arXiv subjects

Jouko Väänänen

Publications and source records attributed to Jouko Väänänen.

53 records · Page 3Linked to original sources

arXiv2016

Quantum Team Logic and Bell's Inequalities

A logical approach to Bell's Inequalities of quantum mechanics has been introduced by Abramsky and Hardy [2]. We point out that the logical Bell's Inequalities of [2] are provable in the probability logic of Fagin, Halpern and Megiddo [4]. Since it is now considered empirically established that quantum mechanics violates Bell's Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell's Inequalities are not provable, and prove a Completeness Theorem for this logic. For this end we generalise the team semantics of dependence logic [7] first to probabilistic team semantics, and then to what we call quantum team semantics.

math.LO↗

arXiv2016

Propositional Logics of Dependence

In this paper, we study logics of dependence on the propositional level. We prove that several interesting propositional logics of dependence, including propositional dependence logic, propositional intuitionistic dependence logic as well as propositional inquisitive logic, are expressively complete and have disjunctive or conjunctive normal forms. We provide deduction systems and prove the completeness theorems for these logics.

math.LO↗

arXiv2014

The Logic of Approximate Dependence

We extend the treatment of functional dependence, the basic concept of dependence logic, to include the possibility of dependence with a limited number of exceptions. We call this approximate dependence. The main result of the paper is a Completeness Theorem for approximate dependence atoms. We point out some problematic features of this which suggests that we should consider multi-teams, not just teams.

math.LO↗

arXiv2013

Positional strategies in long Ehrenfeucht-Fraissé games

We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality \aleph_2 such that the second player has a winning strategy in the Ehrenfeucht-Fraïssé-game of length ω_1 but there is no σ-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist.

math.LO↗

arXiv2013

On Dependence Logic

We give an overview of some developments in dependence and independence logic. This is a tiny selection, intended for a newcomer, from a rapidly growing literature on the topic. Furthermore, we discuss conditional independence atoms and we prove that conditional and non-conditional independence logic are equivalent. Finally, we briefly discuss an application of our logics to belief representation.

math.LO↗

arXiv2013

Dependence Logic with Generalized Quantifiers: Axiomatizations

We prove two completeness results, one for the extension of dependence logic by a monotone generalized quantifier Q with weak interpretation, weak in the meaning that the interpretation of Q varies with the structures. The second result considers the extension of dependence logic where Q is interpreted as "there exists uncountable many." Both of the axiomatizations are shown to be sound and complete for FO(Q) consequences.

math.LO↗

arXiv2012

An Ehrenfeucht-Fraïssé Game for $L_{ω_1ω}$

Ehrenfeucht-Fraisse games are very useful in studying separation and equivalence results in logic. The standard finite Ehrenfeucht-Fraisse game characterizes equivalence in first order logic. The standard Ehrenfeucht-Fraisse game in infinitary logic characterizes equivalence in $L_{\inftyω}$. The logic $L_{ω_1ω}$ is the extension of first order logic with countable conjunctions and disjunctions. There was no Ehrenfeucht-Fraisse game for $L_{ω_1ω}$ in the literature. In this paper we develop an Ehrenfeucht-Fraisse Game for $L_{ω_1ω}$. This game is based on a game for propositional and first order logic introduced by Hella and Vaananen. Unlike the standard Ehrenfeucht-Fraisse games which are modeled solely after the behavior of quantifiers, this new game also takes into account the behavior of connectives in logic. We prove the adequacy theorem for this game. We also apply the new game to prove complexity results about infinite binary strings.

math.LO↗

arXiv2012

Dependence and Independence

We introduce an atomic formula intuitively saying that given variables are independent from given other variables if a third set of variables is kept constant. We contrast this with dependence logic. We show that our independence atom gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence.

cs.LO↗

arXiv2012

On Second-order Characterizability

We investigate the extent of second order characterizable structures by extending Shelah's Main Gap dichotomy to second order logic. For this end we consider a countable complete first order theory T. We show that all sufficiently large models of T have a characterization up to isomorphism in the extension of second order logic obtained by adding a little bit of infinitary logic if and only if T is shallow superstable with NDOP and NOTOP. Our result relies on cardinal arithmetic assumptions. Under weaker assumptions we get consistency results or alternatively results about second order logic with Henkin semantics.

math.LO↗

arXiv2012

The size of a formula as a measure of complexity

We introduce a refinement of the usual Ehrenfeucht-Fra\"ıssé game. The new game will help us make finer distinctions than the traditional one. In particular, it can be used to measure the size formulas needed for expressing a given property. We will give two versions of the game: the first version characterizes the size of formulas in propositional logic, and the second version works for first-order predicate logic.

