Default logic and specification of nonmonotonic reasoning

In this paper constructions leading to the formation of belief sets by agents are studied. The focus is on the situation when possible belief sets are built incrementally in stages. An infinite sequence of theories that represents such a process is called a reasoning trace. A set of reasoning traces describing all possible reasoning scenarios for the agent is called a reasoning frame. Default logic by Reiter is not powerful enough to represent reasoning frames. In the paper a generalization of default logic of Reiter is introduced by allowing infinite sets of justifications. This formalism is called infinitary default logic. In the main result of the paper it is shown that every reasoning frame can be represented by an infinitary default theory. A similar representability result for antichains of theories (belief frames) is also presented.     

Case studies of Z-module reasoning: Proving benchmark theorems from ring theory

A new method, called Z-module reasoning, was formulated for proving and discovering theorems from ring theory. In a case study, the ZMR system designed to implement this method was used to prove the benchmark x ³ ring theorem from associative ring theory. The system proved the theorem quite efficiently. The system was then used to prove the x ⁴ ring theorem from associative ring theory. Again, a proof was produced easily. These proofs, together with the successes in proving other difficult theorems from ring theory suggest that the Z-module reasoning method is useful for solving a class of problems relying on equality reasoning. This paper illustrates the Z-module reasoning method, and analyzes the computer proof of the x ³ ring theorem.This reasearch was supported in part by the Applied Mathematical Sciences subprogram of the office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38.     

Contextual reasoning distilled

In this paper we provide a foundation of a theory of contextual reasoning from the perspective of a theory of knowledge representation. Starting from the so-called metaphor of the box, we firstly show that the mechanisms of contextual reasoning proposed in the literature can be classified into three general forms (called localized reasoning, push and pop, and shifting). Secondly, we provide a justification of this classification, by showing that each mechanism corresponds to operating on a fundamental dimension along which context dependent representations may vary (namely, partiality, approximation and perspective). From the previous analysis, we distill two general principles of a logic of contextual reasoning. Finally, we show that these two principles can be adequately formalized in the framework of MultiContext Systems. In the last part of the paper, we provide a practical illustration of the ideas discussed in the paper by formalising a simple scenario, called the Magic Box problem.     

A logic for reasoning about probabilities

We consider a language for reasoning about probability which allows us to make statements such as “the probability of E₁ is less than ” and “the probability of E₁ is at least twice the probability of E₂,” where E₁ and E₂ are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice, where they may not be measurable. The measurable case is essentially a formalization of (the propositional fragment of) Nilsson's probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by Dempster-Shafer belief functions. In both cases, we provide a complete axiomatization and show that the problem of deciding satisfiability is NP-complete, no worse than that of propositional logic. As a tool for proving our complete axiomatizations, we give a complete axiomatization for reasoning about Boolean combinations of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields.     

OSCAR: a general theory of rationality

Abstract

The enterprise is the construction of a general theory of rationality, and its implementation in an automated reasoning system named OSCAR. The paper describes a general architecture for rational thought. This includes both theoretical reasoning and practical reasoning, and builds in important interconnections between them. It is urged that a sophisticated reasoner must be an introspective reasoner, capable of monitoring its own reasoning and reasoning about it. An introspective reasoner is built on top of a non-introspective reasoner that represents the system's default reasoning strategies. The introspective reasoner engages in practical reasoning about reasoning in order to override these default strategies. The paper concludes with a discussion of some aspects of the default reasoner, including the manner in which reasoning is interest driven, and the structure of defeasible reasoning.     

Fuzzy reasoning based on the extension principle*

According to the operation of decomposition (also known as representation theorem) (Negoita CV, Ralescu, DA. Kybernetes 1975;4:169–174) in fuzzy set theory, the whole fuzziness of an object can be characterized by a sequence of local crisp properties of that object. Hence, any fuzzy reasoning could also be implemented by using a similar idea, i.e., a sequence of precise reasoning. More precisely, we could translate a fuzzy relation “If A then B” of the Generalized Modus Ponens Rule (the most common and widely used interpretation of a fuzzy rule, A, B, are fuzzy sets in a universe of discourse X, and of discourse Y, respectively) into a corresponding precise relation between a subset of P(X) and a subset of P(Y), and then extend this corresponding precise relation to two kinds of transformations between all L-type fuzzy subsets of X and those of Y by using Zadeh's extension principle, where L denotes a complete lattice. In this way, we provide an alternative approach to the existing compositional rule of inference, which performs fuzzy reasoning based on the extension principle. The approach does not depend on the choice of fuzzy implication operator nor on the choice of a t-norm. The detailed reasoning methods, applied in particular to the Generalized Modus Ponens and the Generalized Modus Tollens, are established and their properties are further investigated in this paper. © 2001 John Wiley & Sons, Inc.     

