Nonmonotonicity in the Framework of Parametric Logic

This paper is concerned with the algorithmic learning where the learner is allowed to make finite but bounded number of mind changes. Briefly, in our learning paradigm, a learner is given examples from a recursive function, which the learner attempts to learn by producing programs to compute that function. We say that a team is successful if at least one member of the team learns the target function. The problem, given two teams with bounded number of learners and mind changes whether one team can provably learn more than the other team has been open for the last fifteen years. This paper makes significant progress toward a complete solution of this problem. In the case of error-free learning, this paper solves the open problem. Finally, in the case of EX learning our result shows that there is no team with a≥0 mind changes whose learning power is exactly equal to a single learner with bounded b(≠ a) number of mind changes. In the case of PEX learning we have a positive answer.

We consider the inductive inference model of Gold [15]. Suppose we are given a set of functions that are learnable with certain number of mind changes and errors. What can we consistently predict about those functions if we are allowed fewer mind changes or errors? In [20] we relaxed the notion of exact learning by considering some higher level properties of the input-output behavior of a given function. in this context, a learner produces a program that describes the property of a given function. Can we predict generic properties such as threshold or modality if we allow fewer number of mind changes or errors? These questions were completely answered in [20] when the learner is restricted to a single IIM. In this paper we allow a team of IIMs to collaborate in the learning process. The learning is considered to be successful if any one of the team member succeeds. A motivation for this extension is to understand and characterize properties that are learnable for a given set of functions in a team environment.

Many text mining approaches adopt bag-of-words or n-grams models to represent documents. Looking beyond just the words, i.e., the explicit surface forms, in a document can improve a computer's understanding of text. Being aware of this, researchers have proposed concept-based models that rely on a human-curated knowledge base to incorporate other related concepts in the document representation. But these methods are not desirable when applied to vertical domains (e.g., literature, enterprise, etc.) due to low coverage of in-domain concepts in the general knowledge base and interference from out-of-domain concepts. In this paper, we propose a data-driven model named Latent Keyphrase Inference (LAKI) that represents documents with a vector of closely related domain keyphrases instead of single words or existing concepts in the knowledge base. We show that given a corpus of in-domain documents, topical content units can be learned for each domain keyphrase, which enables a computer to do smart inference to discover latent document keyphrases, going beyond just explicit mentions. Compared with the state-of-art document representation approaches, LAKI fills the gap between bag-of-words and concept-based models by using domain keyphrases as the basic representation unit. It removes dependency on a knowledge base while providing, with keyphrases, readily interpretable representations. When evaluated against 8 other methods on two text mining tasks over two corpora, LAKI outperformed all.

We study the learnability of enumerable families \(\mathcal{L}\) of uniformly recursive languages in dependence on the number of allowed mind changes, i.e., with respect to a well-studied measure of efficiency. We distinguish between exact learnability (\(\mathcal{L}\) has to be inferred w.r.t. \(\mathcal{L}\)) and class preserving learning (\(\mathcal{L}\) has to be inferred w.r.t. some suitable chosen enumeration of all the languages from \(\mathcal{L}\)) as well as between learning from positive and from both, positive and negative data.The measure of efficiency is applied to prove the superiority of class preserving learning algorithms over exact learning. We considerably improve results obtained previously and establish two infinite hierarchies. Furthermore, we separate exact and class preserving learning from positive data that avoids overgeneralization. Finally, language learning with a bounded number of mind changes is completely characterized in terms of recursively generable finite sets. These characterizations offer a new method to handle overgeneralizations and resolve an open question of Mukouchi (1992).KeywordsLanguage LearningRecursive FunctionInductive InferencePositive DataNegative DataThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This talk will be about different kinds of understanding, including especially that provided by deductive and inductive inference. Examples will be drawn from Galileo, Kepler, and Fermat. Applications will be given to the physicist’s problem of inferring the laws of nature, and to the restricted case of inferring sequences in Sloane’s Encyclopedia of Integer Sequences.While there is much theory and understanding of deductive inference, there is relatively little understanding of the process of inductive inference. Inductive inference is about hypothesizing, hopefully inferring, the laws of physics from observation of the data. It is what Kepler did in coming up with his three laws of planetary motion.Once laws are in place, deductive inference can be used to derive their consequences, which then uncovers much of the understanding in physics.This talk will define inductive inference, state some of its open problems, and give applications to a relatively simpler but still largely unexplored task, namely, inferring a large natural class of algorithmically generated integer sequences that includes (as of this writing) 20in Sloane’s Encyclopedia of integer sequences.KeywordsArtificial IntelligenceCommunication NetworkOpen ProblemProblem ComplexityComputer GraphicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

