Group Axioms Research Articles

A description logic based situation calculus

We consider a modified version of the situation calculus built using a two-variable fragment of the first-order logic extended with counting quantifiers. We mention several additional groups of axioms that can be introduced to capture taxonomic reasoning. We show that the regression operator in this framework can be defined similarly to regression in Reiter's version of the situation calculus. Using this new regression operator, we show that the projection and executability problems (the important reasoning tasks in the situation calculus) are decidable in the modified version even if an initial knowledge base is incomplete. We also discuss the complexity of solving the projection problem via regression in this modified language in general. Furthermore, we define description logic based sub-languages of our modified situation calculus. They are based on the description logics $\mathcal{ALCO}(U)$ (or $\mathcal{ALCQO}(U)$ , respectively). We show that in these sub-languages solving the projection problem via regression has better computational complexity than in the general modified situation calculus. We mention possible applications to formalization of Semantic Web services and some connections with reasoning about actions based on description logics.

Formal theory of the comparative relations: its application to the study of quantum similarity and dissimilarity measures and indices

The study proposes a formal theory of comparative relations characterized by a group of abstract definitions and axioms. This abstract characterization allows us to rigorously define in a very general form what a comparative relation is, which can in turn be used as starting point in every case where comparative analysis is used. Some of the consequences of this formalization—when applied to existing results- are analyzed, particularly those related with the consistency of the results of the comparisons. Following, a study was conducted of descriptors of Quantum Quantitative Structure–Property Relationships models proposed by Carbo-Dorca et al. Several quantum comparative indices such as: Carbo, Hodgkin–Richards, Petke, Tanimoto, conjugate Petke and conjugate Hodgkin–Richards were also analyzed. The proposed theory gives a solid theoretical basis for comparative analysis that will have a positive impact on a broad spectrum of disciplines, where the field of chemistry could be one of the most important beneficiaries.

On ranking opportunity distributions

Our results show the intimate relationship between a large group of apparently different rankings of opportunity distributions. First, we provide a set of core basic axioms that are intuitively plausible under any concern for equality or efficiency aspects. Second, we introduce two very opposed views of the problem by incorporating different perspectives of the notion of advantage (better opportunity set) and we provide some axioms that might be defended under such extreme positions. For any of these two different perspectives, we characterize the families of rankings which satisfy the core axioms and the group of axioms corresponding to that view. These characterizations will prove insightful to better understand most of the criteria already introduced in the literature, which are part of the families we study.

Searching for Shortest Single Axioms for Groups of Exponent 6

In this note, we study shortest (with respect to the number of variable occurrences on the left-hand side) possible single axioms for groups of exponent 6 of the form T = x, where T is a term in product only. These investigations were carried out with the automated theorem provers OTTER and Prover9 and the finite first-order model finder Mace4.

Minimal axiom group of similarity-based rough set model

Rough set axiomatization is one aspect of rough set study to characterize rough set theory using dependable and minimal axiom groups. Thus, rough set theory can be studied by logic and axiom system methods. The classical rough set theory is based on equivalence relation, but the rough set theory based on similarity relation has wide applications in the real world. To characterize similarity-based rough set theory, an axiom group named S, consisting of 3 axioms, is proposed. The reliability of the axiom group, which shows that characterizing of rough set theory based on similarity relation is rational, is proved. Simultaneously, the minimization of the axiom group, which requests that each axiom is an equation and independent, is proved. The axiom group is helpful to research rough set theory by logic and axiom system methods.

A new minimal rough set axiom group

Rough set axiomatization is one aspect of rough set study, and the purpose is to characterize rough set theory using independable and minimal axiom groups. Thus, rough set theory can be studied by logic and axiom system methods. To characterize rough set theory, an axiom group named H consisting of 4 axioms, is proposed. That validity of the axiom group in characterizing rough set theory is reasonable, is proved. Simultaneously, the minimization of the axiom group, which requires that each axiom is an inequality and each is independent, is proved. The axiom group is helpful for researching rough set theory by logic and axiom system methods.

EZPAL: Environment for composing constraint axioms by instantiating templates

Many ontology-development tools allow users to supplement frame-based representations with arbitrary logical sentences. However, few users actually take advantage of this opportunity. For example, in the Ontolingua ontology library, only 20% of the ontologies have any user-defined axioms. We believe the difficulty of composing axioms primarily accounts for the lack of axioms in these knowledge bases: Many domain experts cannot translate their thoughts into abstract and symbolic representations. We attempt to remedy the difficulties by identifying groups of axioms that manifest common patterns, creating “templates” that allow users to compose axioms by “filling in the blanks.” We studied axioms in two public ontology libraries, and derived 20 templates that cover 85% of all the user-defined axioms. We describe our methodology for identifying the templates and present examples. We constructed an interface that allows users to create constraints on knowledge bases by “filling in blanks;” our usability testing shows that users could use templates to encode axioms with a success rate similar to that of experts writing directly in an axiom language. Our approach should foster the introduction of axioms and constraints that are currently missing in many ontologies.

