Dynamic Topological Logic (DTL) is a multimodal system for reasoning about
dynamical systems. It is defined semantically and, as such, most of the work
done in the field has been model-theoretic. In particular, the problem of
finding a complete axiomatization for the full language of DTL over the class
of all dynamical systems has proven to be quite elusive.

We propose a logic of interactive proofs as the first and main step towards
an intuitionistic foundation for interactive computation to be obtained via an
interactive analog of the G\"odel-Kolmogorov-Art\"emov definition of
intuitionistic logic as embedded into a classical modal logic of proofs, and of
the Curry-Howard isomorphism between intuitionistic proofs and typed programs.
Our interactive proofs effectuate a persistent epistemic impact in their
intended communities of peer reviewers that consists in the induction of the
(propositional) knowledge of their proof goal by means of the (i

We introduce the notion of normal hyperimaginary and we develop its basic
theory. We present a new proof of Lascar-Pillay's theorem on bounded
hyperimaginaries based on properties of normal hyperimaginaries. However, the
use of Weil's theorem on the structure of compact Hausdorff groups is not
completely eliminated from the proof.

Sufficient conditions are given for the computation of accessing arcs and
arcs that links boundary components of multiply connected domains. The
existence of a not-computably-accessible but computable point on a computably
compact arc is also demonstrated.

We formalize the notion of Herbrand Consistency in an appropriate way for
bounded arithmetics, and show the existence of a finite fragment of ${\rm
I\Delta_0}$ whose Herbrand Consistency is not provable in the thoery ${\rm
I\Delta_0}$. We also show the existence of an ${\rm I\Delta_0}-$derivable
$\Pi_1-$sentence such that ${\rm I\Delta_0}$ cannot prove its Herbrand
Consistency.

We initiate the study of pseudofiniteness in continuous logic. We introduce a
related concept, namely that of pseudocompactness, and investigate the
relationship between the two concepts. We establish some basic properties of
pseudofiniteness and pseudocompactness and provide many examples. We also
investigate the injective-surjective phenomenon for definable endofunctions in
pseudofinite structures.

A set of first-order formulas, whatever the cardinality of the set of
symbols, is equivalent to an independent set.

Under CH we prove that for any tall ideal $\cal I$ on $\omega$ and for any
ordinal $\gamma \leq \omega_1$ there is an ${\cal I}$-ultrafilter (in the sense
of Baumgartner), which belongs to the class ${\cal P}_{\gamma}$ of P-hierarchy
of ultrafilters. Since the class of ${\cal P}_2$ ultrafilters coincides with a
class of P-points, out result generalize theorem of Fla\v{s}kov\'a, which
states that there are ${\cal I}$-ultrafilters which are not P-points.

In this note we exhibit a very simple proof of McNaughton Theorem, almost
right out of the definitions, and at the same time we observe that this theorem
does not depend of Chang's completeness theorem.

Let \Gamma be the generalized random bipartite graph that has two sides Rl
and Rr with edges for every pair of vertices between R1 and Rr but no edges
within each side, where all the edges are randomly colored by three colors P1;
P2; P3. In this paper, we investigate the reducts of \Gamma that preserve Rl
and Rr, and classify the closed permutation subgroups in Sym(Rl)\timesSym(Rr)
containing the group Aut(\Gamma). Our results rely on a combinatorial theorem
of Nesetril-Rodl and the strong finite submodel property of the random
bipartite graph.

We present a logical framework for formalizing connections between finitary
combinatorics and measure theory or ergodic theory that have appeared various
places throughout the literature. We develop the basic syntax and semantics of
this logic and give applications, showing that the method can express the
classic Furstenberg correspondence and to give a short proof of the Szemer\'edi
Regularity Lemma. We also derive some connections between the model-theoretic
notion of stability and the Gowers uniformity norms from combinatorics.

The paper is a contribution to intuitionistic reverse mathematics. Working in
the context of a formal system called Basic Intuitionistic Mathematics BIM, we
formulate a large number of equivalents of the Fan Theorem. We introduce the
class of the closed-and-separable subsets of Baire space and consider the
members of this class that enjoy the so-called Heine-Borel property. The Fan
Theorem is the statement that Cantor space has this property. We prove that the
class of the closed-and-separable subsets of Baire space with the
Heine-Borel-property may be characterized in many different ways.

