The logic of justified belief,explicit knowledge,and conclusive evidence

We present a complete, decidable logic for reasoning about a notion of completely trustworthy (“conclusive”) evidence and its relations to justifiable (implicit) belief and knowledge, as well as to their explicit justifications. This logic makes use of a number of evidence-related notions such as availability, admissibility, and “goodness” of a piece of evidence, and is based on an innovative modification of the Fitting semantics for Artemov?s Justification Logic designed to preempt Gettier-type counterexamples. We combine this with ideas from belief revision and awareness logics to provide an account for explicitly justified (defeasible) knowledge based on conclusive evidence that addresses the problem of (logical) omniscience.     

On the Modal Logic of Jeffrey Conditionalization

We continue the investigations initiated in the recent papers (Brown et al. in The modal logic of Bayesian belief revision, 2017; Gyenis in Standard Bayes logic is not finitely axiomatizable, 2018) where Bayes logics have been introduced to study the general laws of Bayesian belief revision. In Bayesian belief revision a Bayesian agent revises (updates) his prior belief by conditionalizing the prior on some evidence using the Bayes rule. In this paper we take the more general Jeffrey formula as a conditioning device and study the corresponding modal logics that we call Jeffrey logics, focusing mainly on the countable case. The containment relations among these modal logics are determined and it is shown that the logic of Bayes and Jeffrey updating are very close. It is shown that the modal logic of belief revision determined by probabilities on a finite or countably infinite set of elementary propositions is not finitely axiomatizable. The significance of this result is that it clearly indicates that axiomatic approaches to belief revision might be severely limited.     

Distance-based paraconsistent logics

We introduce a general framework that is based on distance semantics and investigate the main properties of the entailment relations that it induces. It is shown that such entailments are particularly useful for non-monotonic reasoning and for drawing rational conclusions from incomplete and inconsistent information. Some applications are considered in the context of belief revision, information integration systems, and consistent query answering for possibly inconsistent databases.     

What’s so Good about a Wise and Knowledgeable Public?

Philosophers have been concerned for some time with the epistemic caliber of the general public, qua the body that is, ultimately, tasked with political decision-making in democratic societies. Unfortunately, the empirical data paint a pretty dismal picture here, indicating that the public tends to be largely ignorant on the issues relevant to governance. To make matters worse, empirical research on how ignorance tends to breed overconfidence suggests that the public will not only lack knowledge on the relevant issues, but also wisdom, in the Socratic sense of an awareness of your ignorance. While increasing the knowledge and wisdom of the public might be thought an obvious remedy, there is, as far as sound political decision-making and action are concerned, nothing particularly valuable about knowledge or wisdom per se. In fact, it might just be that what the public needs is nothing but the most basic epistemic good: true belief.     

A formal system for fuzzy reasoning

A formal system for fuzzy reasoning is described which is capable of dealing rationally with evidence which may be inconsistent and/or involve degrees of belief. The basic idea is that the meaning of each formal sentence should be given by a certain commitment or bet associated with it. Each item of evidence is first expressed in the form of such a (hypothetical) bet, which is then written as a formal sentence in a language related to ?ukasiewicz logic. The sentences may be weighted to express the relative reliability of the various informants. A sentence is considered to “follow” from the evidence if the bet it represents can be offered by a speaker without fear of loss, on the assumption that the bets representing various items of evidence have been offered to him. A detailed account, illustrated by concrete examples, is given of the procedures by which an arbitrary sentence in common language can be translated into a formal sentence. The treatment of inconsistency, degrees of belief, and weights is illustrated by a practical example which is solved in full. It is shown that in most practical cases the computations involved in the process of formal reasoning reduce to a problem in linear programming. In the last section the relation between this system and the procedures advocated by Zadeh is examined. It is shown that, subject to certain modifications in formulas, there is general agreement in the region of overlap.     

Dempster's rule of combination

The theory of belief functions is a generalization of probability theory; a belief function is a set function more general than a probability measure but whose values can still be interpreted as degrees of belief. Dempster's rule of combination is a rule for combining two or more belief functions; when the belief functions combined are based on distinct or “independent” sources of evidence, the rule corresponds intuitively to the pooling of evidence. As a special case, the rule yields a rule of conditioning which generalizes the usual rule for conditioning probability measures. The rule of combination was studied extensively, but only in the case of finite sets of possibilities, in the author's monograph A Mathematical Theory of Evidence. The present paper describes the rule for general, possibly infinite, sets of possibilities. We show that the rule preserves the regularity conditions of continuity and condensability, and we investigate the two distinct generalizations of probabilistic independence which the rule suggests.     

