratified sentence of set theory and decides every 2‐stratified sentence of class theory.

AN ESCAPE FROM VARDANYAN’S THEOREM

AbstractVardanyan’s Theorems [36, 37] state that $\mathsf {QPL}(\mathsf {PA})$ —the quantified provability logic of Peano Arithmetic—is $\Pi ^0_2$ complete, and in particular that this already holds when the language is restricted to a single unary predicate. Moreover, Visser and de Jonge [38] generalized this result to conclude that it is impossible to computably axiomatize the quantified provability logic of a wide class of theories. However, the proof of this fact cannot be performed in a strictly positive signature. The system $\mathsf {QRC_1}$ was previously introduced by the authors [1] as a candidate first-order provability logic. Here we generalize the previously available Kripke soundness and completeness proofs, obtaining constant domain completeness. Then we show that $\mathsf {QRC_1}$ is indeed complete with respect to arithmetical semantics. This is achieved via a Solovay-type construction applied to constant domain Kripke models. As corollaries, we see that $\mathsf {QRC_1}$ is the strictly positive fragment of $\mathsf {QGL}$ and a fragment of $\mathsf {QPL}(\mathsf {PA})$ .

ON 1-INDISCERNIBILITY OF E-COMBINATIONS OF ORDERED THEORIES

In this paper, we investigate properties that are preserved or acquired when combining an arbitrary number of theories or structures. Recently, an interest has been shown in the study of P-combinations (when each structure is distinguished by a separate unary predicate) and E-combinations (when each structure is distinguished by a separate class of equivalence with respect to E). Here we studied the properties of E-combinations of linearly ordered theories. The 1-indiscernibilty and density of a weakly o-minimal E-combination of countably many copies of an almost omega-categorical weakly o-minimal theory in a language that does not contain distinguished constants are established.

Characterizations of monadic NIP

We give several characterizations of when a complete first-order theory T T is monadically NIP, i.e. when expansions of T T by arbitrary unary predicates do not have the independence property. The central characterization is a condition on finite satisfiability of types. Other characterizations include decompositions of models, the behavior of indiscernibles, and a forbidden configuration. As an application, we prove non-structure results for hereditary classes of finite substructures of non-monadically NIP models that eliminate quantifiers.

Pathological examples of structures with o‐minimal open core

AbstractThis paper answers several open questions around structures with o‐minimal open core. We construct an expansion of an o‐minimal structure by a unary predicate such that its open core is a proper o‐minimal expansion of . We give an example of a structure that has an o‐minimal open core and the exchange property, yet defines a function whose graph is dense. Finally, we produce an example of a structure that has an o‐minimal open core and definable Skolem functions, but is not o‐minimal.

Exorcising the phantom zone

In this paper we introduce a language of first-order hybrid logic in which function symbols are interpreted by partial functions and prove a number of completeness results. Syntactically, our language builds on the basic propositional hybrid language, has a primitive unary predicate symbol DEN which tests whether a term denotes or not, and permits satisfaction operators to rigidify predicate and function symbols. Semantically, our system is actualist, allows terms to be undefined, and has no truth-value gaps. But should we follow Fitting and Mendelsohn and rule out domain elements not belonging to any world, or should we tolerate them? Here we explore both options. As we shall see, while the choice makes no difference when it comes to validity, it has consequences for richer logics.

AROUND RUBIN’S “THEORIES OF LINEAR ORDER”

AbstractLet $\mathcal M=(M,<,\ldots)$ be a linearly ordered first-order structure and T its complete theory. We investigate conditions for T that could guarantee that $\mathcal M$ is not much more complex than some colored orders (linear orders with added unary predicates). Motivated by Rubin’s work [5], we label three conditions expressing properties of types of T and/or automorphisms of models of T. We prove several results which indicate the “geometric” simplicity of definable sets in models of theories satisfying these conditions. For example, we prove that the strongest condition characterizes, up to definitional equivalence (inter-definability), theories of colored orders expanded by equivalence relations with convex classes.

Minimal Predicates for Δ-Definability

We consider two kinds of reducibilities on finite families of predicates on a countable set: the definability of predicates and their complements of one family via another by means of existential formulas with parameters and the same definability on isomorphism types of families. Ordered structures of degrees generated by families of unary predicates are described. It is proved that for both reducibilities, there exist continuum many minimal nonzero degrees.

Neo-Davidsonian-Based Event Class Semantic Representation

Abstract Event class is an abstract event that represents a set of events with some common features. There are inherent relations among event classes. Event classes and their relations are the important parts of event knowledge base. We gave a novel framework to represent event class semantic by marriage Neo-Davidsonian event semantic and 6-element event class model. The framework treated the predicate of event class as unary predicate with event class argument ec only. It connected the predicate and the other elements of event class with connecting symbol ⁁. And the framework can’t only represent the taxonomic relations among event classes, but also respectively represent non-taxonomic relations via the connecting symbols ⊄, ⊳, →, ∥ and ⊒. The representation examples indicate the novel framework can represent event classes semantic and different kinds of relations among event classes.

Partial Awareness

We develop a modal logic to capture partial awareness. The logic has three building blocks: objects, properties, and concepts. Properties are unary predicates on objects; concepts are Boolean combinations of properties. We take an agent to be partially aware of a concept if she is aware of the concept without being aware of the properties that define it. The logic allows for quantification over objects and properties, so that the agent can reason about her own unawareness. We then apply the logic to contracts, which we view as syntactic objects that dictate outcomes based on the truth of formulas. We show that when agents are unaware of some relevant properties, referencing concepts that agents are only partially aware of can improve welfare.

CHARACTERIZING DOWNWARDS CLOSED, STRONGLY FIRST-ORDER, RELATIVIZABLE DEPENDENCIES

AbstractIn Team Semantics, a dependency notion is strongly first order if every sentence of the logic obtained by adding the corresponding atoms to First-Order Logic is equivalent to some first-order sentence. In this work it is shown that all nontrivial dependency atoms that are strongly first order, downwards closed, and relativizable (in the sense that the relativizations of the corresponding atoms with respect to some unary predicate are expressible in terms of them) are definable in terms of constancy atoms.Additionally, it is shown that any strongly first-order dependency is safe for any family of downwards closed dependencies, in the sense that every sentence of the logic obtained by adding to First-Order Logic both the strongly first-order dependency and the downwards closed dependencies is equivalent to some sentence of the logic obtained by adding only the downwards closed dependencies.