We study a new proof principle in the context of constructive
Zermelo-Fraenkel set theory based on what we will call "non-deterministic
inductive definitions". We give applications to formal topology as well as a
predicative justification of this principle.

Usually a name of the category is inherited from the name of objects. However
more relevant for a category of objects and morphisms is an algebra of
morphisms. Therefore we prefer to say a category of graphs if every morphism is
a graph. In a monoidal category every morphism can be seen as a graph, and a
partial algebra of morphisms possesses a structure of an operad, operad of
graphs. We consider a monoidal category of operad of graphs with underlying
graphical calculus. If, in particular, there is a single generating objects,
then each morphism is a bi-arity graph.

New concepts of rough natural number systems, recently introduced by the
present author, are used to improve most rough set-theoretical measures in
general Rough Set theory (\textsf{RST}) and measures of mutual consistency of
multiple models of knowledge. In this research paper, the explicit dependence
on the axiomatic theory of granules of \cite{AM99} is reduced and more results
on the measures and representation of the numbers are proved.

We study the spectrum of forcing notions between the iterations of
$\sigma$-closed followed by ccc forcings and the proper forcings. This includes
the hierarchy of $\alpha$-proper forcings for indecomposable countable ordinals
as well as the Axiom A forcings. We focus on the bounded forcing axioms for the
hierarchy of $\alpha$-proper forcings and connect them to a hierarchy of weak
club guessing principles. We show that they are, in a sense, dual to each
other.

We describe the countable ordinals in terms of iterations of Mostowski
collapsings. This gives a proof-theoretic bound of definable countable ordinals
in the Zermelo-Fraenkel's set theory ZF.

We study complexity of the index set of countably categorical theories and
Ehrenfeucht theories in finite languages.

Let K be a subfield of the real field, D be a closed and discrete subset of K
and f : D^n -> K be a function such that f(D^n) is somewhere dense. Then (K,f)
defines the set of integers. We present several applications of this result. We
show that K expanded by predicates for different cyclic multiplicative
subgroups defines the set of integers. Moreover, we prove that every definably
complete expansion of a subfield of the real field satisfies an analogue of the
Baire Category Theorem.

We give a new proof for Godel's second incompleteness theorem, based on
Kolmogorov complexity, Chaitin's incompleteness theorem, and an argument that
resembles the surprise examination paradox. We then go the other way around and
suggest that the second incompleteness theorem gives a possible resolution of
the surprise examination paradox.

Let A be a finite alphabet and let L contained in (A*)^n be an n-variable
language over A. We say that L is regular if it is the language accepted by a
synchronous n-tape finite state automaton, it is quasi-regular if it is
accepted by an asynchronous n-tape automaton, and it is weakly regular if it is
accepted by a non-deterministic asynchronous n-tape automaton. We investigate
the closure properties of the classes of regular, quasi-regular, and weakly
regular languages under first-order logic, and apply these observations to an
open decidability problem in automatic group theory.

In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard
analysis and the finitization of soft analysis statements into hard analysis
statements. One of his main examples was a quasi-finitization of the infinite
pigeonhole principle IPP, arriving at the "finitary" infinite pigeonhole
principle FIPP1. That turned out to not be the proper formulation and so we
proposed an alternative version FIPP2. Tao himself formulated yet another
version FIPP3 in a revised version of his essay.

This is a short survey illustrating some of the essential aspects of the
theory of canonical extensions. In addition some topological results about
canonical extensions of lattices with additional operations in finitely
generated varieties are given. In particular, they are doubly algebraic
lattices and their interval topologies agree with their double Scott topologies
and make them Priestley topological algebras.

In this paper we study a notion of a $\kappa$-covering in connection with
Bernstein sets and other types of nonmeasurability. Our results correspond to
those obtained by Muthuvel and Nowik. We consider also other types of
coverings.

Let $\RR_S$ denote the expansion of the real ordered field by a family of
real-valued functions $S$, where each function in $S$ is defined on a compact
box and is a member of some quasianalytic class which is closed under the
operations of function composition, division by variables, and implicitly
defined functions. It is shown that the first order theory of $\RR_S$ is
decidable if and only if two oracles, called the approximation and precision
oracles for $S$, are decidable.

We show that usage of elementary submodels is a simple but powerful method to
prove theorems, or to simplify proofs in infinite combinatorics. First we
introduce all the necessary concepts of logic, then we prove classical theorems
using elementary submodels. We also present a new proof of Nash-Williams's
theorem on cycle-decomposition of graphs, and finally we obtain some new
decomposition theorems by eliminating GCH from some proofs concerning
bond-faithful decompositions of graphs.

