Chapter 6. Tractable Boolean and Arithmetic Circuits

Tractable Boolean and arithmetic circuits have been studied extensively in AI for over two decades now. These circuits were initially proposed as “compiled objects,” meant to facilitate logical and probabilistic reasoning, as they permit various types of inference to be performed in linear time and a feed-forward fashion like neural networks. In more recent years, the role of tractable circuits has significantly expanded as they became a computational and semantical backbone for some approaches that aim to integrate knowledge, reasoning and learning. In this chapter, we review the foundations of tractable circuits and some associated milestones, while focusing on their core properties and techniques that make them particularly useful for the broad aims of neuro-symbolic AI.

Reasoning under uncertainty within the context of probability education: A case study of preservice mathematics teachers

Interpreting phenomena under uncertainty stands as a substantial cognitive activity in our daily life. Furthermore, in probability education research, there is a need for developing a unified model that involves several probabilistic conceptions. From this aspect, a central inquiry has been raised through this study: how do preservice mathematics teachers (PSMTs) reason under uncertainty? A multiple case study design was operated in which a purposive sample of PSMTs was selected to justify their reasoning in two probabilistic contexts while their responses were coded by NVivo, and corresponding categories were developed. As a result, PSMTs’ probabilistic reasoning was classified into mathematical (M), subjective (S), and outcome-oriented (O). Besides, several biases emerged along with these modes of reasoning. While M thinkers shared equiprobability and insensitivity to prior probability, the prediction bias and the belief of Allah’s willingness were yielded among S thinkers. Also, the causal conception spread among O thinkers.

Methods of Learning the Structure of the Bayesian Network

Sometimes in practice it is necessary to calculate the probability of an uncertain cause, taking into account some observed evidence. For example, we would like to know the probability of a particular disease when we observe the patient’s symptoms. Such problems are often complex with many interrelated variables. There may be many symptoms and even more potential causes. In practice, it is usually possible to obtain only the inverse conditional probability, the probability of evidence giving the cause, the probability of observing the symptoms if the patient has the disease.Intelligent systems must think about their environment. For example, a robot needs to know about the possible outcomes of its actions, and the system of medical experts needs to know what causes what consequences. Intelligent systems began to use probabilistic methods to deal with the uncertainty of the real world. Instead of building a special system of probabilistic reasoning for each new program, we would like a common framework that would allow probabilistic reasoning in any new program without restoring everything from scratch. This justifies the relevance of the developed genetic algorithm. Bayesian networks, which first appeared in the work of Judas Pearl and his colleagues in the late 1980s, offer just such an independent basis for plausible reasoning.This article presents the genetic algorithm for learning the structure of the Bayesian network that searches the space of the graph, uses mutation and crossover operators. The algorithm can be used as a quick way to learn the structure of a Bayesian network with as few constraints as possible.learn the structure of a Bayesian network with as few constraints as possible.

Phenomenal Explanationism vs Conservatism

This chapter situates PE within the context of the broader debate between Epistemic Liberalism (which holds, roughly, that it is reasonable to grant that things are the way they appear to be unless there is reason for doubting it) and Epistemic Conservatism (the view that, roughly, it is not reasonable to grant that things are the way they appear to be unless there is independent reason to think that the appearances are reliable). PE is squarely within the Liberal camp. Therefore, after explaining some of the primary elements of Liberal/Conservative debate in epistemology, two of the primary challenges faced by Liberal views like PE are examined. The first is, again, the problem of bootstrapping, which any theory that allows for immediate justification seems to run into. The second is White’s Bayesian objection to PC (introduced in Chapter 1), according to which Liberalism, and so PE, is flawed because it is incompatible with probabilistic reasoning. It is shown that PE is not troubled by these challenges. The upshot of the chapter is that Liberalism, when exemplified in PE, is victorious over Conservatism.

Causal inference via string diagram surgery

Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endo-functor which performs ‘string diagram surgery’ within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on two well-known toy examples: one where we predict the causal effect of smoking on cancer in the presence of a confounding common cause and where we show that this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature; the other one is an illustration of counterfactual reasoning where the same interventional techniques are used, but now in a ‘twinned’ set-up, with two version of the world – one factual and one counterfactual – joined together via exogenous variables that capture the uncertainties at hand.