My long-standing interest in the notion of logical consequence became urgent when I was working on my recent book on higher-order logic (1991). The most common complaint is that second-order logic is out of bounds. as logic, because its consequence relation is not effective. Even though higher order logic is squarely within the prevailing model-theoretic tradition, its set of logical truths is not recursively enumerable-it is not even in the Kleene hierarchy.

The present work investigates to which extent semantical knowledge can support the learning of basic mathematical concepts. The considered learning criteria are learning characteristic or enumerable indices for languages from positive data where the learner has to converge either syntactically (Ex) or semantically (BC). The considered classes are the classes of all monoids of a given group, all ideals of a given ring or all subspaces of a given vector space. The following is shown: (a) Learnability depends much on the amount of semantic knowledge given at the synthesis of the learner where this knowledge is represented by programs for the algebraic operations, codes for prominent elements of the algebraic structure (like 0 and 1 in fields) and certain parameters (like the dimension of finite dimensional vector spaces). For several natural examples good knowledge of the semantics may enable to keep ordinal mind change bounds while restricted knowledge may either allow only BC-convergence or even not permit learnability at all. (b) A recursive commutative ring is Noetherian iff the class of its ideals is BC-learnable. Such a BC-learner can be synthesized from programs for addition and multiplication. In many Noetherian rings, one can Ex-learn characteristic indices for the ideals with an ordinal bound on the number of mind changes. But there are also some Noetherian rings where it is impossible to Ex-learn the ideals or to learn characteristic indices for them.

Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and “nearly” minimal size. i.e.. within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. Alim-computable functionis (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be “not-so-nearly” minimal size, e.g., to be within a lim-computable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are lim-computable functions as above but for whichnotations forconstructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of lim-computability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the lim-computable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight.

The problem of justifying induction has vexed philosophers for centuries. It has been entangled with issues concerning logic and probability and the philosophy of science. This article proposes a new approach to untangling these issues and resolving the overall problem. This new approach is by way of the perspective of realist ontology. Induction should not be seen as a debased form of logic and the search for inflexible rules for justifying particular sorts of inductive inference should be abandoned. Rather, induction can be justified pragmatically as a general practice because reality possesses ordered patterns. Induction, being dependent on repetition, can sometimes identify aspects of those patterns simply because the instances of repetition really do occur. Thus, confidence in conclusions based on inductive inferences is not properly to be derived from unchanging logical rules governing when and where inductive inferences may be successfully applied, but from the involvement of such inferences with a wider, deeper process in which not only deductive inferences and other inductive inferences are involved, but also a set of relations to broader bodies of relevant knowledge. That is, unlike particular deductive inferences, inductive inferences may not be engaged in isolation from all other information about the world and completely in the abstract. Induction is always connected with a process of retroduction.

The purpose of this paper is to examine some passages of Tarski‘s paper ’On the concept of logical consequence’ and to show that some recent readings of those passages are wrong. John Etchemendy has claimed that in those passages Tarski gave an argument purporting to show that the notion of logical consequence defined by him (as opposed to some pretheoretic notion of logical consequence) possesses certain modal properties. Etchemendy further claims that the argument he attributes to Tarski is fallacious. Some of Etchemendy’s critics have granted him that Tarski did give an argument purporting to show that the defined notion possesses certain modal properties ; but they have claimed that Tarski’s argument was not a fallacious one. I will show that both Etchemendy and his critics are wrong; in the relevant passages, Tarski did not offer (nor did he intend to offer) an argument that the defined notion of logical consequence possesses any modal properties