Revising beliefs on the basis of evidence

Approaches to belief revision most commonly deal with categorical information: an agent has a set of beliefs and the goal is to consistently incorporate a new item of information given by a formula. However, most information about the real world is not categorical. In revision, one may circumvent this fact by assuming that, in some fashion or other, an agent has elected to accept a formula ?, and the task of revision is to consistently incorporate ? into its belief corpus. Nonetheless, it is worth asking whether probabilistic information and noncategorical beliefs may be reconciled with, or even inform, approaches to revision. In this paper, one such account is presented. An agent receives uncertain information as input, and its probabilities on (a finite set of) possible worlds are updated via Bayesian conditioning. A set of formulas among the noncategorical beliefs is identified as the agent’s categorical belief set. The effect of this updating on the belief set is examined with respect to its appropriateness as a revision operator. We show that few of the classical AGM belief revision postulates are satisfied by this approach. Most significantly, though not surprisingly, the success postulate is not guaranteed to hold. However it does hold after a sufficient number of iterations. As well, it proves to be the case that in revising by a formula consistent with the agent’s beliefs, revision does not correspond to expansion. Postulates for iterated revision also examined, and it proves to be the case that most such postulates also do not hold. On the other hand, limiting cases of the presented approach correspond to specific approaches to revision that have appeared in the literature.     

Inference in directed evidential networks based on the transferable belief model

Inference algorithms in directed evidential networks (DEVN) obtain their efficiency by making use of the represented independencies between variables in the model. This can be done using the disjunctive rule of combination (DRC) and the generalized Bayesian theorem (GBT), both proposed by Smets [Ph. Smets, Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem, International Journal of Approximate Reasoning 9 (1993) 1–35]. These rules make possible the use of conditional belief functions for reasoning in directed evidential networks, avoiding the computations of joint belief function on the product space. In this paper, new algorithms based on these two rules are proposed for the propagation of belief functions in singly and multiply directed evidential networks.     

Self-tuning of fuzzy belief rule bases for engineering system safety analysis

A framework for modelling the safety of an engineering system using a fuzzy rule-based evidential reasoning (FURBER) approach has been recently proposed, where a fuzzy rule-base designed on the basis of a belief structure (called a belief rule base) forms a basis in the inference mechanism of FURBER. However, it is difficult to accurately determine the parameters of a fuzzy belief rule base (FBRB) entirely subjectively, in particular for complex systems. As such, there is a need to develop a supporting mechanism that can be used to train in a locally optimal way a FBRB initially built using expert knowledge. In this paper, the methods for self-tuning a FBRB for engineering system safety analysis are investigated on the basis of a previous study. The method consists of a number of single and multiple objective nonlinear optimization models. The above framework is applied to model the system safety of a marine engineering system and the case study is used to demonstrate how the methods can be implemented.     

一个不同时刻加工成本有差异的单机排序问题

考虑一个单机排序问题:一批工件在零时刻到达可加工,加工时不可中断,在某个给定时间区间外的加工工时将招致额外的加工成本;当时间区间为给定参数时,要求确定一个最优加工序,当时间区间为决策变量时,要求找到一个最优序及最优区间位置, 由此来最小化总额外加工成本．文中对各种区间外单位加工工时之额外成本的情况给出了多项式算法, NP-hardness的证明及伪多项式时间算法．     

A theory,implementation and applications of human-like understanding

Operation logic is a formal logic with well-defined formulas as semantic language clauses and with modus ponens rules as a method of reasoning. Operation logic can be implemented on any database management system (as the so-called OLS) having a universal general knowledge database and enabling understanding of data stored in the database. Semantic language clauses have necessary and sufficient properties for being able to describe any process in the world. Semantic language is the deepest level of any natural language, the level of data storing, understanding and reasoning. OLS can be a tool for studying implementation possibilities of human-like consciousness, for building artificial experts and artificial encyclopedias and for constructing semantic mathematical theories of anthropoecosystems (which is such an exact theory that qualitative information can be used with meaning completely defined by the user). In the paper the theory (and complete information enabling implementation) is presented for human-like understanding, topic-focus division of clauses, for human-like problem solving (program synthesis and verification) and for semantic mathematical analyses. Many examples are presented.     

An evidential reasoning based consensus model for multiple attribute group decision analysis problems with interval-valued group consensus requirements

With the aim of modeling multiple attribute group decision analysis problems with group consensus (GC) requirements, a GC based evidential reasoning approach and further an attribute weight based feedback model are sequentially developed based on an evidential reasoning (ER) approach. In real situations, however, giving precise (crisp) assessments for alternatives is often too restrictive and difficult for experts, due to incompleteness or lack of information. Experts may also find it difficult to give appropriate assessments on specific attributes, due to limitation or lack of knowledge, experience and provided data about the problem domain. In this paper, an ER based consensus model (ERCM) is proposed to deal with these situations, in which experts’ assessments are interval-valued rather than precise. Correspondingly, predefined interval-valued GC (IGC) requirements need to be reached after group analysis and discussion within specified times. Also, the process of reaching IGC is accelerated by a feedback mechanism including identification rules at three levels, consisting of the attribute, alternative and global levels, and a suggestion rule. Particularly, recommendations on assessments in the suggestion rule are constructed based on recommendations on their lower and upper bounds detected by the identification rule at a specific level. A preferentially developed industry selection problem is solved by the ERCM to demonstrate its detailed implementation process, validity, and applicability.     