Predicting Physics Achievement of 11th - grade Students through Critical and Logical Thinking Skills

The purpose of this study was to measure the extent to which critical thinking skills and logical thinking skills predicted the achievement in physics of 11th - grade students in Gaza. To this end, (215) students participated in the study. California Critical Thinking Skills Test (CCTST) and Test of Logical Thinking (TOLT), were used as data collection tools in the study. Data were analyzed by multiple stepwise regression analysis. Results indicated that the students’ scores of probabilistic reasoning and proportional reasoning (as logical thinking skills), and inference (as critical thinking skill) were significant predictors of physics achievement scores, explaining (15.7%) of the variance of physics achievement scores. In addition, findings indicated that the best predictor was probabilistic reasoning, explaining (9.2 %) of the variance. The study recommended that a physics curriculum should involve thinking skills, which predict students’ achievement in physics.

The Model Counting Competition 2020

Many computational problems in modern society account to probabilistic reasoning, statistics, and combinatorics. A variety of these real-world questions can be solved by representing the question in (Boolean) formulas and associating the number of models of the formula directly with the answer to the question. Since there has been an increasing interest in practical problem solving for model counting over the past years, the Model Counting Competition was conceived in fall 2019. The competition aims to foster applications, identify new challenging benchmarks, and promote new solvers and improve established solvers for the model counting problem and versions thereof. We hope that the results can be a good indicator of the current feasibility of model counting and spark many new applications. In this article, we report on details of the Model Counting Competition 2020, about carrying out the competition, and the results. The competition encompassed three versions of the model counting problem, which we evaluated in separate tracks. The first track featured the model counting problem, which asks for the number of models of a given Boolean formula. On the second track, we challenged developers to submit programs that solve the weighted model counting problem. The last track was dedicated to projected model counting. In total, we received a surprising number of nine solvers in 34 versions from eight groups.

Base rate neglect and conservatism in probabilistic reasoning: Insights from eliciting full distributions

Bayesian statistics offers a normative description for how a person should combine their original beliefs (i.e., their priors) in light of new evidence (i.e., the likelihood). Previous research suggests that people tend to under-weight both their prior (base rate neglect) and the likelihood (conservatism), although this varies by individual and situation. Yet this work generally elicits people's knowledge as single point estimates (e.g., x has 5% probability of occurring) rather than as a full distribution. Here we demonstrate the utility of eliciting and fitting full distributions when studying these questions. Across three experiments, we found substantial variation in the extent to which people showed base rate neglect and conservatism, which our method allowed us to measure for the first time simultaneously at the level of the individual. We found that while most people tended to disregard the base rate, they did so less when the prior was made explicit. Although many individuals were conservative, there was no apparent systematic relationship between base rate neglect and conservatism within individuals. We suggest that this method shows great potential for studying human probabilistic reasoning.

Probabilistic Thinking: A Principle-Based Concept Analysis in the Context of Uncertainty in Chronic Illness

Background and PurposeThe Reconceptualized Uncertainty in Illness Theory (RUIT) includes the concept of “probabilistic thinking” intending to explain the positive reappraisal of uncertainty in chronic illness. However, the description of the concept is vague, thereby limiting the understanding of the theory. Thus, the aim was to develop a theoretical definition of probabilistic thinking in order to increase the explanatory value of RUIT.MethodsWe conducted a principle-based concept analysis by means of a conceptually driven literature search. Methods consisted of database, dictionary, lexicon, and free web searching as well as citation tracking. We analyzed the concept in terms of (a) epistemology, (b) pragmatics, (c) logic, and (d) linguistics.ResultsThe final data set included 27 publications, 14 of them from nursing. (a) Probabilistic thinking is a coping strategy to handle uncertainty. It involves a focus on either possibilities (in nursing) or probabilities (in other disciplines). (b) There is a lack of operationalization in nursing, though three measurements focusing the handling of probabilities are offered in psychology. (c) Nursing authors interpreting probabilistic thinking as accepted uncertainty lacked logical appropriateness, since probability negotiates uncertainty. (d) Probabilistic thinking is used synonymously with positive thinking and probabilistic reasoning.Implications for PracticeNurses working with chronically ill patients should consider the findings for the application of RUIT. They should recognize whether uncertainty is perceived as a danger and encourage probabilistic thinking. Efforts are necessary to achieve a common language between nursing and other disciplines in order to avoid misunderstandings in clinical practice and research.