We study which cardinals are characterizable by a Scott sentence, in the
sense that $\phi_M$ characterizes $\kappa$, if it has a model of size $\kappa$,
but not of $\kappa^+$. We show that if $\aleph_\alpha$ is characterizable by a
Scott sentence and $\beta<\omega_1$, then $\aleph_{\alpha+\beta}$ is
characterizable by a Scott sentence.

Using the proof-program (Curry-Howard) correspondence, we give a new method
to obtain models of ZF and relative consistency results. We show the relative
consistency of ZF + DC + some unusual properties for the power set of R.

We show that $\omega$-categorical rings with NIP are nilpotent-by-finite. We
prove that an $\omega$-categorical group with NIP and fsg is
nilpotent-by-finite. We also notice that an $\omega$-categorical group with at
least one strongly regular type is abelian. Moreover, we get that each
$\omega$-categorical, characteristically simple $p$-group with NIP has an
infinite, definable abelian subgroup. Assuming additionally the existence of a
non-algebraic, generically stable over $\emptyset$ type, such a group is
abelian.

Let $(X_n,d_n),\,n\in\Bbb N$ be a sequence of pseudo-metric spaces, $p\ge 1$.
For $x,y\in\prod_{n\in\Bbb N}X_n$, let $(x,y)\in E((X_n)_{n\in\Bbb
N};p)\Leftrightarrow\sum_{n\in\Bbb N}d_n(x(n),y(n))^p<+\infty$. For Borel
reducibility between equivalence relations $E((X_n)_{n\in\Bbb N};p)$, we show
it is closely related to finitely H\"older($\alpha$) embeddability between
pseudo-metric spaces.

In [P. Majumdar, S. K. Samanta, Similarity measure of soft sets, New
Mathematics and Natural Computation 4(1)(2008) 1-12], the authors use matrix
representation based distances of soft sets to introduce matching function and
distance based similarity measures. We first give counterexamples to show that
their Definition 2.7 and Lemma 3.5(3) contain errors, then improve their Lemma
4.4 making it a corllary of our result. The fundamental assumption of Majumdar
et al has been shown to be flawed. This motivates us to introduce set
operations based measures.

Let $X$ be a Polish space, $d$ a pseudo-metric on $X$. If
$\{(u,v):d(u,v)<\delta\}$ is ${\bf\Pi}^1_1$ for each $\delta>0$, we show that
either $(X,d)$ is separable or there are $\delta>0$ and a perfect set
$C\subseteq X$ such that $d(u,v)\ge\delta$ for distinct $u,v\in C$.

Granting this dichotomy, we characterize the positions of $\ell_p$-like and
$c_0$-like equivalence relations in the Borel reducibility hierarchy.

We show that it is relatively consistent with ZFC that 2^omega is arbitrarily
large and every sequence s=(s_i:i<omega_2) of infinite cardinals with
s_i<=2^omega is the cardinal sequence of some locally compact scattered space.

We study the problem of computing conditional probabilities, a fundamental
operation in statistics and machine learning. In the elementary discrete
setting, a ratio of probabilities defines conditional probability. In the
abstract setting, conditional probability is defined axiomatically and the
search for more constructive definitions is the subject of a rich literature in
probability theory and statistics.

In this note we will introduce a class of search problems, called nested
Polynomial Local Search (nPLS) problems, and show that definable NP search
problems, i.e., $\Sigma^b_1$-definable functions in $T^2_2$ are characterized
in terms of the nested PLS.

Quantum logic generalizes, and in dimension one coincides with, Boolean
logic. We show that the satisfiability problem of quantum logic formulas is
NP-complete in dimension two as well. For higher higher-dimensional spaces R^d
and C^d with d>2 fixed, we establish quantum satisfiability to be polynomial
time equivalent to the real feasibility of a multivariate quartic polynomial
equation: a problem well-known complete for the counterpart of NP in the
Blum-Shub-Smale model of computation lying (probably strictly) between
classical NP and PSPACE.

We show that the set of codes for Ramsey positive analytic sets is
$\mathbf{\Sigma}^1_2$-complete. This is a one projective-step higher analogue
of the Hurewicz theorem saying that the set of codes for uncountable analytic
sets is $\mathbf{\Sigma}^1_1$-complete. This shows a close resemblance between
the Sacks forcing and the Mathias forcing. In particular, we get that the
$\sigma$-ideal of Ramsey null sets is not ZFC-correct. This solves a problem
posed by Ikegami, Pawlikowski and Zapletal.

