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Lottery Semantics 30 Dependence and Independence in Logic Juha Kontinen and Jouko Väänänen ESSLLI 2010 CPH Juha Kontinen and Jouko Väänänen (eds.) Proceedings of Dependence and Independence in Logic ESSLLI 2010 Workshop Copenhagen, August 16-20, 2010 ii Preface Dependence and independence are common phenomena, wherever one looks: ecological systems, astronomy, human history, stock markets - but what is their role in logic and - turning the tables - what is the logic of these concepts? The possibility of nesting quantifiers, thus expressing patterns of dependence and independence between variables, accounts for much of the expressive power of first order logic. However, first order logic is not capable of expressing all such patterns, and as a consequence various generalizations - such as branching quantifiers, or the various variants of independence-friendly logic - have been introduced during the last fifty years. Dependence logic is a recent formalism, which brings to the forefront the very concept of dependence, isolating it from the notion of quantifier and making it one of the primitive elements of the language. It can also be added to other logics, such as modal logic. This has opened up an opportunity to develop logical tools for the study of complex forms of dependence, with applications to computer science, philosophy, linguistics, game theory and mathematics. Recently there has been an increasing interest in this topic, especially among young researchers. The goal of this workshop is to provide an opportunity for researchers to further explore the very notions of dependence and independence and their role in formal logic, in particular with regard to logics of imperfect information. iii iv Programme Committee Wilfrid Hodges (British Academy) Tapani Hyttinen (University of Helsinki) Juha Kontinen (University of Helsinki) Michaá Krynicki (Cardinal Stefan WyszyĔski University) Jouko Väänänen (University of Helsinki and University of Amsterdam) v vi Contents Invited lectures Samson Abramsky Relational Hidden Variables and Non-Locality 1 Xavier Caicedo On imperfect information logic as [0,1]-valued logic 2 Lauri Hella Constraint Satisfaction Problems, Partially Ordered Connectives and Dependence Logic 3 Contributed talks Pietro Galliani Epistemic Operators and Uniform Definability in Dependence Logic 4 Pietro Galliani and Allen L. Mann Lottery Semantics 30 Theo M.V. Janssen A Comparison of Independence Friendly logic and Dependence logic 55 Jarmo Kontinen Coherence and computational complexity of quantifier-free dependence logic formulas 58 Antti Kuusisto Logics of Imperfect Information without Identity 78 Peter Lohmann and Heribert Vollmer Complexity Results for Modal Dependence Logic 93 vii Denis Paperno Quantifier Independence and Downward Monotonicity 107 Fan Yang Expressing Second-order Sentences in Intuitionistic Dependence Logic 118 viii Relational Hidden Variables and Non-Locality Samson Abramsky Oxford University Computing Laboratory Abstract We use a simple relational framework to develop the key notions and results on hidden variables and non-locality. The extensive literature on these topics in the foundations of quantum mechanics is couched in terms of probabilistic models, and properties such as locality and no-signalling are formulated probabilistically. We show that to a remarkable extent, the main structure of the theory, through the major No-Go theorems and beyond, survives intact under the replacement of probability distributions by mere relations. In particular, probabilistic notions of independence are replaced by purely logical ones. We also study the relationships between quantum systems, probabilis- tic models and relational models. Probabilistic models can be reduced to relational ones by the ‘possibilistic collapse’, in which non-zero probabili- ties are conflated to (possible) truth. We show that all the independence properties we study are preserved by the possibilistic collapse, in the sense that if the property in its probabilistic form is satisfied by the probabilis- tic model, then the relational version of the property will be satisfied by its possibilistic collapse. More surprisingly, we also show a lifting prop- erty: if a relational model satisfies one of the independence properties, then there is a probabilistic model whose possibilistic collapse gives rise to the relational model, and which satisfies the probabilistic version of the property. These probabilistic models are constructed in a canonical fashion by a form of maximal entropy or indifference principle. 