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  • 1.
    Ahlman, Ove
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Limit laws and automorphism groups of random nonrigid structures2015In: Journal of Logic and Analysis, ISSN 1759-9008, E-ISSN 1759-9008, Vol. 7, no 2, p. 1-53, article id 1Article in journal (Refereed)
    Abstract [en]

    A systematic study is made, for an arbitrary finite relational language with at least one symbol of arity at least 2, of classes of nonrigid finite structures. The well known results that almost all finite structures are rigid and that the class of finite structures has a zero-one law are, in the present context, the first layer in a hierarchy of classes of finite structures with increasingly more complex automorphism groups. Such a hierarchy can be defined in more than one way. For example, the kth level of the hierarchy can consist of all structures having at least k elements which are moved by some automorphism. Or we can consider, for any finite group G, all finite structures M such that G is a subgroup of the group of automorphisms of M; in this case the "hierarchy" is a partial order. In both cases, as well as variants of them, each "level" satisfies a logical limit law, but not a zero-one law (unless k = 0 or G is trivial). Moreover, the number of (labelled or unlabelled) n-element structures in one place of the hierarchy divided by the number of n-element structures in another place always converges to a rational number or to infinity as n -> infinity. All instances of the respective result are proved by an essentially uniform argument.

  • 2.
    Ahlman, Ove
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    On sets with rank one in simple homogeneous structures2015In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 228, p. 223-250Article in journal (Refereed)
    Abstract [en]

    We study definable sets D of SU-rank 1 in Meq, where M is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such D can be seen as a 'canonically embedded structure', which inherits all relations on D which are definable in Meq, and has no other definable relations. Our results imply that if no relation symbol of the language of M has arity higher than 2, then there is a close relationship between triviality of dependence and D being a reduct of a binary random structure. Somewhat more precisely: (a) if for every n≥2, every n-type p(x1,...,xn) which is realized in D is determined by its sub-2-types q(xi,xj)⊆p, then the algebraic closure restricted to D is trivial; (b) if M has trivial dependence, then D is a reduct of a binary random structure.

  • 3.
    Ahlman, Ove
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Random l-colourable structures with a pregeometry2017In: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 63, no 1-2, p. 32-58Article in journal (Refereed)
    Abstract [en]

    We study finite -colourable structures with an underlying pregeometry. The probability measure that is usedcorresponds to a process of generating such structures by which colours are first randomly assigned to all1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions aresatisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure,where we now forget the specific colouring of the generating process, has a given property. With this measurewe get the following results: (1) A zero-one law. (2) The set of sentences with asymptotic probability 1 has anexplicit axiomatisation which is presented. (3) There is a formula ξ (x, y) (not directly speaking about colours)such that, with asymptotic probability 1, the relation “there is an -colouring which assigns the same colourto x and y” is defined by ξ (x, y). (4) With asymptotic probability 1, an -colourable structure has a unique-colouring (up to permutation of the colours).

  • 4. Cherlin, G.
    et al.
    Djordjevic, Marko
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
    Hrushovski, E.
    A note on orthogonality and stable embeddedness2005In: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 70, no 4, p. 1359-1364Article in journal (Refereed)
    Abstract [en]

    Orthogonality between two stably embedded definable sets is preserved under the addition of constants.

  • 5.
    Djordjevic, Marko
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
    Entropy of formulas2006Report (Other (popular science, discussion, etc.))
  • 6.
    Djordjevic, Marko
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
    Finite satisfiability and N-0-categorical structures with trivial dependence2006In: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 71, no 3, p. 810-830Article in journal (Refereed)
  • 7.
    Djordjevic, Marko
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
    Finite variable logic, stability and finite models2001In: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 66, no 2, p. 837-858Article in journal (Refereed)
  • 8.
    Djordjevic, Marko
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
    On first-order sentences without finite models2004In: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 69, no 2, p. 329-339Article in journal (Refereed)
  • 9.
    Djordjevic, Marko
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Stability theory in finite variable logic2000Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis studies finite variable theories. To be more precise. complete Ln -theories, where Ln is the set of formulas in a first order language L in which at most n distinct variables occur. These need not be complete in the usual first order sense. We use ideas from infinite model theory, in particular stability theory, to define a class of complete Ln -theories which, as we show, has a tractable model theory, also with respect to finite models.