Quantum Mereotopology

While mereotopology – the theory of boundaries, contact and separation built up on a mereological foundation – has found fruitful applications in the realm of qualitative spatial reasoning, it faces problems when its methods are extended to deal with those varieties of spatial and non-spatial reasoning which involve a factor of granularity. This is because granularity cannot easily be represented within a mereology-based framework. We sketch how this problem can be solved by means of a theory of granular partitions, a theory general enough to comprehend not only the familiar sorts of spatial partitions but also a range of coarse-grained partitions of other, non-spatial sorts. We then show how these same methods can be extended to apply to finite sequences of granular partitions evolving over time, or to what we shall call coarse- and fine-grained histories.     

A theory of defeasible reasoning

Reasoning can lead not only to the adoption of beliefs, but also to the retraction of beliefs. In philosophy, this is described by saying that reasoning is defeasible. My ultimate objective is the construction of a general theory of reasoning and its implementation in an automated reasoner capable of both deductive and defeasible reasoning. the resulting system is named “OSCAR.” This article addresses some of the theoretical underpinnings of OSCAR. This article extends my earlier theory in two directions. First, it addresses the question of what the criteria of adequacy should be for a defeasible reasoner. Second, it extends the theory to accommodate reasons of varying strengths.     

Self-defeating arguments

An argument is self-defeating when it contains defeaters for some of its own defeasible lines. It is shown that the obvious rules for defeat among arguments do not handle self-defeating arguments correctly. It turns out that they constitute a pervasive phenomenon that threatens to cripple defeasible reasoning, leading to almost all defeasible reasoning being defeated by unexpected interactions with self-defeating arguments. This leads to some important changes in the general theory of defeasible reasoning.     

Handling Transitive Relations in First-Order Automated Reasoning

We present a number of alternative ways of handling transitive binary relations that commonly occur in first-order problems, in particular equivalence relations, total orders, and transitive relations in general. We show how such relations can be discovered syntactically in an input theory, and how they can be expressed in alternative ways. We experimentally evaluate different such ways on problems from the TPTP, using resolution-based reasoning tools as well as instance-based tools. Our conclusions are that (1) it is beneficial to consider different treatments of binary relations as a user, and that (2) reasoning tools could benefit from using a preprocessor or even built-in support for certain types of binary relations.

     

Practical Reflection for Sequent Logics

It is well-known that adding reflective reasoning can tremendously increase the power of a proof assistant. In order for this theoretical increase of power to become accessible to users in practice, the proof assistant needs to provide a great deal of infrastructure to support reflective reasoning. In this paper we explore the problem of creating a practical implementation of such a support layer.Our implementation takes a specification of a logical theory (which is identical to how it would be specified if we were simply going to reason within this logical theory, instead of reflecting it) and automatically generates the necessary definitions, lemmas, and proofs that are needed to enable the reflected meta-reasoning in the provided theory.One of the key features of our approach is that the structure of a logic is preserved when it is reflected. In particular, all variables, including meta-variables, are preserved in the reflected representation. This also allows the preservation of proof automation—there is a structure-preserving one-to-one map from proof steps in the original logic to proof step in the reflected logic.To enable reasoning about terms with sequent context variables, we develop a principle for context induction, called teleportation.This work is fully implemented in the MetaPRL theorem prover.     

Cataloguing/analogizing: A nonmonotonic view

Reasoning deductively under incomplete information is nonmonotonic in nature since the arrival of additional information may invalidate or reverse previously obtained conclusions. It amounts to apply generic default rules in an appropriate way to a particular (partially described) situation. This type of nonmonotonic reasoning can only provide plausible conclusions. Analogical reasoning is another form of commonly used reasoning that yields brittle conclusions. It is nondeductive in nature and proceeds by putting particular situations in parallel. Analogical reasoning also exhibits nonmonotonic features, as investigated in this paper when particular situations may be incompletely stated. The paper reconsiders the pattern of plausible reasoning proposed by Polya, “a and b are analogous, a is true, then b true is more credible,'' from a nonmonotonic reasoning point of view. A representation of the statement “a and b are analogous” in terms of nonmonotonic consequences relations is presented. This representation is then related to a logical definition of analogical proportions, i.e. statements of the form “a is to b as c is to d” that has been recently proposed and extended to other types of proportions. Remarkably enough, semantic equivalence between conditional objects of the form “b given a,” which have been shown as being at the root of nonmonotonic reasoning, constitutes another type of noticeable proportions. By offering a parallel between two important forms of commonsense reasoning, this paper enriches the comparison between nonmonotonic reasoning and analogical reasoning that is not often made. © 2011 Wiley Periodicals, Inc.     

Reverse triple Ⅰ method of fuzzy reasoning

A theory of reverse triple I method with sustention degree is presented by using the implication operator R0 in every step of the fuzzy reasoning. Its computation formulas of supremum for fuzzy modus ponens and infimum for fuzzy modus tollens are given respectively. Moreover, through the generalization of this problem, the corresponding formulas of α-reverse triple I method with sustention degree are also obtained. In addition, the theory of reverse triple I method with restriction degree is proposed as well by using the operator R0, and the computation formulas of infimum for fuzzy modus ponens and supremum for fuzzy modus tollens are shown.     

