The usual resource interpretation of linear logic says that variables have to be used exactly once. However, there are models of linear logic where this interpretation is too restrictive. In this work we show how in probabilistic models of linear logic the correct resource interpretation should be sampling, i.e. the linear arrow should be read as "the output may only sample once from its input". We accommodate this new interpretation by defining a multilanguage syntax and its categorical semantics that bridges the Markov kernel and linear logic interpretations of probabilistic programs.
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We propose a new sheaf semantics for secure information flow over a space of abstract behaviors, based on synthetic domain theory: security classes are open/closed partitions, types are sheaves, and redaction of sensitive information corresponds to restricting a sheaf to a closed subspace. Our security-aware computational model satisfies termination-insensitive noninterference automatically, and therefore constitutes an intrinsic alternative to state of the art extrinsic/relational models of noninterference. Our semantics is the latest application of Sterling and Harper's recent re-interpretation of phase distinctions and noninterference in programming languages in terms of Artin gluing and topos-theoretic open/closed modalities. Prior applications include parametricity for ML modules, the proof of normalization for cubical type theory by Sterling and Angiuli, and the cost-aware logical framework of Niu et al. In this paper we employ the phase distinction perspective twice: first to reconstruct the syntax and semantics of secure information flow as a lattice of phase distinctions between "higher" and "lower" security, and second to verify the computational adequacy of our sheaf semantics vis-\`a-vis an extension of Abadi et al.'s dependency core calculus with a construct for declassifying termination channels.
Machine learning-based methods have achieved successful applications in machinery fault diagnosis. However, the main limitation that exists for these methods is that they operate as a black box and are generally not interpretable. This paper proposes a novel neural network structure, called temporal logic neural network (TLNN), in which the neurons of the network are logic propositions. More importantly, the network can be described and interpreted as a weighted signal temporal logic. TLNN not only keeps the nice properties of traditional neuron networks but also provides a formal interpretation of itself with formal language. Experiments with real datasets show the proposed neural network can obtain highly accurate fault diagnosis results with good computation efficiency. Additionally, the embedded formal language of the neuron network can provide explanations about the decision process, thus achieve interpretable fault diagnosis.
Category theory can be used to state formulas in First-Order Logic without using set membership. Several notable results in logic such as proof of the continuum hypothesis can be elegantly rewritten in category theory. We propose in this paper a reformulation of the usual set-theoretical semantics of the description logic $\mathcal{ALC}$ by using categorical language. In this setting, ALC concepts are represented as objects, concept subsumptions as arrows, and memberships as logical quantifiers over objects and arrows of categories. Such a category-theoretical semantics provides a more modular representation of the semantics of $\mathcal{ALC}$ and a new way to design algorithms for reasoning.
Many forms of dependence manifest themselves over time, with behavior of variables in dynamical systems as a paradigmatic example. This paper studies temporal dependence in dynamical systems from a logical perspective, by extending a minimal modal base logic of static functional dependencies. We define a logic for dynamical systems with single time steps, provide a complete axiomatic proof calculus, and show the decidability of the satisfiability problem for a substantial fragment. The system comes in two guises: modal and first-order, that naturally complement each other. Next, we consider a timed semantics for our logic, as an intermediate between state spaces and temporal universes for the unfoldings of a dynamical system. We prove completeness and decidability by combining techniques from dynamic-epistemic logic and modal logic of functional dependencies with complex terms for objects. Also, we extend these results to the timed logic with functional symbols and term identity. Finally, we conclude with a brief outlook on how the system proposed here connects with richer temporal logics of system behavior, and with dynamic topological logic.
The interpretability of deep neural networks has attracted increasing attention in recent years, and several methods have been created to interpret the "black box" model. Fundamental limitations remain, however, that impede the pace of understanding the networks, especially the extraction of understandable semantic space. In this work, we introduce the framework of semantic explainable AI (S-XAI), which utilizes row-centered principal component analysis to obtain the common traits from the best combination of superpixels discovered by a genetic algorithm, and extracts understandable semantic spaces on the basis of discovered semantically sensitive neurons and visualization techniques. Statistical interpretation of the semantic space is also provided, and the concept of semantic probability is proposed for the first time. Our experimental results demonstrate that S-XAI is effective in providing a semantic interpretation for the CNN, and offers broad usage, including trustworthiness assessment and semantic sample searching.