We denote by Conc L the semilattice of all finitely generated congruences of
a lattice L. For varieties (i.e., equational classes) V and W of lattices such
that V is contained neither in W nor its dual, and such that every simple
member of W contains a prime interval, we prove that there exists a bounded
lattice A in V with at most aleph 2 elements such that Conc A is not isomorphic
to Conc B for any B in W. The bound aleph 2 is optimal. As a corollary of our
results, there are continuously many congruence classes of locally finite
varieties of (bounded) modular lattices.

This paper introduces a refinement of the sequent calculus approach called
cirquent calculus. While in Gentzen-style proof trees sibling (or cousin, etc.)
sequents are disjoint sequences of formulas, in cirquent calculus they are
permitted to share elements. Explicitly allowing or disallowing shared
resources and thus taking to a more subtle level the resource-awareness
intuitions underlying substructural logics, cirquent calculus offers much
greater flexibility and power than sequent calculus does.

We analyze the degree-structure induced by large reducibilities under the
Axiom of Determinacy. This generalizes the analysis of Borel reducibilities
given in references [1], [6] and [5] e.g. to the projective levels.

We show that Vopenka's Principle and Vopenka cardinals are indestructible
under reverse Easton forcing iterations of increasingly directed-closed partial
orders, without the need for any preparatory forcing. As a consequence, we are
able to prove the relative consistency of these large cardinal axioms with a
variety of statements known to be independent of ZFC, such as the generalised
continuum hypothesis, the existence of a definable well-order of the universe,
and the existence of morasses at many cardinals.

In reference [8] we have considered a wide class of "well-behaved"
reducibilities for sets of reals. In this paper we continue with the study of
Borel reducibilities by proving a dichotomy theorem for the degree-structures
induced by good Borel reducibilities. This extends and improves the results of
[8] allowing to deal with a larger class of notions of reduction (including,
among others, the Baire class $\xi$ functions).

We study various notions of "tameness" for definably complete expansions of
ordered fields. We mainly study structures with locally o-minimal open core,
d-minimal structures, and dense pairs of d-minimal structures.

The \emph{stationary set splitting game} is a game of perfect information of
length $\omega_{1}$ between two players, \unspls and \spl, in which \unspls
chooses stationarily many countable ordinals and \spls tries to continuously
divide them into two stationary pieces. We show that it is possible in ZFC to
force a winning strategy for either player, or for neither.

We consider the expansion of the real field by the group of rational points
of an elliptic curve over the rational numbers. We prove a completeness result,
followed by a quantifier elimination result. Moreover we show that open sets
definable in that structure are semialgebraic.

Our main theorem is about iterated forcing for making the continuum larger
than aleph_2. We present a generalization of math.LO/0303294 which is dealing
with oracles for random, etc., replacing aleph_1, aleph_2 by lambda,lambda^+
(starting with lambda=lambda^{<lambda}>aleph_1). Well, instead of properness we
demand absolute c.c.c. So we get, e.g. the continuum is lambda^+ but we can get
cov(meagre)=lambda. We give some applications.

A structure M is pregeometric if the algebraic closure is a pregeometry in
all M' elementarily equivalent to M. We define a generalisation: structures
with an existential matroid. The main examples are superstable groups of U-rank
a power of omega and d-minimal expansion of fields. Ultraproducts of
pregeometric structures expanding a field, while not pregeometric in general,
do have an unique existential matroid.

We present a way of topologizing sets of Galois types over structures in
abstract elementary classes with amalgamation. In the elementary case, the
topologies thus produced refine the syntactic topologies familiar from first
order logic. We exhibit a number of natural correspondences between the
model-theoretic properties of classes and their constituent models and the
topological properties of the associated spaces. Tameness of Galois types, in
particular, emerges as a topological separation principle.

We present a way to organize a constructive development of the theory of
Banach algebras, inspired by works of Cohen, de Bruijn and Bishop. We
illustrate this by giving elementary proofs of Wiener's result on the inverse
of Fourier series and Wiener's Tauberian Theorem, in a sequel to this paper we
show how this can be used in a localic, or point-free, description of the
spectrum of a Banach algebra.

This paper began as a generalization of a part of the author's PhD thesis
about ACFA and ended up with a characterization of groups definable in T_A.