1 1 Workshop on Dependence and Independence in logic European Summer School on Logic, Language and Information ESSLLI 16 - 20 August, 2010 Copenhagen, Denmark On imperfect information logic as [0,1]-valued logic. Xavier Caicedo Universidad de los Andes, Bogot´a Imperfect information logic may be given a [0,1]-valued semantics in finite mod- els by means of the expected value of an optimal pair of mixed strategies for the associated games, so that a sentence ' which is undetermined in a model M gets an intermediate value 0 < M (') < 1 (Sevenster-Sandu, Galliani). It remains the problem whether for each rational r and sentence ' the class fM : M finite and M (') ≥ rg is definable in the ordinary sense in imperfect information logic (that is, whether it belongs to NP ). In very general grounds it may be shown that there is no [0,1]-valued semantics in all models extending the partial f0,1g-valued game semantics of imperfect information logic such that M(:') = 1 − M(');M(' _ ) = maxfM(');M( )g and fM : M (') ≥ rg is 1 in Σ1 for each rational r: 1 2 Constraint Satisfaction Problems, Partially Ordered Connectives and Dependence Logic Lauri Hella University of Tampere Joint work with Merlijn Sevenster and Tero Tulenheimo Abstract: A constraint satisfaction problem (CSP) is the problem of deciding whether there is an assignment of a set of variables in a fixed domain such that the assignment satisfies given constraints. Every CSP can be expressed alternatively as a homomorphism problem: for a given finite relational structure A, Csp(A) is the class of all structures B that can be homomorphically mapped into A. The infamous Dichotomy Conjecture of Feder and Vardi ([1]) states that for all A, the problem Csp(A) is either NP-complete or it is in PTIME. In their seminal work [1] on the descriptive complexity of CSP, Feder and Vardi 1 introduced a natural fragment of Σ1, which they call Monotone Monadic SNP (MM- SNP). They showed that every CSP is expressible in MMSNP, and moreover, the Dichotomy Conjecture holds for CSP if and only if it holds for MMSNP. In the article [2], we established connections between partially ordereded connec- tives, CSP and MMSNP. In our first main result, we characterize SNP (the fragment 1 ~ of Σ1 consisting of formulas X ~x φ, where φ is quantifier free) in terms of partially ordered connectives. More precisely∃ ∀ , we prove that SNP D[QF], where D[QF] is the set of all formulas with a single partially ordered connectiv≡ e applied to a matrix of quantifier free formulas. In our second main result we give a similar character- ization for MMSNP: MMSNP C1[NEQF]. Here C is a minor variant of the set of partially ordered connectives,≡ and the subscript 1 refers to the restriction that each row of the connective has only one universal quantifier. Furthermore, NEQF is the set of equality and quantifier free formulas such that all relation symbols of the vocabulary appear only negatively. In the talk, I will first give a survey on CSP, MMSNP, SNP and partially or- dered connectives. Then I will go through the main results in [2]. Finally, I will consider some new ideas concerning certain fragments of dependence logic, and their relationship to partially ordered connectives. References: 1. Feder, T. and Vardi M. (1998) The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM Journal on Computing, 28, 57–104. 2. Hella, L., Sevenster, M. and Tulenheimo, T. (2008) Partially ordered con- nectives and monadic monotone strict NP. Journal of Logic, Language and Information, 17, 323–344. 3 Epistemic Operators and Uniform Definability in Dependence Logic∗ Pietro Galliani ILLC University of Amsterdam The Netherlands ([email protected]) Abstract The concept of uniform definability of operators in the language of Dependence Logic is formally defined, and two families of operators δα and quantifiers ∀α, for α ranging over the cardinal numbers, are investigated. In particular, it is proved that for any finite n and m, all ∀n, ∀m, δn and δm are reciprocally uniformly definable, but neither of them is uniformly definable in Dependence Logic: this answers an open problem by Kontinen and V¨a¨an¨anen ([12]) about whether the ∀1 quantifier is uniformly definable in Dependence Logic. A more direct proof than that of ([12]) is also found of the fact that the ∀1 quantifier (and, consequently, all ∀n and δn quantifiers) do not increase the expressive power of Dependence Logic, even on open formulas; furthermore, an interpretation of the δn operators in terms of partial announcements is proved to hold for the Game Theoretic Semantics of Dependence Logic, and the Ehrenfeucht-Fra¨ıss´egame for Dependence Logic is adapted to the logics D(⊔, ∀α), where ⊔ is the classical disjunction. Finally, a criterion for uniform definability in Dependence Logic is proved. 1 Dependence Logic Logics of imperfect information ([6], [7], [15], [14]) are extensions of First Order Logic whose Game Theoretic Semantics may be derived from
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