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Decision support is the science and associated practice that consist in providing recommendations to decision makers facing problems, based on available theoretical knowledge and empirical data. Although this activity is often seen as being concerned with solving mathematical problems and conceiving algorithms, it is essentially an empirical and socially framed activity, where interactions between clients and analysts, and between them and concerned third parties, play a crucial role. Since the 80s, two concepts have structured the literature devoted to analysing this aspect of decision support: validity and legitimacy. Whereas validity is focused on the interactions between the client and the analyst, legitimacy refers to the broader picture: the organisational context, the overall problem situation, the environment, culture, history. Despite its importance, this concept has not received the attention it deserves in the literature in decision support. The present paper aims at filling this gap. For that purpose, we review the literature in other disciplines relevant to elaborate a concept of legitimacy useful in decision support contexts. Based on this review, we propose a general theory of legitimacy, adapted to decision support contexts, encompassing the relevant contributions we found in the literature. According to this general theory, a legitimate decision support intervention is one for which the decision support provider produces a justification that satisfies two conditions: (i) it effectively convinces the decision support provider's interlocutors (effectiveness condition) and (ii) it is organised around the active elicitation of as many and as diverse counterarguments as possible (truthfulness condition). Despite its conceptual simplicity, legitimacy, understood in this sense, is a very exacting requirement, opening ambitious research avenues that we delineate.

We prove two theorems related to the Central Limit Theorem (CLT) for Martin-L\"of Random (MLR) sequences. Martin-L\"of randomness attempts to capture what it means for a sequence of bits to be "truly random". By contrast, CLTs do not make assertions about the behavior of a single random sequence, but only on the distributional behavior of a sequence of random variables. Semantically, we usually interpret CLTs as assertions about the collective behavior of infinitely many sequences. Yet, our intuition is that if a sequence of bits is "truly random", then it should provide a "source of randomness" for which CLT-type results should hold. We tackle this difficulty by using a sampling scheme that generates an infinite number of samples from a single binary sequence. We show that when we apply this scheme to a Martin-L\"of random sequence, the empirical moments and cumulative density functions (CDF) of these samples tend to their corresponding counterparts for the normal distribution. We also prove the well known almost sure central limit theorem (ASCLT), which provides an alternative, albeit less intuitive, answer to this question. Both results are also generalized for Schnorr random sequences.

We give a nearly-linear time reduction that encodes any linear program as a 2-commodity flow problem with only a small blow-up in size. Under mild assumptions similar to those employed by modern fast solvers for linear programs, our reduction causes only a polylogarithmic multiplicative increase in the size of the program and runs in nearly-linear time. Our reduction applies to high-accuracy approximation algorithms and exact algorithms. Given an approximate solution to the 2-commodity flow problem, we can extract a solution to the linear program in linear time with only a polynomial factor increase in the error. This implies that any algorithm that solves the 2-commodity flow problem can solve linear programs in essentially the same time. Given a directed graph with edge capacities and two source-sink pairs, the goal of the 2-commodity flow problem is to maximize the sum of the flows routed between the two source-sink pairs subject to edge capacities and flow conservation. A 2-commodity flow can be directly written as a linear program, and thus we establish a nearly-tight equivalence between these two classes of problems. Our proof follows the outline of Itai's polynomial-time reduction of a linear program to a 2-commodity flow problem (JACM'78). Itai's reduction shows that exactly solving 2-commodity flow and exactly solving linear programming are polynomial-time equivalent. We improve Itai's reduction to nearly preserve the problem representation size in each step. In addition, we establish an error bound for approximately solving each intermediate problem in the reduction, and show that the accumulated error is polynomially bounded. We remark that our reduction does not run in strongly polynomial time and that it is open whether 2-commodity flow and linear programming are equivalent in strongly polynomial time.

Reasoning over natural language is a long-standing goal for the research community. However, studies have shown that existing language models are inadequate in reasoning. To address the issue, we present POET, a new pre-training paradigm. Through pre-training language models with programs and their execution results, POET empowers language models to harvest the reasoning knowledge possessed in program executors via a data-driven approach. POET is conceptually simple and can be instantiated by different kinds of programs. In this paper, we show three empirically powerful instances, i.e., POET-Math, POET-Logic, and POET-SQL. Experimental results on six benchmarks demonstrate that POET can significantly boost model performance on natural language reasoning, such as numerical reasoning, logical reasoning, and multi-hop reasoning. Taking the DROP benchmark as a representative example, POET improves the F1 metric of BART from 69.2% to 80.6%. Furthermore, POET shines in giant language models, pushing the F1 metric of T5-11B to 87.6% and achieving a new state-of-the-art performance on DROP. POET opens a new gate on reasoning-enhancement pre-training and we hope our analysis would shed light on the future research of reasoning like program executors.