    The three main properties of such theories that we consider are (1) a finite bound on the number of Ln -types, (2) an amalgamation property and (3) stability. We prove that any complete Ln -theory with an infinite model and with properties (l),(2) and (3) has an infinite model M which is ω-categorical and ω-stable from which it follows that it has arbitrarily large finite models. In fact, M almost admits elimination of quantifiers, in the sense that there exists an expansion of M by finitely many new n-ary relation symbols which admits elimination of quantifiers. This together with the stability of M allows us to obtain finer information about complete Ln -theories with properties (1)-(3).

    We show that there exists a recursive function f : ω2 → ω such that every theory T as above has a finite model of size at most f(n, |Snn(T)|), where Snn(T) is the set of Ln -types of T in n free variables.

    Then we derive some results about forking in stable structures where there exists n < ω such that any type (with any number of free variables) over ω is determined by its subtypes with at most n free variables. We use this to give a different proof of a result due to Lachlan. saying that in a stable structure which almost admits elimination of quantifiers every strictly minimal set is indiscernible.

    Finally. using the theory of stable structures which admit elimination of quantifiers, we show how to construct new (finite and infinite) models of Ln -theories T with an infinite model and properties (l)-(3). Moreover, every sufficiently saturated model of T which is Ln -elementarily embeddable in a stable structure which almost admits elimination of quantifiers can be constructed in this way and the amount of saturation that is needed can be effectively computed from |Snn(T)|.

  • 10.
    Djordjevic, Marko
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
    The finite submodel property and ω-categorical expansions of pregeometries2006In: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 139, no 1-3, p. 201-229Article in journal (Refereed)
    Abstract [en]

    We prove, by a probabilistic argument, that a class of ω-categorical structures, on which algebraic closure defines a pregeometry, has the finite submodel property. This class includes any expansion of a pure set or of a vector space, projective space or affine space over a finite field such that the new relations are sufficiently independent of each other and over the original structure. In particular, the random graph belongs to this class, since it is a sufficiently independent expansion of an infinite set, with no structure. The class also contains structures for which the pregeometry given by algebraic closure is non-trivial.

  • 11.
    Djordjevic, Vera
    et al.
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
    Gorbow, Paul
    Partial stability in simple theories2006Report (Other (popular scientific, debate etc.))
  • 12.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    A limit law of almost l-partite graphs2013In: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 78, no 3, p. 911-936Article in journal (Refereed)
    Abstract [en]

    For integers l >= 1, d >= 0 we study (undirected) graphs with vertices 1,..., n such that the vertices can be partitioned into l parts such that every vertex has at most d neighbours in its own part. The set of all such graphs is denoted P-n (l, d). We prove a labelled first-order limitlaw, i.e., for every first-order sentence phi, the proportion of graphs in P-n (l, d) that satisfy phi converges as n -> infinity. By combining this result with a result of Hundack, Promel and Steger [12] we also prove that if 1 <= s(1) <=...<= s(1) are integers, then Forb(A(I),s(1),...,s(l)) has alabelled first-order limit law, where Forb (A(I),s(1),...,s(l)) denotes the set of all graphs with vertices 1,..., n, for some n, in which there is no subgraph isomorphic to the complete (l + 1)-partite graph with parts of sizes 1, S-1,..., S-l. In the course of doing this we also prove that there exists a first-order formula depending only on l and d, such that the proportion of g e P (I, d) with the following property approaches 1 as n ->infinity: there is a unique partition of {1,..., n} into l parts such that every vertex has at most d neighbours in its own part, and this partition, viewed as an equivalence relation, is defined by xi.