In this study, the researchers used a classical Delphi method to re-examine the conceptual framework, definition, and knowledge base of the field. Seventeen engaged scholars, each representing the expert agricultural education community, reached consensus on defining the field of study, 10 knowledge domains, and 67 knowledge objects. The Delphi panel agreed, “Agricultural education–2010, as a field of study, integrates social and behavioral sciences with the natural and applied science of agriculture, renewable natural resources, and environment. The knowledge base for agricultural education–2010 includes planning and needs assessment; curriculum development; learning theory; instructional design; delivery strategies; evaluation; research methods and tools; scholarship and writing; history, philosophy, and ethics; and contextual applications, culture, and diversity—all effecting continual improvement. Agricultural education empowers people to think more critically, to perform more skillfully, to communicate more clearly, to plan and affect change more efficiently, to solve problems more creatively, and to act based on principles—all of which involves vital choices and consequences in a global society.” The Delphi panel concurred on 10 knowledge domains composed of 67 core knowledge objects consisting of fundamental and powerful concepts, knowledge, paradigms, skills, and/or theories that are essential for building professional practice in agricultural education.

Many connections have been established between learning and logic, or learning and topology, or logic and topology. Still, the connections are not at the heart of these fields. Each of them is fairly independent of the others when attention is restricted to basic notions and main results. We show that connections can actually be made at a fundamental level, and result in a parametrized logic that needs topological notions for its early developments, and notions from learning theory for interpretation and applicability.One of the key properties of first-order logic is that the classical notion of logical consequence is compact. We generalize the notion of logical consequence, and we generalize compactness to β-weak compactness where β is an ordinal. The effect is to stratify the set of generalized logical consequences of a theory into levels, and levels into layers. Deduction corresponds to the lower layer of the first level above the underlying theory, learning with less than β mind changes to layer β of the first level, and learning in the limit to the first layer of the second level. Refinements of Borel-like hierarchies provide the topological tools needed to develop the framework.KeywordsTopological SpaceLogical ConsequenceCommon FrameworkInductive Logic ProgrammingDeductive LogicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Regional innovation strategies rank at the top of public policy agendas today. There is a widespread consensus in both academic and policy circles that standardized “best practice” innovation policy models suffer from severe limitations and major shortcomings. The recent literature is replete with claims that regional innovation policies should be placebased and context-sensitive, taking into consideration the specificities of regions and their distinctive preconditions and capacities for innovation. Various conceptual approaches and theories support such a view. This paper discusses two concepts that have a particularly strong potential for informing customized regional innovation policies: the regional innovation system (RIS) approach and the knowledge base concept. The RIS literature highlights the importance of the organizational and institutional setting of a region and suggests that system deficiencies or failures should constitute the starting point for designing regional innovation policies. The differentiated knowledge base approach stresses that regional industries can differ strongly in their underlying knowledge bases and, as a consequence, in their policy needs. We elaborate on the policy implications that originate from these concepts and argue that tailor-made regional innovation policies should consider both region-specific institutional set-ups and knowledge bases. Focusing on peripheral regions, we outline how such an integrated framework can inform customized regional innovation policies.

Recent research has shown the need to present knowledge to students in a manner consistent with the characteristics of the human brain, especially during learning mathematics, which contributes to the development of achievement and inductive reasoning. The current study aimed to develop achievement and inductive reasoning by employing brain-compatible learning theory in teaching and learning mathematics for sixth-grade students in Oman. The sample included (74) students, who were divided into two equal groups: experimental and control. Study data were collected using mathematical achievement tests and inductive reasoning. There were significant differences between the two groups in the post-application of the achievement and inductive reasoning tests in favor of the experimental group, which indicates that teaching mathematics according to the theory of learning compatible with the brain had a significant impact on the development of achievement and inductive reasoning among the participants. The study recommended the need to develop and teach mathematics curricula in a way that allows for the practice of many activities and exercises that contribute to the development of students’ achievement and inductive reasoning skills.

The supervaluationist theory of vagueness provides a notion of logical consequence that is akin to classical consequence. In the absence of a definitely operator, supervaluationist consequence coincides with classical consequence. In the presence of ‘definitely’, however, supervaluationist logic gives raise to counterexamples to classically valid patterns of inference. Foes of supervaluationism emphasize the last result to argue against the supervaluationist theory. This paper shows a way in which we might obtain systems of deduction adequate for supervaluationist consequence based on systems of deduction adequate for classical consequence. Deductions on the systems obtained this way adopt a completely classical form with the exception of a single step. The paper reviews (at least part of) the discussion on the non-classicality of supervaluationist logic under the light of this result.KeywordsVaguenessSupervaluationismGlobal ValidityDeductive SystemsLogical Consequence