A minimal axiom group for rough set based on quasi-ordering

Rough set axiomatization is one aspect of rough set study to characterize rough set theory using dependable and minimal axiom groups. Thus, rough set theory can be studied by logic and axiom system methods. The classic rough set theory is based on equivalent relation, but rough set theory based on reflexive and transitive relation (called quasi-ordering) has wide applications in the real world. To characterize topological rough set theory, an axiom group named RT, consisting of 4 axioms, is proposed. It is proved that the axiom group reliability in characterizing rough set theory based on similar relation is reasonable. Simultaneously, the minimization of the axiom group, which requires that each axiom is an equation and each is independent, is proved. The axiom group is helpful for researching rough set theory by logic and axiom system methods.

A handbook of mathematical discourse

algebra See algebra. abstraction An abstraction of a concept C is a concept C ′ that includes all instances of C and that is constructed by taking as axioms certain assertions that are true of all instances of C. C may already be defined mathematically, in which case the abstraction is typically a legitimate generalization of C. In other cases, C may be a familiar concept or property that has not been given a mathematical definition. In that case, the mathematical definition may allow instances of the abstract version of C that were not originally thought of as being part of C.ion An abstraction of a concept C is a concept C ′ that includes all instances of C and that is constructed by taking as axioms certain assertions that are true of all instances of C. C may already be defined mathematically, in which case the abstraction is typically a legitimate generalization of C. In other cases, C may be a familiar concept or property that has not been given a mathematical definition. In that case, the mathematical definition may allow instances of the abstract version of C that were not originally thought of as being part of C. Example 1 The concept of “group” is historically an abstraction of the concept of the set of all symmetries of an object. The group axioms are all true assertions about symmetries when the binary operation is taken to be composition of symmetries. Example 2 The -δ definition of continuous function is historically an abstraction of the intuitive idea that mathematicians had about functions that there was no “break” in the output. This abstraction became the standard definition of “continuous”, but allowed functions to be called continuous that were not contemplated before the definition was introduced. Other examples are given under model and in Remark 2 under free variable. See also the discussions under definition, generalization and representation. Citations: (31), (270).

Divergence measure between fuzzy sets

In this paper we propose a way of measuring the difference between two fuzzy sets by means of a function which we will call divergence. We define this concept by means of a group of natural axioms and we study in detail the most important classes of such measures, those which have the local property.

Choice Without Beliefs

We provide an axiomatic foundation for decision making in a complex environment. We do not assume that the decision maker has complete structural knowledge of the environment. Instead the agent knows the set of actions he can take, he formulates preferences directly on the actions, and chooses according to these preferences. On the basis of experience he modifies these preferences according to a systematic procedure. Our axioms are imposed on this procedure, rather than directly on the choice itself. The axioms consist of a group of natural structural restrictions and a group of independence axioms. Our main result is an axiomatic foundation for a set of simple adaptive learning procedures that include the replicator dynamic.

Group Axioms for Iteration

Iteration theories provide a sound and complete axiomatization of the equational properties of the iteration (or fixed point) operation in many models of theoretical computer science including ordered and metric structures, trees and synchronization trees. All known equational axiomatizations of iteration theories consist of a small set of equational axioms for Conway theories and a complicated equation scheme, the commutative identity. Here we associate an identity with each finite semigroup. We prove that the set consisting of the Conway identities and the group identities associated with the finite (simple) groups is complete. Moreover, we prove that the Conway identities and a subcollection of the semigroup identities associated with a subclass of the finite semigroups is complete iff each finite (simple) group divides one of the semigroups in the subclass. We also formulate a conjecture and study its consequences. The results are a generalization of Krob's axiomatization of the equational theory of the regular sets.