Universal coding of integers~(UCI) is a class of variable-length code, such that the ratio of the expected codeword length to $\max\{1,H(P)\}$ is within a constant factor, where $H(P)$ is the Shannon entropy of the decreasing probability distribution $P$. However, if we consider the ratio of the expected codeword length to $H(P)$, the ratio tends to infinity by using UCI, when $H(P)$ tends to zero. To solve this issue, this paper introduces a class of codes, termed generalized universal coding of integers~(GUCI), such that the ratio of the expected codeword length to $H(P)$ is within a constant factor $K$. First, the definition of GUCI is proposed and the coding structure of GUCI is introduced. Next, we propose a class of GUCI $\mathcal{C}$ to achieve the expansion factor $K_{\mathcal{C}}=2$ and show that the optimal GUCI is in the range $1\leq K_{\mathcal{C}}^{*}\leq 2$. Then, by comparing UCI and GUCI, we show that when the entropy is very large or $P(0)$ is not large, there are also cases where the average codeword length of GUCI is shorter. Finally, the asymptotically optimal GUCI is presented.
Factor analysis is often used to assess whether a single univariate latent variable is sufficient to explain most of the covariance among a set of indicators for some underlying construct. When evidence suggests that a single factor is adequate, research often proceeds by using a univariate summary of the indicators in subsequent research. Implicit in such practices is the assumption that it is the underlying latent, rather than the indicators, that is causally efficacious. The assumption that the indicators do not have effects on anything subsequent, and that they are themselves only affected by antecedents through the underlying latent is a strong assumption, effectively imposing a structural interpretation on the latent factor model. In this paper, we show that this structural assumption has empirically testable implications, even though the latent variable itself is unobserved. We develop a statistical test to potentially reject the structural interpretation of a latent factor model. We apply this test to data concerning associations between the Satisfaction-with-Life-Scale and subsequent all-cause mortality, which provides strong evidence against a structural interpretation for a univariate latent underlying the scale. Discussion is given to the implications of this result for the development, evaluation, and use of measures and for the use of factor analysis itself.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
Machine-learning models have demonstrated great success in learning complex patterns that enable them to make predictions about unobserved data. In addition to using models for prediction, the ability to interpret what a model has learned is receiving an increasing amount of attention. However, this increased focus has led to considerable confusion about the notion of interpretability. In particular, it is unclear how the wide array of proposed interpretation methods are related, and what common concepts can be used to evaluate them. We aim to address these concerns by defining interpretability in the context of machine learning and introducing the Predictive, Descriptive, Relevant (PDR) framework for discussing interpretations. The PDR framework provides three overarching desiderata for evaluation: predictive accuracy, descriptive accuracy and relevancy, with relevancy judged relative to a human audience. Moreover, to help manage the deluge of interpretation methods, we introduce a categorization of existing techniques into model-based and post-hoc categories, with sub-groups including sparsity, modularity and simulatability. To demonstrate how practitioners can use the PDR framework to evaluate and understand interpretations, we provide numerous real-world examples. These examples highlight the often under-appreciated role played by human audiences in discussions of interpretability. Finally, based on our framework, we discuss limitations of existing methods and directions for future work. We hope that this work will provide a common vocabulary that will make it easier for both practitioners and researchers to discuss and choose from the full range of interpretation methods.
In structure learning, the output is generally a structure that is used as supervision information to achieve good performance. Considering the interpretation of deep learning models has raised extended attention these years, it will be beneficial if we can learn an interpretable structure from deep learning models. In this paper, we focus on Recurrent Neural Networks (RNNs) whose inner mechanism is still not clearly understood. We find that Finite State Automaton (FSA) that processes sequential data has more interpretable inner mechanism and can be learned from RNNs as the interpretable structure. We propose two methods to learn FSA from RNN based on two different clustering methods. We first give the graphical illustration of FSA for human beings to follow, which shows the interpretability. From the FSA's point of view, we then analyze how the performance of RNNs are affected by the number of gates, as well as the semantic meaning behind the transition of numerical hidden states. Our results suggest that RNNs with simple gated structure such as Minimal Gated Unit (MGU) is more desirable and the transitions in FSA leading to specific classification result are associated with corresponding words which are understandable by human beings.
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