An interpretability improvement for fuzzy rule bases obtained by the iterative rule learning approach

Interpretability is one of the key concepts in many of the applications using the fuzzy rule-based approach. It is well known that there are many different criteria around this concept, the complexity being one of them. In this paper, we focus our efforts in reducing the complexity of the fuzzy rule sets. One of the most interesting approaches for learning fuzzy rules is the iterative rule learning approach. It is mainly characterized by obtaining rules covering few examples in final stages, being in most cases useless to represent the knowledge. This behavior is due to the specificity of the extracted rules, which eventually creates more complex set of rules. Thus, we propose a modified version of the iterative rule learning algorithm in order to extract simple rules relaxing this natural trend. The main idea is to change the rule extraction process to be able to obtain more general rules, using pruned searching spaces together with a knowledge simplification scheme able to replace learned rules. The experimental results prove that this purpose is achieved. The new proposal reduces the complexity at both, the rule and rule base levels, maintaining the accuracy regarding to previous versions of the algorithm.     

A simple logic for reasoning about incomplete knowledge

The semantics of modal logics for reasoning about belief or knowledge is often described in terms of accessibility relations, which is too expressive to account for mere epistemic states of an agent. This paper proposes a simple logic whose atoms express epistemic attitudes about formulae expressed in another basic propositional language, and that allows for conjunctions, disjunctions and negations of belief or knowledge statements. It allows an agent to reason about what is known about the beliefs held by another agent. This simple epistemic logic borrows its syntax and axioms from the modal logic KD. It uses only a fragment of the S5 language, which makes it a two-tiered propositional logic rather than as an extension thereof. Its semantics is given in terms of epistemic states understood as subsets of mutually exclusive propositional interpretations. Our approach offers a logical grounding to uncertainty theories like possibility theory and belief functions. In fact, we define the most basic logic for possibility theory as shown by a completeness proof that does not rely on accessibility relations.     

Constructive agents

Brouwer’s ideas of construction, proof, and inquiry in mathematics are more widely applicable. On a well-known philosophical view, intuitionistic logic is a general account of meaning and reasoning for natural language and epistemology. In this brief discussion piece, I go one step further, and discuss how intuitionistic semantics fits with information update and belief revision in agency. In the process, I define a number of new logical systems that give rise to several open problems.     

A branch-and-bound algorithm of the single machine schedule with sequence-dependent setup times for minimizing maximum tardiness

This paper addresses the NP-hard problem of scheduling N jobs on a single machine with due dates, sequence-dependent setup times and no preemption where the objective is to minimize the maximum tardiness. An algorithm based on branch-and-bound permutation schemes is developed including the implementation of lower and upper bounding procedures, and three dominance rules. Computational experiments demonstrate the effectiveness of the algorithm. In the experiments, the impacts of control parameters to generate test instances on algorithm performance (CPU times) are studied by statistics methods.     

An analysis of Klimov's problem with parallel servers

We introduce a family of undiscounted branching bandits on parallel servers under controls which can impose priorities between customer classes. This family can be used to model a wide range of multi-class queueing scheduling problems, with the capacity to incorporate problem features such as machine breakdowns, complex patterns of preemption/non-preemption and semi-Markov extensions. An index policy (which we call Klimov's rule) is developed which is optimal in the particular case of a single server. An expression for its cost suboptimality is given for parallel servers. Under additional conditions on the nature of the stochastic evolution of the systems concerned, the policy is shown to be asymptotically optimal in a heavy traffic limit. These general results are utilised to develop an analysis of the index policy for a parallel server version of Klimov's classical M/GI/1 system with Bernoulli feedback. This work was supported by the Engineering and Physical Research Council through the award of grant GR/M09308. The author would also like to express his appreciation to Professor I. C. Paschalidis for helpful discussions on Klimov's problem and to Professors J. Niño-Mora and G. Weiss for many discussions and much encouragement     

Constructions for Permutation Codes in Powerline Communications

A permutation array (or code) of length n and distance d is a set Γ of permutations from some fixed set of n symbols such that the Hamming distance between each distinct x, y ∈ Γ is at least d. One motivation for coding with permutations is powerline communication. After summarizing known results, it is shown here that certain families of polynomials over finite fields give rise to permutation arrays. Additionally, several new computational constructions are given, often making use of automorphism groups. Finally, a recursive construction for permutation arrays is presented, using and motivating the more general notion of codes with constant weight composition.     

Belief functions combination without the assumption of independence of the information sources

This paper considers the problem of combining belief functions obtained from not necessarily independent sources of information. It introduces two combination rules for the situation in which no assumption is made about the dependence of the information sources. These two rules are based on cautious combinations of plausibility and commonality functions, respectively. The paper studies the properties of these rules and their connection with Dempster’s rules of conditioning and combination and the minimum rule of possibility theory.