Computer and Human Reasoning: Single Implicative Axioms for Groups and for Abelian Groups

We use additive notation, +, 0, ',-, for Abelian groups, and multiplicative notation, , e, l, /, for ordinary groups. Throughout this note, and / are binary operations rather than abbreviations for, e.g., x + y' and x Xy-1 One might think it trivial, given (2), to obtain a single axiom in terms of product and inverse, by simply rewriting a/,(3 to a * l Doing so gives a single axiom, but then * is not product, and -1 is not inverse. The same situation holds for the Abelian case. Another curious fact is that there is no single equational axiom for groups or for Abelian groups in terms of the three standard operations of product, inverse, and the identity element [8]. Single equational axioms in terms of product and inverse have been reported by Neumann [5] and others [3, 2]. In this note we consider single amplicative axioms, that is, axioms of the form ct = ,X3 y = b. For Abelian groups, an axiom of this type with five variables was

Computer and Human Reasoning: Single Implicative Axioms for Groups and for Abelian Groups

Computer and Human Reasoning: Single Implicative Axioms for Groups and for Abelian Groups

The shortest single axioms for groups of exponent 4

We study equations of the form ( α = x), which are single axioms for groups of exponent 4, where α is a term in product only. Every such α must have at least nine variable occurrences, and there are exactly three such α of this size, up to variable renaming and mirroring. These terms were found by an exhaustive search through all terms of this form. Automated techniques were used in two ways: to eliminate many α by verifying that ( α = x) is ture in some nongroup, and to verify that the group axioms do indeed follow from the successful ( α = x). We also present an improvement on Neumann's scheme for single axioms for varieties of groups.

On Infinite Locally Finite Groups

AbstractIf G is a group such that every infinite subset of G contains a commuting pair of elements then G is centre-by-finite. This result is due to B. H. Neumann. From this it can be shown that if G is infinite and such that for every pair X, Y of infinite subsets of G there is some x in X and some y in Y that commute, then G is abelian. It is natural to ask if results of this type would hold with other properties replacing commutativity. It may well be that group axioms are restrictive enough to provide meaningful affirmative results for most of the properties. We prove the following result which is of similar nature.If G is a group such that for each positive integer n and for every n infinite subset X1,...,Xn of G there exist elements xi of Xii = 1,... ,n, such that the subgroup generated by {x1,... ,xn} is finite, then G is locally finite.

COMBINING CCS AND PETRI NETS VIA STRUCTURAL AXIOMS

Many approaches have been developed with the aim of capturing the advantages of both process algebras and Petri nets in terms of modularity and structure on one side and faithful description of concurrency on the other. A natural way of merging these different models is to express the semantics of process algebras in terms of Petri nets. In this work we present a modular construction of operational models for CCS via different groups of structural axioms. To express them, we use Equational Type Logic (ETL), a formalism based on conditional axioms on typed algebras. Typed algebras can be used with profit to present transition systems where both states and transitions have algebraic structure, as opposed to the usual SOS approach where only states have structure. We build an algebra in which different types give different views of the language. In fact, different subalgebras live together in the same structure and are related by axioms. They represent: i) the transition system of CCS; ii) an unfolded version of it; iii) a net for CCS and its marking graph; and iv) a folded version of the latter with the same states as i). The model is completely compositional, since CCS operations are defined on all state representations. We also present axioms which directly establish the relation between interleaving and truly concurrent semantics for CCS. Finally, some related work is discussed and the relation of our models with two previous proposals is shown in detail.

Theories of truth and truth-value gaps

The fact that a group of axioms use the word ‘true’ does not guarantee that that group of axioms yields a theory of truth. For Davidson the derivability of certain biconditionals from the axioms is what guarantees this. We argue that the test does not work. In particular, we argue that if the object language has truth-value gaps, the result of applying Davidson's definition of a theory of truth is that no correct theory of truth for the language is possible.

Single axioms for groups and Abelian groups with various operations

This paper summarizes the results of an investigation into single axioms for groups, both ordinary and Abelian, with each of following six sets of operations: {product, inverse}, {division}, {double division, identity}, {double division, inverse}, {division, identity}, and {division, inverse}. In all but two of the twelve corresponding theories, we present either the first single axioms known to us or single axioms shorter than those previously known to us. The automated theorem-proving program OTTER was used extensively to construct sets of candidate axioms and to search for and find proofs that given candidate axioms are in fact single axioms.

Single axioms for groups

We study equations of the form (α=x) which are single axioms for group theory. Earlier examples of such were found by Neumann and McCune. We prove some lower bounds on the complexity of these α, showing that McCune's examples are the shortest possible. We also show that no such α can have only two distinct variables. We do produce an α with only three distinct variables, refuting a conjecture of Neumann. Automated reasoning techniques are used both positively (searching for and verifying single axioms) and negatively (proving that various candidate (α=x) hold in some nongroup and are, hence, not single axioms).