Both logic programming in general, and Prolog in particular, have a long and fascinating history, intermingled with that of many disciplines they inherited from or catalyzed. A large body of research has been gathered over the last 50 years, supported by many Prolog implementations. Many implementations are still actively developed, while new ones keep appearing. Often, the features added by different systems were motivated by the interdisciplinary needs of programmers and implementors, yielding systems that, while sharing the "classic" core language, and, in particular, the main aspects of the ISO-Prolog standard, also depart from each other in other aspects. This obviously poses challenges for code portability. The field has also inspired many related, but quite different languages that have created their own communities. This article aims at integrating and applying the main lessons learned in the process of evolution of Prolog. It is structured into three major parts. Firstly, we overview the evolution of Prolog systems and the community approximately up to the ISO standard, considering both the main historic developments and the motivations behind several Prolog implementations, as well as other logic programming languages influenced by Prolog. Then, we discuss the Prolog implementations that are most active after the appearance of the standard: their visions, goals, commonalities, and incompatibilities. Finally, we perform a SWOT analysis in order to better identify the potential of Prolog, and propose future directions along which Prolog might continue to add useful features, interfaces, libraries, and tools, while at the same time improving compatibility between implementations.

A generalization of L{\"u}roth's theorem expresses that every transcendence degree 1 subfield of the rational function field is a simple extension. In this note we show that a classical proof of this theorem also holds to prove this generalization.

While models for Visual Question Answering (VQA) have steadily improved over the years, interacting with one quickly reveals that these models lack consistency. For instance, if a model answers "red" to "What color is the balloon?", it might answer "no" if asked, "Is the balloon red?". These responses violate simple notions of entailment and raise questions about how effectively VQA models ground language. In this work, we introduce a dataset, ConVQA, and metrics that enable quantitative evaluation of consistency in VQA. For a given observable fact in an image (e.g. the balloon's color), we generate a set of logically consistent question-answer (QA) pairs (e.g. Is the balloon red?) and also collect a human-annotated set of common-sense based consistent QA pairs (e.g. Is the balloon the same color as tomato sauce?). Further, we propose a consistency-improving data augmentation module, a Consistency Teacher Module (CTM). CTM automatically generates entailed (or similar-intent) questions for a source QA pair and fine-tunes the VQA model if the VQA's answer to the entailed question is consistent with the source QA pair. We demonstrate that our CTM-based training improves the consistency of VQA models on the ConVQA datasets and is a strong baseline for further research.

We introduce GQA, a new dataset for real-world visual reasoning and compositional question answering, seeking to address key shortcomings of previous VQA datasets. We have developed a strong and robust question engine that leverages scene graph structures to create 22M diverse reasoning questions, all come with functional programs that represent their semantics. We use the programs to gain tight control over the answer distribution and present a new tunable smoothing technique to mitigate question biases. Accompanying the dataset is a suite of new metrics that evaluate essential qualities such as consistency, grounding and plausibility. An extensive analysis is performed for baselines as well as state-of-the-art models, providing fine-grained results for different question types and topologies. Whereas a blind LSTM obtains mere 42.1%, and strong VQA models achieve 54.1%, human performance tops at 89.3%, offering ample opportunity for new research to explore. We strongly hope GQA will provide an enabling resource for the next generation of models with enhanced robustness, improved consistency, and deeper semantic understanding for images and language.

Although neural network approaches achieve remarkable success on a variety of NLP tasks, many of them struggle to answer questions that require commonsense knowledge. We believe the main reason is the lack of commonsense connections between concepts. To remedy this, we provide a simple and effective method that leverages external commonsense knowledge base such as ConceptNet. We pre-train direct and indirect relational functions between concepts, and show that these pre-trained functions could be easily added to existing neural network models. Results show that incorporating commonsense-based function improves the state-of-the-art on two question answering tasks that require commonsense reasoning. Further analysis shows that our system discovers and leverages useful evidences from an external commonsense knowledge base, which is missing in existing neural network models and help derive the correct answer.

We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.