The thesis concerns minimal formulae in ACFA of the form "p lies on an
algebraic curve A and s(x)=f(x)" for some dominant rational function f from A
to s(A), where s is the automorphism. These are shown to be uniform in the
Zilber trichotomy, and the pairs (A,f) that fall into each of the three cases
are characterized. These characterizations are definable in families.

We study closed choice principles for different spaces. Given information
about what does not constitute a solution, closed choice determines a solution.
We show that with closed choice one can characterize several models of
hypercomputation in a uniform framework using Weihrauch reducibility.

Given an ideal $I$ on $\omega$ let $a(I) $ ($\bar{a}(I)$) be minimum of the
cardinalities of infinite (uncountable) maximal $I$-almost disjoint subsets of
$[{\omega}]^{\omega}$, and denote $b_I$ and$d_I$ the unbounding and dominating
numbers of $(\omega^\omega,\le_I)$.

Arguments on PL,(=piecewise linear) topology work over any ordered field in
the same way as over the real field, and those on differential topology do over
a real closed field R in an o-minimal structure that expands (R,<,0,1,+,cdot).
One of the most fundamental properties of definable sets is that a compact
definable set in R^n is definably homeomorphic to a polyhedron (see [v]). We
show uniqueness of the polyhedron up to PL homeomorphisms (o-minimal
Hauptvermutung). Hence a compact definable topological manifold admits uniquely
a PL manifold structure and is, so to say, tame.

The central topic of this work is the categories of modules over unital
quantales. The main categorical properties are established and a special class
of operators, called Q-module transforms, is defined. Such operators - that
turn out to be precisely the homomorphisms between free objects in those
categories - find concrete applications in two different branches of image
processing, namely fuzzy image compression and mathematical morphology.

We describe representation theorems for local and perfect MV-algebras in
terms of ultraproducts involving the unit interval [0,1]. Furthermore, we give
a representation of local Abelian lattice-ordered groups with strong unit as
quasi-constant functions on an ultraproduct of the reals. All the above
theorems are proved to have a uniform version, depending only on the
cardinality of the algebra to be embedded, as well as a definable construction
in ZFC. The paper contains both known and new results and provides a complete
overview of representation theorems for such classes.

Let U denote the Urysohn sphere and consider U as a metric structure in the
empty continuous signature. We prove that every definable function from U^n to
U is either a projection function or else has relatively compact range. As a
consequence, we prove that many functions natural to the study of the Urysohn
sphere are not definable. We end with further topological information on the
range of the definable function in case it is compact.

The aim of this paper is to investigate the soundness of the equivalence
between conditional statements (P $\Rightarrow$ Q) and ($\neg$ P or Q) in Logic
With Verbs, as well as its Boolean Algebraic Structure.

We investigate the notion of independence, which is at the basis of many,
seemingly unrelated, properties of logic like Rational Monotony in
non-monotonic logics, and interpolation theorems.

We presents an independence relation on sets, one can define dimension by it,
assuming that we have an abstract elementary class with a forking notion that
satisfies the axioms of a good frame minus stability.

Let R be an o-minimal field with a proper convex subring V. We axiomatize the
class of all structures (R,V) such that k_ind, the corresponding residue field
with structure induced from R via the residue map, is o-minimal. More
precisely, in previous work it was shown that certain first order conditions on
(R,V) are sufficient for the o-minimality of k_ind. Here we prove that these
conditions are also necessary.

It is known that exactly eight varieties of Heyting algebras have a
model-completion, but no concrete axiomatisation of these model-completions
were known by now except for the trivial variety (reduced to the one-point
algebra) and the variety of Boolean algebras. For each of the six remaining
varieties we introduce two axioms and show that 1) these axioms are satisfied
by all the algebras in the model-completion, and 2) all the algebras in this
variety satisfying these two axioms have a certain embedding property.

This is an exposition of much of Sections VI.3 and XVIII.3 of "Proper and
Improper Forcing", including preservations for "no random reals over V", "reals
of V form a non-meager set", "every dense open set contains a dense open set in
V", weak bounding, and weak $\omega^\omega$-bounding. The current version of
part I covering Sections VI.1 and VI.2 is available from the author.

We introduce a family of rank functions and related notions of total
transcendence for Galois types in abstract elementary classes. We focus, in
particular, on abstract elementary classes satisfying the condition know as
tameness (currently suspected to be a necessary condition for the development
of a reasonable classification theory) where the connections between stability
and total transcendence are most evident.

