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24
On the Restraining Power of Guards
 Journal of Symbolic Logic
, 1998
"... Guarded fragments of firstorder logic were recently introduced by Andr'eka, van Benthem and N'emeti; they consist of relational firstorder formulae whose quantifiers are appropriately relativized by atoms. These fragments are interesting because they extend in a natural way many proposit ..."
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Cited by 153 (3 self)
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Guarded fragments of firstorder logic were recently introduced by Andr'eka, van Benthem and N'emeti; they consist of relational firstorder formulae whose quantifiers are appropriately relativized by atoms. These fragments are interesting because they extend in a natural way many propositional modal logics, because they have useful modeltheoretic properties and especially because they are decidable classes that avoid the usual syntactic restrictions (on the arity of relation symbols, the quantifier pattern or the number of variables) of almost all other known decidable fragments of firstorder logic. Here, we investigate the computational complexity of these fragments. We prove that the satisfiability problems for the guarded fragment (GF) and the loosely guarded fragment (LGF) of firstorder logic are complete for deterministic double exponential time. For the subfragments that have only a bounded number of variables or only relation symbols of bounded arity, satisfiability is EXPTI...
Extending partial automorphisms and the profinite topology on free groups
 Tran. AMS
, 2000
"... Abstract. A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C1 and C2 are structures in C, C1 finite, C1 ⊆ C2, and p1,p2,...,pn are partial automorphisms of C1 extending to automorphisms of C2, then there exist a finite structure C3 in C and ..."
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Cited by 41 (0 self)
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Abstract. A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C1 and C2 are structures in C, C1 finite, C1 ⊆ C2, and p1,p2,...,pn are partial automorphisms of C1 extending to automorphisms of C2, then there exist a finite structure C3 in C and automorphisms α1,α2,...,αn of C3 extending the pi. We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and Zalesskiĭ stating that a finite product of finitely generated subgroups is closed for this topology. 1.
Finite conformal hypergraph covers and Gaifman cliques in finite structures
 Bull. Symbolic Logic
, 2002
"... We provide a canonical construction of conformal covers for finite hypergraphs and present two immediate applications to the finite model theory of relational structures. In the setting of relational structures, conformal covers serve to construct guarded bisimilar companion structures that avoid al ..."
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Cited by 19 (14 self)
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We provide a canonical construction of conformal covers for finite hypergraphs and present two immediate applications to the finite model theory of relational structures. In the setting of relational structures, conformal covers serve to construct guarded bisimilar companion structures that avoid all incidental Gaifman cliques – thus serving as a partial analogue in finite model theory for the usually infinite guarded unravellings. In hypergraph theoretic terms, we show that every finite hypergraph admits a bisimilar cover by a finite conformal hypergraph. In terms of relational structures, we show that every finite relational structure admits a guarded bisimilar cover by a finite structure whose Gaifman cliques are guarded. One of our applications answers an open question about a clique constrained strengthening of the extension property for partial automorphisms (EPPA) of Hrushovski, Herwig and Lascar. A second application provides an alternative proof of the finite model property (FMP) for the clique guarded fragment of firstorder logic CGF, by reducing (finite) satisfiability in CGF to (finite) satisfiability in the guarded fragment, GF.
Relation Algebras with nDimensional Relational Bases
 Annals of Pure and Applied Logic
, 1999
"... We study relation algebras with ndimensional relational bases in the sense of Maddux. Fix n with 3 n !. Write Bn for the class of nonassociative algebras with an n dimensional relational basis, and RAn for the variety generated by Bn . We de ne a notion of representation for algebras in RAn , ..."
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Cited by 10 (4 self)
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We study relation algebras with ndimensional relational bases in the sense of Maddux. Fix n with 3 n !. Write Bn for the class of nonassociative algebras with an n dimensional relational basis, and RAn for the variety generated by Bn . We de ne a notion of representation for algebras in RAn , and use it to give an explicit (hence recursive) equational axiomatisation of RAn , and to reprove Maddux's result that RAn is canonical. We show that the algebras in Bn are precisely those that have a complete representation. Then we prove that whenever 4 n < l !, RA l is not nitely axiomatisable over RAn . This con rms a conjecture of Maddux. We also prove that Bn is elementary for n = 3; 4 only.