math.LO↗

arXiv2012

Axiomatizing first order consequences in dependence logic

Dependence logic, introduced in [8], cannot be axiomatized. However, first-order consequences of dependence logic sentences can be axiomatized, and this is what we shall do in this paper. We give an explicit axiomatization and prove the respective Completeness Theorem.

math.LO↗

arXiv2004

Recursive logic frames

We define the concept of a logic frame, which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive logic frame is called recursively (countably) compact, if every recursive (respectively, countable) finitely consistent theory has a model. We show that for logic frames built from the cardinality quantifiers ''there exists at least lambda'' recursive compactness always implies countable compactness. On the other hand we show that a recursively compact extension need not be countably compact.

math.LO↗

arXiv2002

More on the Ehrenfencht-Fraisse game of length omega_1

Let A and B be two first order structures of the same relational vocabulary L. The Ehrenfeucht-Fraisse-game of length gamma of A and B denoted by EFG_gamma(A,B) is defined as follows: There are two players called for all and exists. First for all plays x_0 and then exists plays y_0. After this for all plays x_1, and exists plays y_1, and so on. Eventually a sequence <(x_beta,y_beta): beta has been played. The rules of the game say that both players have to play elements of A cup B. Moreover, if for all plays his x_beta in A (B), then exists has to play his y_beta in B (A). Thus the sequence < (x_beta,y_beta):beta determines a relation pi subseteq AxB. Player exists wins this round of the game if pi is a partial isomorphism. Otherwise for all wins. The game EFG_gamma^delta (A,B) is defined similarly except that the players play sequences of length<delta at a time. Theorem 1: The following statements are equiconsistent relative to ZFC: (A) There is a weakly compact cardinal. (B) CH and EF_{omega_1}(A,B) is determined for all models A,B of cardinality aleph_2 . Theorem 2: Assume that 2^omega <2^{omega_3} and T is a countable complete first order theory. Suppose that one of (i)-(iii) below holds. Then there are A,B models T of power omega_3 such that for all cardinals 1<theta<=omega_3, EF^theta_{omega_1}(A,B) is non-determined. [(i)] T is unstable. [(ii)] T is superstable with DOP or OTOP. [(iii)] T is stable and unsuperstable and 2^omega <= omega_3.

math.LO↗

arXiv2000

A Note on Extensions of Infinitary Logic

We show that a strong form of the so called Lindstrom's Theorem fails to generalize to extensions of L_{kappa,omega} and L_{kappa,kappa}: For weakly compact kappa there is no strongest extension of L_{kappa,omega} with the (kappa,kappa)-compactness property and the Lowenheim-Skolem theorem down to kappa. With an additional set-theoretic assumption, there is no strongest extension of L_{kappa,kappa} with the (kappa,kappa)-compactness property and the Lowenheim-Skolem theorem down to <kappa.

math.LO↗

arXiv1997

Stationary sets and infinitary logic

Let K^0_lambda be the class of structures < lambda,<,A>, where A subseteq lambda is disjoint from a club, and let K^1_lambda be the class of structures < lambda,<,A>, where A subseteq lambda contains a club. We prove that if lambda = lambda^{< kappa} is regular, then no sentence of L_{lambda^+ kappa} separates K^0_lambda and K^1_lambda. On the other hand, we prove that if lambda = mu^+, mu = mu^{< mu}, and a forcing axiom holds (and aleph_1^L= aleph_1 if mu = aleph_0), then there is a sentence of L_{lambda lambda} which separates K^0_lambda and K^1_lambda .

math.LO↗

arXiv1993

The Ehrenfeucht-Fraisse-game of length omega_1

Let (A) and (B) be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fra{i}sse-game of length omega_1 of A and B which we denote by G_{omega_1}(A,B). This game is like the ordinary Ehrenfeucht-Fraisse-game of L_{omega omega} except that there are omega_1 moves. It is clear that G_{omega_1}(A,B) is determined if A and B are of cardinality <= aleph_1. We prove the following results: Theorem A: If V=L, then there are models A and B of cardinality aleph_2 such that the game G_{omega_1}(A,B) is non-determined. Theorem B: If it is consistent that there is a measurable cardinal, then it is consistent that G_{omega_1}(A,B) is determined for all A and B of cardinality <= aleph_2. Theorem C: For any kappa >= aleph_3 there are A and B of cardinality kappa such that the game G_{omega_1}(A,B) is non-determined.

math.LO↗

arXiv1993

On the number of automorphism of uncontable models

Let s(A) denote the number of automorphisms of a model A of power omega_1. We derive a necessary and sufficient condition in terms of trees for the existence of an A with omega_1 < s(A) < 2^{omega_1}. We study the sufficiency of some conditions for s(A)=2^{omega_1}. These conditions are analogous to conditions studied by D.Kueker in connection with countable models.

math.LO↗