We give sufficient conditions for a predicate P in a complete theory T to be
stably embedded: P with its induced 0-definable structure has "finite rank", P
has NIP in T and P is 1-stably embedded. This generalizes recent work by Hasson
and Onshuus in the case where P is o-minimal in T.

We show that graphs generated by collapsible pushdown systems of level 2 are
tree-automatic. Even when we allow $\epsilon$-contractions and add a
reachability predicate (with regular constraints) for pairs of configurations,
the structures remain tree-automatic. Hence, their FO theories are decidable,
even when expanded by a reachability predicate. As a corollary, we obtain the
tree-automaticity of the second level of the Caucal-hierarchy.

This paper shows that the metatheory of the classical, first-order predicate
calculus is subject to paradox. It is shown that an interpretation M of the
language of the calculus is definable within this metatheory such that: a
formula of the calculus F(x) is satisfied at a certain denumerable sequence s
of elements of the domain of M if and only if F(x) is not satisfied at s. Since
the conclusion is absurd, the hypothesis that the metatheory provides a
reliable account of the calculus should be rejected.

This paper provides a first example of a model theoretically well behaved
structure consisting of a proper o-minimal expansion of the real field and a
dense multiplicative subgroup of finite rank. Under certain Schanuel
conditions, a quantifier elimination result will be shown for the real field
with an irrational power function and a dense multiplicative subgroup of finite
rank whose elements are algebraic over the field generated by the irrational
power.

For a countable, weakly minimal theory, we show that the Schroeder-Bernstein
property (any two elementarily bi-embeddable models are isomorphic) is
equivalent to both a condition on orbits of rank 1 types and the property that
the theory has no infinite collection of pairwise bi-embeddable, pairwise
nonisomorphic models. We conclude that for countable weakly minimal theories,
the Schroeder-Bernstein property is absolute between transitive models of ZFC.

This is the third installment in a series of papers on algebraic set theory.
In it, we develop a uniform approach to sheaf models of constructive set
theories based on ideas from categorical logic. The key notion is that of a
"predicative category with small maps" which axiomatises the idea of a category
of classes and class morphisms, together with a selected class of maps whose
fibres are sets (in some axiomatic set theory). The main result of the present
paper is that such predicative categories with small maps are stable under
internal sheaves.

We prove a uniformly computable version of de Finetti's theorem on
exchangeable sequences of real random variables. As a consequence, exchangeable
stochastic processes in probabilistic functional programming languages can be
automatically rewritten as procedures that do not modify non-local state. Along
the way, we prove that a distribution on the unit interval is computable if and
only if its moments are uniformly computable.

The basic result of this note is a statement about the existence of families
of partitions of the set of natural numbers with some favourable properties,
the n-optimal matrices of partitions. We use this to improve a decomposition
result for strongly homogeneous Souslin trees. The latter is in turn applied to
separate strong notions of rigidity of Souslin trees, thereby answering a
considerable portion of a question of Fuchs and Hamkins.

It is known that a definably compact group G is an extension of a compact Lie
group L by a divisible torsion-free normal subgroup. We show that the o-minimal
higher homotopy groups of G are isomorphic to the corresponding higher homotopy
groups of L. As a consequence, we obtain that all abelian definably compact
groups of a given dimension are definably homotopy equivalent, and that their
universal cover are contractible.

This paper studies convex sets categorically, namely as algebras of a
distribution monad. It is shown that convex sets occur in two dual adjunctions,
namely one with preframes via the Boolean truth values {0,1} as dualising
object, and one with effect algebras via the (real) unit interval [0,1] as
dualising object. These effect algebras are of interest in the foundations of
quantum mechanics.

In this note we derive a property of maximal ideal-independent subsets of
boolean algebras which has corollaries regarding the continuum cardinals p and
s_mm(P(omega)/fin).

A dictionary is a set of finite words over some finite alphabet X. The
omega-power of a dictionary V is the set of infinite words obtained by infinite
concatenation of words in V. Lecomte studied in [Omega-powers and descriptive
set theory, JSL 2005] the complexity of the set of dictionaries whose
associated omega-powers have a given complexity.

In a previous work we introduced an elementary method to analyze the
periodicity of a generating function defined by a single equation y=G(x,y).
This was based on deriving a single set-equation Y = Gammma(Y) defining the
spectrum of the generating function. This paper focuses on extending the
analysis of periodicity to generating functions defined by a system of
equations y = G(x,y).

