The distribution semantics for normal programs with function symbols

The distribution semantics integrates logic programming and probability theory using a possible worlds approach. Its intuitiveness and simplicity have made it the most widely used semantics for probabilistic logic programming, with successful applications in many domains. When the program has function symbols, the semantics was defined for special cases: either the program has to be definite or the queries must have a finite number of finite explanations. In this paper we show that it is possible to define the semantics for all programs. We also show that this definition coincides with that of Sato and Kameya on positive programs. Moreover, we highlight possible approaches for inference, both exact and approximate.     

Objective Bayesian probabilistic logic

This paper develops connections between objective Bayesian epistemology—which holds that the strengths of an agent's beliefs should be representable by probabilities, should be calibrated with evidence of empirical probability, and should otherwise be equivocal—and probabilistic logic. After introducing objective Bayesian epistemology over propositional languages, the formalism is extended to handle predicate languages. A rather general probabilistic logic is formulated and then given a natural semantics in terms of objective Bayesian epistemology. The machinery of objective Bayesian nets and objective credal nets is introduced and this machinery is applied to provide a calculus for probabilistic logic that meshes with the objective Bayesian semantics.     

Nonmonotonic probabilistic logics under variable-strength inheritance with overriding: Complexity,algorithms, and implementation

In previous work, I have introduced nonmonotonic probabilistic logics under variable-strength inheritance with overriding. They are formalisms for probabilistic reasoning from sets of strict logical, default logical, and default probabilistic sentences, which are parameterized through a value λ ∈ [0, 1] that describes the strength of the inheritance of default probabilistic knowledge. In this paper, I continue this line of research. I give a precise picture of the complexity of deciding consistency of strength λ and of computing tight consequences of strength λ. Furthermore, I present algorithms for these tasks, which are based on reductions to the standard problems of deciding satisfiability and of computing tight logical consequences in model–theoretic probabilistic logic. Finally, I describe the system nmproblog, which includes a prototype implementation of these algorithms.     

An anytime deduction algorithm for the probabilistic logic and entailment problems

We study two basic problems of probabilistic reasoning: the probabilistic logic and the probabilistic entailment problems. The first one can be defined as follows. Given a set of logical sentences and probabilities that these sentences are true, the aim is to determine whether these probabilities are consistent or not. Given a consistent set of logical sentences and probabilities, the probabilistic entailment problem consists in determining the range of the possible values of the probability associated with additional sentences while maintaining a consistent set of sentences and probabilities.This paper proposes a general approach based on an anytime deduction method that allows the follow-up of the reasoning when checking consistency for the probabilistic logic problem or when determining the probability intervals for the probabilistic entailment problem. Considering a series of subsets of sentences and probabilities, the approach proceeds by computing increasingly narrow probability intervals that either show a contradiction or that contain the tightest entailed probability interval. Computational experience have been conducted to compare the proposed anytime deduction method, called ad-psat with an exact one, psatcol, using column generation techniques, both with respect to the range of the probability intervals and the computing times.     

Probabilistic stit logic and its decomposition

We define an extension of stit logic that encompasses subjective probabilities representing beliefs about simultaneous choice exertion of other agents. This semantics enables us to express that an agent sees to it that a condition obtains under a minimal chance of success. We first define the fragment of XSTIT where choice exertion is not collective. Then we add lower bounds for the probability of effects to the stit syntax, and define the semantics of the newly formed stit operator in terms of subjective probabilities concerning choice exertion of other agents. We show how the resulting probabilistic stit logic faithfully generalizes the non-probabilistic XSTIT fragment. In a second step we analyze the defined probabilistic stit logic by decomposing it into an XSTIT fragment and a purely epistemic fragment. The resulting epistemic logic for grades of believes is a weak modal logic with a neighborhood semantics combining probabilistic and modal logic theory.     

An epistemic model of an agent who does not reflect on reasoning processes

This paper introduces an epistemic model of a boundedly rational agent under the two assumptions that (i) the agent’s reasoning process is in accordance with the model but (ii) the agent does not reflect on these reasoning processes. For such a concept of bounded rationality a semantic interpretation by the possible world semantics of the Kripke (1963) type is no longer available because the definition of knowledge in these possible world semantics implies that the agent knows all valid statements of the model. The key to my alternative semantic approach is the extension of the method of truth tables, first introduced for the propositional logic by Wittgenstein (1922), to an epistemic logic so that I can determine the truth value of epistemic statements for all relevant truth conditions. In my syntactic approach I define an epistemic logic–consisting of the classical calculus of propositional logic plus two knowledge axioms–that does not include the inference rule of necessitation, which claims that an agent knows all theorems of the logic. As my main formal result I derive a determination theorem linking my semantic with my syntactic approach. The difference between my approach and existing knowledge models is illustrated in a game-theoretic application concerning the epistemic justification of iterative solution concepts.     