Random Orderings and Unique Ergodicity of Automorphism Groups
, 2012
"... We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, ..."
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Cited by 10 (1 self)
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We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner and Weiss’s example of the group of all permutations of the integers. We also contrast these results to those for certain special classes of graphs and metric spaces in which such random orderings can be found that are not uniform.
Generic Automorphisms of the Universal Partial Order
 Proc. Amer. Math. Soc
, 1939
"... We show that the countable universalhomogeneous partial order (P;!) has a generic automorphism in the sense of [8], namely that it lies in a comeagre conjugacy class of Aut(P; !). For this purpose, we work with `determined' partial finite automorphisms that need not be automorphisms of fin ..."
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Cited by 6 (0 self)
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We show that the countable universalhomogeneous partial order (P;!) has a generic automorphism in the sense of [8], namely that it lies in a comeagre conjugacy class of Aut(P; !). For this purpose, we work with `determined' partial finite automorphisms that need not be automorphisms of finite substructures (as in the proofs of similar results for other countable homogeneous structures) but are nevertheless sufficient to characterize the isomorphism type of the union of their orbits. 1 Introduction The definition of generic given in [8] as applied to automorphisms g of a countable first order structure was that g should lie in a comeagre conjugacy class (where the automorphism group of the structure is endowed with the natural topology). A sufficient condition for the existence of generics is that the family P of finite partial automorphisms of the structure should have the amalgamation property. This property is however false in general, and a weaker condition, that P should h...
Reconstruction of homogeneous relational structures
 J. Symb. Logic
"... This paper contains a result on the reconstruction of certain homogeneous transitive ωcategorical structures from their automorphism group. The structures treated are relational. In the proof it is shown that their automorphism group contains a generic pair (in a slightly nonstandard sense, ..."
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Cited by 6 (2 self)
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This paper contains a result on the reconstruction of certain homogeneous transitive ωcategorical structures from their automorphism group. The structures treated are relational. In the proof it is shown that their automorphism group contains a generic pair (in a slightly nonstandard sense,
On Groupoids and Hypergraphs
, 2012
"... We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (subgroupoid), and only counts transitions between colour classes (cosets). These groupoi ..."
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Cited by 3 (3 self)
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We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (subgroupoid), and only counts transitions between colour classes (cosets). These groupoids are employed towards a generic construction method for finite hypergraphs that realise specified overlap patterns and avoid small cyclic configurations. The constructions are based on reduced products with groupoids generated by the elementary local extension steps, and can be made to preserve the symmetries of the given overlap pattern. In particular, we obtain highly symmetric, finite hypergraph coverings without short cycles. The groupoids and their application in reduced products are sufficiently generic to be applicable to other constructions that are specified in terms of local glueing operations and require global finite closure.
Finite Conformal Hypergraph Covers, With Two Applications
, 2001
"... Introduction The main construction in this paper is presented in terms of hypergraphs, i.e., structures consisting of just a universe together with a collection of subsets of the universe. The main motivation behind the construction, however, arises in the context of ordinary relational structures; ..."
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Cited by 2 (1 self)
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Introduction The main construction in this paper is presented in terms of hypergraphs, i.e., structures consisting of just a universe together with a collection of subsets of the universe. The main motivation behind the construction, however, arises in the context of ordinary relational structures; and here primarily their model theory with respect to guarded logics and extension properties for partial automorphisms. When looking at hypergraphs rather than relational structures we mainly abstract away from the actual relational information and only retain { as a hypergraph structure { the subsets covered by relational edges, together with Department of Computing, Imperial College, London SW7 2BZ, UK; imh@doc.ic.ac.uk; www.doc.ic.ac.uk/~imh y Department of Computer Science, University of Wales Swansea, SA2 8PP, UK; m.otto@swan.ac.uk; wwwcompsci.swan.ac.uk/~csmartin 1 their intersection pattern. Note that this ltered vi