  • 13.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Asymptotic probabilities of extension properties and random l-colourable structures2012In: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 163, no 4, p. 391-438Article in journal (Refereed)
    Abstract [en]

    We consider a set of finite structures such that all members of have the same universe, the cardinality of which approaches as n. Each structure in may have a nontrivial underlying pregeometry and on each we consider a probability measure, either the uniform measure, or what we call the dimension conditional measure. The main questions are: What conditions imply that for every extension axiom φ, compatible with the defining properties of , the probability that φ is true in a member of approaches 1 as n? And what conditions imply that this is not the case, possibly in the strong sense that the mentioned probability approaches 0 for some φ?

    If each is the set of structures with universe {1,…,n}, in a fixed relational language, in which certain “forbidden” structures cannot be weakly embedded and has the disjoint amalgamation property, then there is a condition (concerning the set of forbidden structures) which, if we consider the uniform measure, gives a dichotomy; i.e., the condition holds if and only if the answer to the first question is ‘yes’. In general, we do not obtain a dichotomy, but we do obtain a condition guaranteeing that the answer is ‘yes’ for the first question, as well as condition guaranteeing that the answer is ‘no’; and we give examples showing that in the gap between these conditions the answer may be either ‘yes’ or ‘no’. This analysis is made for both the uniform measure and for the dimension conditional measure. The later measure has a closer relation to random generation of structures and is more “generous” with respect to satisfiability of extension axioms.

    Random l-coloured structures fall naturally into the framework discussed so far, but random l-colourable structures need further considerations. It is not the case that every extension axiom compatible with the class of l-colourable structures almost surely holds in an l-colourable structure. But a more restricted set of extension axioms turns out to hold almost surely, which allows us to prove a zero–one law for random l-colourable structures, using a probability measure which is derived from the dimension conditional measure, and, after further combinatorial considerations, also for the uniform probability measure.

  • 14.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Binary primitive homogeneous simple structures2017In: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 82, no 1, p. 183-207Article in journal (Refereed)
    Abstract [en]

    Suppose that M is countable, binary, primitive, homogeneous, and simple. We prove that the SU-rank of the complete theory of M is 1 and hence 1-based. It follows that M is a random structure. The conclusion that M is a random structure does not hold if the binarity condition is removed, as witnessed by the generic tetrahedron-free 3-hypergraph. However, to show that the generic tetrahedron-free 3-hypergraph is 1-based requires some work (it is known that it has the other properties) since this notion is defined in terms of imaginary elements. This is partly why we also characterize equivalence relations which are definable without parameters in the context of omega-categorical structures with degenerate algebraic closure. Another reason is that such characterizations may be useful in future research about simple (nonbinary) homogeneous structures.

  • 15.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Binary simple homogeneous structures2018In: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 169, no 12, p. 1335-1368Article in journal (Refereed)
    Abstract [en]

    We describe all binary simple homogeneous structures M in terms of ∅-definable equivalence relations on M, which “coordinatize” M and control dividing, and extension properties that respect these equivalence relations.

  • 16.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Binary simple homogeneous structures are supersimple with finite rank2016In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 144, no 4, p. 1745-1759Article in journal (Refereed)
    Abstract [en]

    Suppose that M is an infinite structure with finite relational vocabulary such that every relation symbol has arity at most 2. If M is simple and homogeneous, then its complete theory is supersimple with finite SU-rank which cannot exceed the number of complete 2-types over the empty set.

  • 17.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Entropy of formulas2009In: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 48, no 6, p. 515-522Article in journal (Refereed)
    Abstract [en]

    A probability distribution can be given to the set of isomorphism classes of models with universe {1, ..., n} of a sentence in first-order logic. We study the entropy of this distribution and derive a result from the 0-1 law for first-order sentences.