In the effective topos there exists a chain-complete distributive lattice
with a monotone and progressive endomap which does not have a fixed point.
Consequently, the Bourbaki-Witt theorem and Tarski's fixed-point theorem for
chain-complete lattices do not have constructive (topos-valid) proofs.

We are interested in examples of a.e.c. with amalgamation having some
(extreme) behaviour concerning types. Note we deal with k being sequence-local,
i.e. local for increasing chains of length a regular cardinal. For any cardinal
theta>= aleph_0 we construct an a.e.c. with amalgamation k with L.S.T.(k) =
theta, |tau_K| = theta such that {kappa : kappa is a regular cardinal and K is
not (2^kappa, kappa)-sequence-local} is maximal. In fact we have a direct
characterization of this class of cardinals: the regular kappa such that there
is no uniform kappa^+-complete ultrafilter.

We introduce an analog of the theory of Borel equivalence relations in which
we study equivalence relations that are decidable by an infinite time Turing
machine. The Borel reductions are replaced by the more general class of
infinite time computable functions. Many basic aspects of the classical theory
remain intact, with the added bonus that it becomes sensible to study some
special equivalence relations whose complexity is beyond Borel or even
analytic.

We study the notion of dp-minimality, beginning by providing several
essential facts, establishing several equivalent definitions, and comparing
dp-minimality to other minimality notions. The rest of the paper is dedicated
to examples. We establish via a simple proof that any weakly o-minimal theory
is dp-minimal and then give an example of a weakly o-minimal group not obtained
by adding traces of externally definable sets.

Let K be a henselian valued field of characteristic 0. Then K admits a
definable partition on each piece of which the leading term of a polynomial in
one variable can be computed as a definable function of the leading term of a
linear map. Two applications are given: first, a constructive quantifier
elimination relative to the leading terms, suggesting a relative decision
procedure; second, a presentation of every definable subset of K as the
pullback of a definable set in the leading terms subjected to a linear
translation.

We propose a new, game-theoretic, approach to the idealized forcing, in terms
of fusion games. This generalizes the classical approach to the Sacks and the
Miller forcing. For definable ($\mathbf{\Pi}^1_1$ on $\mathbf{\Sigma}^1_1)
$\sigma$-ideals we show that if a $\sigma$-ideal is generated by closed sets,
then it is generated by closed sets in all forcing extensions. We also prove an
infinite-dimensional version of the Solecki dichotomy for analytic sets.

We introduce a simplified construction for the countable support
non-elementary proper (nep) iteration. We also present some basic nep notation
and theory in a self contained way.

In Logic, non reflexivity translates into the absence of the identity axiom.
This opens the field to the treatment of many language phenomena, like
fallacies. Ludics, a frame invented by J-Y Girard, because it is founded on
loci (adresses) and not on formulae, allows such a treatment.

We study two different types of (maximal) almost disjoint families: very mad
families and (maximal) cofinitary groups. For the very mad families we prove
the basic existence results. We prove that MA implies there exist many pairwise
orthogonal families, and that CH implies that for any very mad family there is
one orthogonal to it. Finally we prove that the axiom of constructibility
implies that there exists a coanalytic very mad family. Cofinitary groups have
a natural action on the natural numbers.

Let K be a variety of (commutative, integral) residuated lattices. The
substructural logic usually associated with K is an algebraizable logic that
has K as its equivalent algebraic semantics, and is a logic that preserves
truth, i.e., 1 is the only truth value preserved by the inferences of the
logic. In this paper we introduce another logic associated with K, namely the
logic that preserves degrees of truth, in the sense that it preserves lower
bounds of truth values in inferences. We study this second logic mainly from
the point of view of abstract algebraic logic.

A dialectical rough set theory focussed on the relation between roughly
equivalent objects and classical objects was introduced in \cite{AM699} by the
present author. The focus of our investigation is on elucidating the minimal
conditions on the nature of granularity, underlying semantic domain and nature
of the general rough set theories (RST) involved for possible extension of the
semantics to more general RST on a paradigm. On this basis we also formulate a
program in dialectical rough set theory.

We prove that a countable simple unidimensional theory that eliminates
hyperimaginaries is supersimple. This solves a problem of Shelah in the more
general context of simple theories under weak assumptions.