Declarative Formalization of Reasoning Strategies: A Case Study on Heuristic Nonlinear Planning

We study the declarative formalization of reasoning strategies by presenting declarative formalizations of: (1) the SNLP algorithm for nonlinear planning, and (2) a particular algorithm for blocks world nonlinear planning proposed in this paper. The formal models of a heuristic forward chaining planner, which can take advantage of declarative formalizations of action selection strategies, and of a reasoning strategy based planner, which can use declarative formalizations of reasoning strategies, are proposed. The effectiveness of these systems is studied from formal and empirical points of view. Empirical results showing how the use of declarative formalizations of reasoning strategies can reduce the amount of search required for solving planning problems (with respect to state of the art planning systems) are presented.     

解释模型类理论及其极小三I-算法

首先在多类（many-sorted）一阶形式系统Lukms、Gödms,∏ms和L^*ms中通过引入多类一阶模糊语言Lms的解释模型类及基于解释模型类的α-逻辑有效公式的概念,建立了多类一阶模糊语言的解释模型类理论;然后,基于上述理论探讨了模糊推理算法（CRI及三I算法）与其理论Г-推理的关系,从而进一步奠定了模糊推理的理论基础,同时得到一种新型的模糊推理算法,称为极小三I算法。     

On Construction and Evaluation of Analogical Arguments for Persuasive Reasoning

ABSTRACT

Analogical reasoning is a complex process based on a comparison between two pairs of concepts or states of affairs (aka. the source and the target) for characterizing certain features from one to another. Arguments which employ this process to support their claims are called analogical arguments. Our goals are to study the structure and the computation for their defeasibility in light of the argumentation theory. Our proposed assumption-based argumentation with predicate similarity ABA^(p) framework can be seen as an extension of assumption-based argumentation framework (ABA), in which not only assumptions can be used but also similarity of predicates is used to support a claim. ABA ^(p) labels each argument tree with an analogical degree and different ways to aggregate numerical values are studied toward gullible/skeptical characteristics in agent reasoning. The acceptability of analogical arguments is evaluated w.r.t. the semantics of abstract argumentation. Finally, we demonstrate that ABA ^(p) captures the argumentation scheme for argument from analogy and provides an explanation when it is used for persuasion.     

Set theory for verification. II: Induction and recursion

A theory of recursive definitions has been mechanized in Isabelle's Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs, and other computational reasoning. Inductively defined sets are expressed as least fixedpoints, applying the Knaster-Tarski theorem over a suitable set.Recursive functions are defined by well-founded recursion and its derivatives, such as transfinite recursion.Recursive data structures are expressed by applying the Knaster-Tarski theorem to a set, such asV , that is closed under Cartesian product and disjoint sum.Worked examples include the transitive closure of a relation, lists, variable-branching trees, and mutually recursive trees and forests. The Schröder-Bernstein theorem and the soundness of propositional logic are proved in Isabelle sessions.     

On Closure Under Stuttering

For over a decade, researchers in formal methods have tried to create formalisms that permit natural specification of systems and allow mathematical reasoning about their correctness. The availability of fully automated reasoning tools enables non-experts to use formal methods effectively—their responsibility reduces to specifying the model and expressing the desired properties. Thus, it is essential that these properties be represented in a language that is easy to use, sufficiently expressive and succinct. Linear-time temporal logic (LTL) is a formalism that has been used extensively by researchers for program specification and verification. One of the desired properties of LTL formulas is closure under stuttering. That is, we do not want the interpretation of formulas to change over traces where some states are repeated. This property is important from both practical and theoretical prospectives; all properties which are closed under stuttering can be expressed in LTL_–X—a fragment of LTL without the next operator. However, it is often difficult to express properties in this fragment of LTL. Further, determining whether a given LTL property is closed under stuttering is PSPACE-complete. In this paper, we introduce a notion of edges of LTL formulas and present a formal theory of closure under stuttering. Edges allow natural modelling of systems with events. Our theory enables syntactic reasoning about whether the resulting properties are closed under stuttering. Finally, we apply the theory to the pattern-based approach of specifying temporal formulas.     

A Symbolic Approach To Uncertainty Management

In this paper, we present a new symbolic approach to deal with the uncertainty encountered in common-sense reasoning. This approach enables us to represent the uncertainty by using linguistic expressions of the interval [Certain, Totally uncertain]. The original uncertainty scale that we use here, presents some advantages over other scales in the representation and in the management of the uncertainty. The axioms of our theory are inspired by Shannon's entropy theory and built on the substrate of a symbolic many-valued logic. So, the uncertainty management in the symbolic logic framework leads to new generalizations of classical inference rules.