  • 18.
    Koponen, Vera
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
    Gregory Cherlin and Ehud Hrushovski. Finite structures with few types. Annals of Mathematics Studies. Princeton University Press, 2003, vi + 196pp.2008In: The Bulletin of Symbolic Logic, Vol. 14, no 1, p. 114-116Article, book review (Other (popular scientific, debate etc.))
  • 19.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Homogeneous 1-based structures and interpretability in random structures2017In: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 63, no 1-2, p. 6-18Article in journal (Refereed)
    Abstract [en]

    Let V be a finite relational vocabulary in which no symbol has arity greater than 2. Let math formula be countable V-structure which is homogeneous, simple and 1-based. The first main result says that if math formula is, in addition, primitive, then it is strongly interpretable in a random structure. The second main result, which generalizes the first, implies (without the assumption on primitivity) that if math formula is “coordinatized” by a set with SU-rank 1 and there is no definable (without parameters) nontrivial equivalence relation on M with only finite classes, then math formula is strongly interpretable in a random structure.

  • 20.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
    Independence and the finite submodel property2009In: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 158, no 1-2, p. 58-79Article in journal (Refereed)
    Abstract [en]

    We study a class c of aleph(0)-categorical simple structures such that every M in c has uncomplicated forking behavior and such that definable relations in M which do not cause forking are independent in a sense that is made precise; we call structures in c independent. The SU-rank of such M may be n for any natural number n > 0. The most well-known unstable member of c is the random graph, which has SU-rank one. The main result is that for every strongly independent structure M in e, if a sentence phi is true in M then phi is true in a finite substructure of M. The same conclusion holds for every structure in c with SU-rank one: so in this case the word 'strongly' can be removed. A probability theoretic argument is involved and it requires sufficient independence between relations which do not cause forking. A stable structure M belongs to c if and only if it is aleph(0)-categorical, aleph(0)-stable and every definable strictly minimal Subset of M-eq is indiscernible.

     

  • 21.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    On Constraints And Dividing In Ternary Homogeneous Structures2018In: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 83, no 4, p. 1691-1721Article in journal (Refereed)
    Abstract [en]

    Let M be ternary, homogeneous and simple. We prove that if M is finitely constrained, then it is supersimple with finite SU-rank and dependence is k-trivial for some k < omega and for finite sets of real elements. Now suppose that, in addition, M is supersimple with SU-rank 1. If M is finitely constrained then algebraic closure in M is trivial. We also find connections between the nature of the constraints of M, the nature of the amalgamations allowed by the age of M, and the nature of definable equivalence relations. A key method of proof is to "extract" constraints (of M) from instances of dividing and from definable equivalence relations. Finally, we give new examples, including an uncountable family, of ternary homogeneous supersimple structures of SU-rank 1.

  • 22.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Random graphs with bounded maximum degree: asymptotic structure and a logical limit law2012In: Discrete mathematics and theoretical computer science (Online), ISSN 1462-7264, E-ISSN 1365-8050, Vol. 14, no 2, p. 229-254Article in journal (Refereed)
  • 23.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Some connections between finite and infinite model theory2011In: Finite and Algorithmic Model Theory / [ed] Javier Esparza, Christian Michaux, Charles Steinhorn, Cambridge University Press, 2011, p. 109-139Chapter in book (Other academic)
  • 24.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Typical automorphism groups of finite nonrigid structures2015In: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 54, no 5-6, p. 571-586Article in journal (Refereed)
  • 25.
    Koponen, Vera
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Hyttinen, Tapani
    Helsingfors Universitet.
    On compactness of logics that can express properties of symmetry or connectivity2015In: Studia Logica: An International Journal for Symbolic Logic, ISSN 0039-3215, E-ISSN 1572-8730, Vol. 103, no 1, p. 1-20Article in journal (Refereed)
    Abstract [en]

    A condition, in two variants, is given such that if a property P satisfies this condition, then every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The result is used to prove that for a number of natural properties P speaking about automorphism groups or connectivity, every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The basic idea underlying the results and examples presented here is that it is possible to construct a countable first-order theory T such that every model of T has a very rich automorphism group, but every finite subset T' of T has a model which is rigid.

1 - 25 of 25
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