A representation theorem and applications to measure selection and noninformative priors

We introduce a set of transformations on the set of all probability distributions over a finite state space, and show that these transformations are the only ones that preserve certain elementary probabilistic relationships. This result provides a new perspective on a variety of probabilistic inference problems in which invariance considerations play a role. Two particular applications we consider in this paper are the development of an equivariance-based approach to the problem of measure selection, and a new justification for Haldane’s prior as the distribution that encodes prior ignorance about the parameter of a multinomial distribution.     

Tractability frontiers in probabilistic team semantics and existential second-order logic over the reals

Probabilistic team semantics is a framework for logical analysis of probabilistic dependencies. Our focus is on the axiomatizability, complexity, and expressivity of probabilistic inclusion logic and its extensions. We identify a natural fragment of existential second-order logic with additive real arithmetic that captures exactly the expressivity of probabilistic inclusion logic. We furthermore relate these formalisms to linear programming, and doing so obtain PTIME data complexity for the logics. Moreover, on finite structures, we show that the full existential second-order logic with additive real arithmetic can only express NP properties. Lastly, we present a sound and complete axiomatization for probabilistic inclusion logic at the atomic level.     

Unifying hidden-variable problems from quantum mechanics by logics of dependence and independence

We study hidden-variable models from quantum mechanics and their abstractions in purely probabilistic and relational frameworks by means of logics of dependence and independence, which are based on team semantics. We show that common desirable properties of hidden-variable models can be defined in an elegant and concise way in dependence and independence logic. The relationship between different properties and their simultaneous realisability can thus be formulated and proven on a purely logical level, as problems of entailment and satisfiability of logical formulae. Connections between probabilistic and relational entailment in dependence and independence logic allow us to simplify proofs. In many cases, we can establish results on both probabilistic and relational hidden-variable models by a single proof, because one case implies the other, depending on purely syntactic criteria. We also discuss the ‘no-go’ theorems by Bell and Kochen-Specker and provide a purely logical variant of the latter, introducing non-contextual choice as a team-semantical property.     

A Shared Framework for Consequence Operations and Abstract Model Theory

In this paper we develop an abstract theory of adequacy. In the same way as the theory of consequence operations is a general theory of logic, this theory of adequacy is a general theory of the interactions and connections between consequence operations and its sound and complete semantics. Addition of axioms for the connectives of propositional logic to the basic axioms of consequence operations yields a unifying framework for different systems of classical propositional logic. We present an abstract model-theoretical semantics based on model mappings and theory mappings. Between the classes of models and theories, i.e., the set of sentences verified by a model, it obtains a connection that is well-known within algebra as Galois correspondence. Many basic semantical properties can be derived from this observation. A sentence A is a semantical consequence of T if every model of T is also a model of A. A model mapping is adequate for a consequence operation if its semantical inference operation is identical with the consequence operation. We study how properties of an adequate model mapping reflect the properties of the consequence operation and vice versa. In particular, we show how every concept of the theory of consequence operations can be formulated semantically.     

Dual‐Context Sequent Calculus and Strict Implication

We introduce a dual‐context style sequent calculus which is complete with respectto Kripke semantics where implication is interpreted as strict implication in the modal logic K. The cut‐elimination theorem for this calculus is proved by a variant of Gentzen's method.     

Probabilistic abductive logic programming using Dirichlet priors

Probabilistic programming is an area of research that aims to develop general inference algorithms for probabilistic models expressed as probabilistic programs whose execution corresponds to inferring the parameters of those models. In this paper, we introduce a probabilistic programming language (PPL) based on abductive logic programming for performing inference in probabilistic models involving categorical distributions with Dirichlet priors. We encode these models as abductive logic programs enriched with probabilistic definitions and queries, and show how to execute and compile them to boolean formulas. Using the latter, we perform generalized inference using one of two proposed Markov Chain Monte Carlo (MCMC) sampling algorithms: an adaptation of uncollapsed Gibbs sampling from related work and a novel collapsed Gibbs sampling (CGS). We show that CGS converges faster than the uncollapsed version on a latent Dirichlet allocation (LDA) task using synthetic data. On similar data, we compare our PPL with LDA-specific algorithms and other PPLs. We find that all methods, except one, perform similarly and that the more expressive the PPL, the slower it is. We illustrate applications of our PPL on real data in two variants of LDA models (Seed and Cluster LDA), and in the repeated insertion model (RIM). In the latter, our PPL yields similar conclusions to inference with EM for Mallows models.     

