A non-interleaving process calculus for multi-party synchronisation

We introduce the wire calculus. Its dynamic features are inspired by Milner's CCS: a unary prefix operation, binary choice and a standard recursion construct. Instead of an interleaving parallel composition operator there are operators for synchronisation along a common boundary and non-communicating parallel composition. The (operational) semantics is a labelled transition system obtained with SOS rules. Bisimilarity is a congruence with respect to the operators of the language. Quotienting terms by bisimilarity results in a compact closed category

Redundancy, Deduction Schemes, and Minimum-Size Bases for Association Rules

Association rules are among the most widely employed data analysis methods in the field of Data Mining. An association rule is a form of partial implication between two sets of binary variables. In the most common approach, association rules are parameterized by a lower bound on their confidence, which is the empirical conditional probability of their consequent given the antecedent, and/or by some other parameter bounds such as "support" or deviation from independence. We study here notions of redundancy among association rules from a fundamental perspective. We see each transaction in a dataset as an interpretation (or model) in the propositional logic sense, and consider existing notions of redundancy, that is, of logical entailment, among association rules, of the form "any dataset in which this first rule holds must obey also that second rule, therefore the second is redundant". We discuss several existing alternative definitions of redundancy between association rules and provide new characterizations and relationships among them. We show that the main alternatives we discuss correspond actually to just two variants, which differ in the treatment of full-confidence implications. For each of these two notions of redundancy, we provide a sound and complete deduction calculus, and we show how to construct complete bases (that is, axiomatizations) of absolutely minimum size in terms of the number of rules. We explore finally an approach to redundancy with respect to several association rules, and fully characterize its simplest case of two partial premises.Comment: LMCS accepted pape

Issues in stochastic ocean modeling

The general theory of stochastic differential equations is presented in this chapter, including the theoretical background on how measured statistics from time series can be used to develop a stochastic parameterization. The general rules of stochastic calculus, including the important and often overlooked differences between Ito and Stratonovich calculus, are mentioned, and references are provided in which more detail may be found. We discuss how Stratonovich calculus is usually appropriate for fluid systems, whereas Ito calculus is often appropriate for data assimilation. We also discuss some common numerical pitfalls awaiting the unwary modeler, and warn against unsophisticated random number generators. Finally, we offer a selection of examples showing the importance of the variability of unresolved scales in an ocean model and, by citation, a variety of methods that have been employed

Root-theoretic Young diagrams and Schubert Calculus

A longstanding problem in algebraic combinatorics is to find nonnegative combinatorial rules for the Schubert calculus of generalized flag varieties; that is, for the structure constants of their cohomology rings with respect to the Schubert basis. There are several natural choices of combinatorial indexing sets for the Schubert basis classes. This thesis examines a number of Schubert calculus problems from the common lens of root-theoretic Young diagrams (RYDs). In terms of RYDs, we present nonnegative Schubert calculus rules for the (co)adjoint varieties of classical Lie type. Using these we give polytopal descriptions of the set of nonzero Schubert structure constants for the (co)adjoint varieties where the RYDs are all planar, and suggest a connection between planarity of the RYDs and polytopality of the nonzero Schubert structure constants. This is joint work with A. Yong. For the family of (nonmaximal) isotropic Grassmannians, we characterize the RYDs and give a bijection between RYDs and the k-strict partitions of A. Buch, A. Kresch and H. Tamvakis. We apply this bijection to show that the (co)adjoint Schubert calculus rules agree with the Pieri rules of A. Buch, A. Kresch and H. Tamvakis, which is needed for the proofs of the (co)adjoint rules. We also use RYDs to study the Belkale-Kumar deformation of the ordinary cup product on cohomology of generalized flag varieties. This product structure was introduced by P. Belkale and S. Kumar and used to study a generalization of the Horn problem. A structure constant of the Belkale-Kumar product is either zero or equal to the corresponding Schubert structure constant, hence the Belkale-Kumar product captures a certain subset of the Schubert structure constants. We give a new formula (after that of A. Knutson and K. Purbhoo) in terms of RYDs for the Belkale-Kumar product on flag varieties of type A. We also extend this formula outside of type A to the (co)adjoint varieties of classical type. With O. Pechenik, we introduce a new deformed product structure on the cohomology of generalized flag varieties, whose nonzero structure constants can be understood in terms of projections to smaller flag varieties. We draw comparisons with the ordinary cup product and the Belkale-Kumar product

The Economy of Happiness

Happiness in philosophical ethics and utility or satisfaction in economics have much in common. The paper investigates the ethical economy of happiness as a joint topic of ethical and economic theory. It shows that limits of the calculus of utility maximization also apply to concepts of the greatest happiness in philosophy: it is impossible to distinguish the utility or happiness maximizing life strategy. The paper discusses the problem of inter-personal comparisons of happiness and satisfaction and the relevance of the theory of material value qualities developed by Max Scheler’s non-formal, material value ethics for the theory of goods, private and public. Ethics and economics are concerned with rules and duties. It is, however, also necessary to develop a theory of goods and values. Reflections are also made on the relationship between fact and value. Since there are side-effects of facts or experiences on our values, the naturalist fallacy of deriving value statements from experience seems to be less a fallacy than is usually assumed since Hume.

Classical Logic through the Looking-Glass

In Lewis Carroll’s Through the Looking Glass and What Alice Found There, Alice enters through a mirror into the realm reflected. It is, of course, left-right reversed but this is only the start of the fun and games when Alice explores the world on the other side of the mirror. Borrowing, if only in part, Carroll’s theme of inversion, my aim is to take a look at classical logic in something of an inverted way, or, to be more exact, in three somewhat inverted ways. Firstly, I come at proof of the completeness of classical logic in the Lindenbaum-Henkin style backwards: I take for granted the existence of a set Σ for which it holds, for some formula φ, that ψ !in Σ if, and only if, Σu{ψ} |- φ  then read off the rules of inference governing connectives and quantifiers that most directly yield the desired (classical) semantic properties. We thus obtain general elimination rules and what I have elsewhere called general introduction rules. Secondly, the same approach lets us read off a different set of rules: those of the cut-free sequent calculus S' of (Smullyan, 1968). Smullyan uses this calculus in proving the Craig-Lyndon interpolation theorem for first-order logic (without identity and function symbols). By attending very carefully to the steps in Smullyan’s proof, we obtain a strengthening: if φ |- ψ,  /|- ¬φ and /|- ψ then there is an interpolant χ, a formula employing only the non-logical vocabulary common to φ and ψ, such that φ entails χ in the first-order version of Kleene’s 3-valued logic and χ entails ψ in the first-order version of Graham Priest’s Logic of Paradox. The result, which is hidden from view in natural deduction formulations of classical logic, extends, I believe, to firstorder logic with identity. Thirdly, we look at a contraction-free “approximation” to classical propositional logic. Adding the general introduction rules for negation or the conditional leads to Contraction being a derived rule, apparently blurring the distinction between structural and operational rules