The Structure of First-Order Causality

Game semantics describe the interactive behavior of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in first-order propositional logic. One of the main difficulties that has to be faced during the elaboration of this kind of semantics is to characterize definable strategies, that is strategies which actually behave like a proof. This is usually done by restricting the model to strategies satisfying subtle combinatorial conditions, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task, which requires to combine advanced tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of the monoidal category of definable strategies of our model, by the means of generators and relations: those strategies can be generated from a finite set of atomic strategies and the equality between strategies admits a finite axiomatization, this equational structure corresponding to a polarized variation of the notion of bialgebra. This work thus bridges algebra and denotational semantics in order to reveal the structure of dependencies induced by first-order quantifiers, and lays the foundations for a mechanized analysis of causality in programming languages

The Combinatorics of Causality

We introduce and explore the notion of "spaces of input histories", a broad family of combinatorial objects which can be used to model input-dependent, dynamical causal order. We motivate our definition with reference to traditional partial order- and preorder-based notions of causal order, adopted by the majority of previous literature on the subject, and we proceed to explore the novel landscape of combinatorial complexity made available by our generalisation of those notions. In the process, we discover that the fine-grained structure of causality is significantly more complex than we might have previously believed: in the simplest case of binary inputs, the number of available "causally complete" spaces grows from 7 on 2 events, to 2644 on 3 events, to an unknown number on 4 events (likely around a billion). For perspective, previous literature on non-locality and contextuality used a single one of the 2644 available spaces on 3 events, work on definite causality used 19 spaces, derived from partial orders, and work on indefinite causality used only 6 more, for a grand total of 25. This paper is the first instalment in a trilogy: the sheaf-theoretic treatment of causal distributions is detailed in Part 2, "The Topology of Causality" [arXiv:2303.07148], while the polytopes formed by the associated empirical models are studied in Part 3, "The Geometry of Causality" [arXiv:2303.09017]. An exhaustive classification of the 2644 causally complete spaces on 3 events with binary inputs is provided in the supplementary work "Classification of causally complete spaces on 3 events with binary inputs", together with the algorithm used for the classification and partial results from the ongoing search on 4 events.Comment: Originally Part 1 of "The Topology and Geometry of Causality" [arXiv:2206.08911v2], which it now replaces. Part 2 of the trilogy is now published as "The Topology of Causality" [arXiv:2303.07148] and Part 3 is now published as "The Geometry of Causality" [arXiv:2303.09017

Wave propagation in axion electrodynamics

In this paper, the axion contribution to the electromagnetic wave propagation is studied. First we show how the axion electrodynamics model can be embedded into a premetric formalism of Maxwell electrodynamics. In this formalism, the axion field is not an arbitrary added Chern-Simon term of the Lagrangian, but emerges in a natural way as an irreducible part of a general constitutive tensor.We show that in order to represent the axion contribution to the wave propagation it is necessary to go beyond the geometric approximation, which is usually used in the premetric formalism. We derive a covariant dispersion relation for the axion modified electrodynamics. The wave propagation in this model is studied for an axion field with timelike, spacelike and null derivative covectors. The birefringence effect emerges in all these classes as a signal of Lorentz violation. This effect is however completely different from the ordinary birefringence appearing in classical optics and in premetric electrodynamics. The axion field does not simple double the ordinary light cone structure. In fact, it modifies the global topological structure of light cones surfaces. In CFJ-electrodynamics, such a modification results in violation of causality. In addition, the optical metrics in axion electrodynamics are not pseudo-Riemannian. In fact, for all types of the axion field, they are even non-Finslerian

Finite sample sign-based procedures in linear and non-linear statistical models: with applications to Granger causality analysis