On elementary logics for quantitative dependencies

We define and study logics in the framework of probabilistic team semantics and over metafinite structures. Our work is paralleled by the recent development of novel axiomatizable and tractable logics in team semantics that are closed under the Boolean negation. Our logics employ new probabilistic atoms that resemble so-called extended atoms from the team semantics literature. We also define counterparts of our logics over metafinite structures and show that all of our logics can be translated into functional fixed point logic implying a polynomial time upper bound for data complexity with respect to BSS-computations.     

A semantic hierarchy for intuitionistic logic

Brouwer’s views on the foundations of mathematics have inspired the study of intuitionistic logic, including the study of the intuitionistic propositional calculus and its extensions. The theory of these systems has become an independent branch of logic with connections to lattice theory, topology, modal logic, and other areas. This paper aims to present a modern account of semantics for intuitionistic propositional systems. The guiding idea is that of a hierarchy of semantics, organized by increasing generality: from the least general Kripke semantics on through Beth semantics, topological semantics, Dragalin semantics, and finally to the most general algebraic semantics. While the Kripke, topological, and algebraic semantics have been extensively studied, the Beth and Dragalin semantics have received less attention. We bring Beth and Dragalin semantics to the fore, relating them to the concept of a nucleus from pointfree topology, which provides a unifying perspective on the semantic hierarchy.     

Divergences without probability vectors and their applications

In general, divergences and measures of information are defined for probability vectors. However, in some cases, divergences are ‘informally’ used to measure the discrepancy between vectors, which are not necessarily probability vectors. In this paper we examine whether divergences with nonprobability vectors in their arguments share the properties of probabilistic or information theoretic divergences. The results indicate that divergences with nonprobability vectors share, under some conditions, some of the properties of probabilistic or information theoretic divergences and therefore can be considered and used as information measures. We then use these divergences in the problem of actuarial graduation of mortality rates. Copyright © 2009 John Wiley & Sons, Ltd.     

A granularity-based framework of deduction, induction, and abduction

In this paper, we propose a granularity-based framework of deduction, induction, and abduction using variable precision rough set models proposed by Ziarko and measure-based semantics for modal logic proposed by Murai et al. The proposed framework is based on α-level fuzzy measure models on the basis of background knowledge, as described in the paper. In the proposed framework, deduction, induction, and abduction are characterized as reasoning processes based on typical situations about the facts and rules used in these processes. Using variable precision rough set models, we consider β-lower approximation of truth sets of nonmodal sentences as typical situations of the given facts and rules, instead of the truth sets of the sentences as correct representations of the facts and rules. Moreover, we represent deduction, induction, and abduction as relationships between typical situations.     

Independence logic and abstract independence relations

We continue the work on the relations between independence logic and the model‐theoretic analysis of independence, generalizing the results of 15 to the framework of abstract independence relations for an arbitrary AEC. We give a model‐theoretic interpretation of the independence atom and characterize under which conditions we can prove a completeness result with respect to the deductive system that axiomatizes independence in team semantics and statistics.     

Formative Use of Intuitive Analysis of Variance

Students’ informal inferential reasoning (IIR) is often inconsistent with the normative logic underlying formal statistical methods such as Analysis of Variance (ANOVA), even after instruction. In two experiments reported here, student's IIR was assessed using an intuitive ANOVA task at the beginning and end of a statistics course. In both experiments, students were provided feedback regarding the normative logic underlying ANOVA and how their reasoning compared with it. Additionally, students in Experiment 2 were given an assignment in which they analyzed and interpreted other students’ performance on the intuitive ANOVA task. Results indicate that the feedback combined with the assignment (which required active explanation of both normative and non-normative reasoning applied to the task) led to more normative inferential reasoning at the end of the course, whereas providing feedback alone did not. Implications are discussed for using the intuitive ANOVA task as a formative classroom tool to help students improve their conceptual understanding of ANOVA.