We carry out some of Galois's work in the setting of an arbitrary first-order
theory T. We replace the ambient algebraically closed field by a large model M
of T, replace fields by definably closed subsets of M, assume that T codes
finite sets, and obtain the fundamental duality of Galois theory matching
subgroups of the Galois group of L over F with intermediate extensions.

We carry out a model-theoretic analysis of the Heisenberg algebra. To this
end, a geometric structure is associated to the Heisenberg algebra and is shown
to be a Zariski geometry. Furthermore, this Zariski geometry is shown to be
non-classical, in the sense that it is not interpretable in an algebraically
closed field. On assuming self-adjointness of the position and momentum
operators, one obtains a discrete substructure of which the original Zariski
geometry is seen as the complexification.

We show some basic facts about dp-minimal ordered structures. The main
results are : dp-minimal groups are abelian-by-finite-exponent, in a divisible
ordered dp-minimal group, any infinite set has non-empty interior, and any
theory of pure tree is dp-minimal.

We show that the continuum hypothesis implies that every measure preserving
near-action of a group on a standard Borel probability $(X,\mu)$ has a
pointwise implementation by Borel measure preserving automorphisms.

In a previous paper, we introduced a way of constructing a forcing along a
simplified gap-1 morass such that the forcing satisfies a chain condition. Now,
we generalize this to gap-2 morasses. As an application, we prove that GCH is
consistent with the existence of a 0-dimensional Hausdorff topology on
$\omega_3$ which has spread $\omega_1$.

It is well-known that a topological group can be represented as a group of
isometries of a reflexive Banach space if and only if its topology is induced
by weakly almost periodic functions (see
\cite{Shtern:CompactSemitopologicalSemigroups},
\cite{Megrelishvili:OperatorTopologies} and
\cite{Megrelishvili:TopologicalTransformations}). We show that for a metrisable
group this is equivalent to the property that its metric is uniformly
equivalent to a stable metric in the sense of Krivine and Maurey (see
\cite{Krivine-Maurey:EspacesDeBanachStables}).

We announce some new results regarding the classification problem for
separable von Neumann algebras. Our results are obtained by applying the notion
of Borel reducibility and Hjorth's theory of turbulence to the isomorphism
relation for separable von Neumann algebras.

The characteristic sequence of hypergraphs $<P_n : n<\omega>$ associated to a
formula $\phi(x;y)$, introduced in [arXiv:0908.4111], is defined by
$P_n(y_1,... y_n) = (\exists x) \bigwedge_{i\leq n} \phi(x;y_i)$. This paper
continues the study of characteristic sequences, showing that graph-theoretic
techniques, notably Szemer\'edi's celebrated regularity lemma, can be naturally
applied to the study of model-theoretic complexity via the characteristic
sequence.

We exhibit canonical middle-inverse Choice maps within categorical
(Free-Variable) Theory of Primitive Recursion as well as in Theory of partial
PR maps over the Theory of Primitive Recursion with predicate abstraction.
Using these choice-maps, defined by mu-recursion, we address the Consistency
problem for a minimal Quantified extension Q of latter two theories: We prove,
that Q's exists-defined mu-operator coincides on PR predicates with that
inherited from theory of partial PR maps.

In this paper, the connections between model theory and the theory of
infinite permutation groups are used to study the n-existence and the
n-uniqueness for n-amalgamation problems of stable theories. We show that, for
any n>1, there exists a stable theory having (k+1)-existence and k-uniqueness,
for every k<n+1, but that does not have neither (n+2)-existence nor
(n+1)-uniqueness. In particular, this generalizes the example, for n=2, due to
E.Hrushovski given in [3].

We investigate the topological complexity of non Borel recognizable tree
languages with regard to the difference hierarchy of analytic sets. We show
that, for each integer $n \geq 1$, there is a $D_{\omega^n}({\bf
\Sigma}^1_1)$-complete tree language L_n accepted by a (non deterministic)
Muller tree automaton. On the other hand, we prove that a tree language
accepted by an unambiguous B\"uchi tree automaton must be Borel. Then we
consider the game tree languages $W_{(i,k)}$, for Mostowski-Rabin indices $(i,
k)$.

We prove that if $V=L$ then there is a $\Pi^1_1$ maximal orthogonal (i.e.
mutually singular) set of measures on Cantor space. This provides a natural
counterpoint to the well-known Theorem of Preiss and Rataj that no analytic set
of measures can be maximal orthogonal.