This thesis consists of three essays on hypothesis testing and Granger causality analysis. The two main topics under consideration are: (1) exact point-optimal sign-based inference in linear and non-linear predictive regressions with a financial application; and (2) sign-based measures of causality in the Granger sense with an economics application. These essays can be regarded as an extension to the sign-based procedures proposed by Dufour and Taamouti (2010a). The distinction is that in our study the predictors are stochastic and the signs may exhibit serial dependence. As a consequence, the task of obtaining feasible test statistics and measures of Granger causality is more challenging. Therefore, in each essay we either impose an assumption on the sign process or propose tools with which the entire dependence structure of the signs can feasibly be modeled. The three essays are summarized below. In the first chapter, we acknowledge that the predictors of stock returns (e.g. dividend-price ratio, earnings-price ratio, etc.) are often persistent, with innovations that are highly correlated with the disturbances of the predictive regressions. This generally leads to invalid inference using the conventional T-test. Therefore, we propose point-optimal sign-based tests in the context of linear and non-linear models that are valid in the presence of stochastic regressors. In order to obtain feasible test statistics, we impose an assumption on the dependence structure of the signs; namely, we assume that the signs follow a finite order Markov process. The proposed tests are exact,distribution-free, and robust against heteroskedasticity of unknown form. Furthermore, they may be inverted to build confidence regions for the parameters of the regression function. Point-optimal sign-based tests depend on the alternative hypothesis, which in practice is unknown. Therefore, a problem exists: that of finding an alternative which maximizes the power. To choose the alternative, we adopt the adaptive approach based on the split-sample technique suggested by Dufour and Taamouti (2010a). We present a Monte Carlo study to assess the performance of the proposed "quasi"-point-optimal sign test by comparing its size and power to those of certain existing tests that are intended to be robust against heteroskedasticity. The results show that our procedures outperform the other tests. Finally, we consider an empirical application to illustrate the usefulness of the proposed tests for testing the predictability of stock returns. In the second chapter, we relax the assumption imposed earlier on the dependence structure of the signs. We had provided a caveat that to obtain feasible test statistics, the Markovian assumption must be imposed on the signs. In this essay, we extend the flexibility of the exact point-optimal sign-based tests proposed in the first chapter, by considering the entire dependence structure of the signs and building feasible test statistics based on pair copula constructions of the sign process. In a Monte Carlo study, we compare the performance of the proposed "quasi"-point-optimal sign tests based on pair copula constructions by comparing its size and power to those of certain existing tests that are intended to be robust against heteroskedasticity. The simulation results maintain the superiority of our procedures to existing popular tests. In the third chapter, we propose sign-based measures of Granger causality based on the Kullback-Leibler distance that quantify the degree of causalities. Furthermore, we show that by using bound-type procedures, Granger non-causality tests between random variables can be developed as a byproduct of the sign-based measures. The tests are exact, distribution-free and robust against heteroskedasticity of unknown form. Additionally, as in the first chapter, we impose a Markovian assumption on the sign process to obtain feasible measures and tests of causality. To estimate the sign-based measures, we suggest the use of vector autoregressive sieve bootstrap to reduce the bias and obtain bias-corrected estimators. Furthermore, we discuss the validity of the bootstrap technique. A Monte Carlo simulation study reveals that the bootstrap bias-corrected estimator of the causality measures produce the desired outcome. Furthermore, the tests of Granger non-causality based on the signs perform well in terms of size control and power. Finally, an empirical application is considered to illustrate the practical relevance of the sign-based causality measures and tests

Does Oil Predict Gold? A Nonparametric Causality-in-Quantiles Approach

This paper examines the predictive power of oil price for gold price using the novel nonparametric causality-in-quantiles testing approach. The study uses weekly data over the April 1983-August 2016 period for both the spot and 1-month to 12-month futures markets. The new approach, the causality-in-quantile, allows one to test for causality-in-mean and causality-in-variance when there may be no causality in the first moment but higher order interdependencies may exist. The tests are preferred over the linear Granger causality test that might be subject to misleading results due to misspecification. Contrary to no predictability results obtained under misspecified linear structure, the nonparametric causality-in-quantiles test shows that oil price has a weak predictive power for the gold price. Moreover, the causality-in-variance tests obtain strong support for the predictive capacity of oil for gold market volatility. The results underline the importance of accounting for nonlinearity in the analysis of causality from oil to gold