scholarly journals Normal forms of conditional knowledge bases respecting system P-entailments and signature renamings

2021 ◽  
Author(s):  
Christoph Beierle ◽  
Jonas Haldimann
Keyword(s):  
Normal Form ◽  
Knowledge Base ◽  
Normal Forms ◽  
Nonmonotonic Reasoning ◽  
Knowledge Bases ◽  
The Other ◽  
Conditional Knowledge ◽  
System P ◽  
The One ◽  
Model Equivalence

AbstractConditionals are defeasible rules of the form If A then usually B, and they play a central role in many approaches to nonmonotonic reasoning. Normal forms of conditional knowledge bases consisting of a set of such conditionals are useful to create, process, and compare the knowledge represented by them. In this article, we propose several new normal forms for conditional knowledge bases. Compared to the previously introduced antecedent normal form, the reduced antecedent normal form (RANF) represents conditional knowledge with significantly fewer conditionals by taking nonmonotonic entailments licenced by system P into account. The renaming normal form(ρNF) addresses equivalences among conditional knowledge bases induced by renamings of the underlying signature. Combining the concept of renaming normal form with other normal forms yields the renaming antecedent normal form (ρ ANF) and the renaming reduced antecedent normal form (ρ RANF). For all newly introduced normal forms, we show their key properties regarding, existence, uniqueness, model equivalence, and inferential equivalence, and we develop algorithms transforming every conditional knowledge base into an equivalent knowledge base being in the respective normal form. For the most succinct normal form, the ρ RANF, we present an algorithm KBρra systematically generating knowledge bases over a given signature in ρ RANF. We show that the generated knowledge bases are consistent, pairwise not antecedentwise equivalent, and pairwise not equivalent under signature renaming. Furthermore, the algorithm is complete in the sense that, when taking signature renamings and model equivalence into account, every consistent knowledge base is generated. Observing that normalizing the set of all knowledge bases over a signature Σ to ρ RANF yields exactly the same result as KBρra (Σ), highlights the interrelationship between normal form transformations on the one hand and systematically generating knowledge bases in normal form on the other hand.

scholarly journals A Complete Map of Conditional Knowledge Bases in Different Normal Forms and Their Induced System P Inference Relations Over Small Signatures

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Christoph Beierle ◽  
Jonas Haldimann ◽  
Steven Kutsch
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Knowledge Bases ◽  
Predominant Role ◽  
New Normal ◽  
Conditional Knowledge ◽  
System P

Conditional knowledge bases consisting of qualitativeconditionals play a predominant role in knowledge representationand reasoning. In this paper, we develop a full map of allconsistent conditional knowledge bases over a small signature indifferent normal forms. We introduce two new normal formsthat take the induced system P inference relation into account,the system P normal form (SPNF) and the renaming SPNF(ρSPNF) considering additionally renamings of theunderlying signature. For a two-element signature, we systematicallygenerate and compare all consistent knowledge bases in ANF,RANF, SPNF, and their renaming counterparts, as well as allcomplete system P inference relations induced by conditionalknowledge bases.

Syntax Splitting = Relevance + Independence: New Postulates for Nonmonotonic Reasoning From Conditional Belief Bases

2020 ◽  
Author(s):  
Gabriele Kern-Isberner ◽  
Christoph Beierle ◽  
Gerhard Brewka
Keyword(s):  
Knowledge Base ◽  
Belief Revision ◽  
Inductive Inference ◽  
Nonmonotonic Reasoning ◽  
Irrelevant Information ◽  
Knowledge Bases ◽  
P System ◽  
Conditional Belief ◽  
Conditional Knowledge ◽  
System P

Syntax splitting, first introduced by Parikh in 1999, is a natural and desirable property of KR systems. Syntax splitting combines two aspects: it requires that the outcome of a certain epistemic operation should only depend on relevant parts of the underlying knowledge base, where relevance is given a syntactic interpretation (relevance). It also requires that strengthening antecedents by irrelevant information should have no influence on the obtained conclusions (independence). In the context of belief revision the study of syntax splitting already proved useful and led to numerous new insights. In this paper we analyse syntax splitting in a different setting, namely nonmonotonic reasoning based on conditional knowledge bases. More precisely, we analyse inductive inference operators which, like system P, system Z, or the more recent c-inference, generate an inference relation from a conditional knowledge base. We axiomatize the two aforementioned aspects of syntax splitting, relevance and independence, as properties of such inductive inference operators. Our main results show that system P and system Z, whilst satisfying relevance, fail to satisfy independence. C-inference, in contrast, turns out to satisfy both relevance and independence and thus fully complies with syntax splitting.

scholarly journals Nonmonotonic reasoning from conditional knowledge bases with system W

2021 ◽  
Author(s):  
Christian Komo ◽  
Christoph Beierle
Keyword(s):  
Upper Bound ◽  
Constraint Satisfaction Problem ◽  
Nonmonotonic Reasoning ◽  
Knowledge Bases ◽  
Upper And Lower Bounds ◽  
Major Objective ◽  
Ranking Models ◽  
Conditional Knowledge ◽  
System P ◽  
Account System

AbstractFor nonmonotonic reasoning in the context of a knowledge base $\mathcal {R}$ R containing conditionals of the form If A then usually B, system P provides generally accepted axioms. Inference solely based on system P, however, is inherently skeptical because it coincides with reasoning that takes all ranking models of $\mathcal {R}$ R into account. System Z uses only the unique minimal ranking model of $\mathcal {R}$ R , and c-inference, realized via a complex constraint satisfaction problem, takes all c-representations of $\mathcal {R}$ R into account. C-representations constitute the subset of all ranking models of $\mathcal {R}$ R that are obtained by assigning non-negative integer impacts to each conditional in $\mathcal {R}$ R and summing up, for every world, the impacts of all conditionals falsified by that world. While system Z and c-inference license in general different sets of desirable entailments, the first major objective of this article is to present system W. System W fully captures and strictly extends both system Z and c-inference. Moreover, system W can be represented by a single strict partial order on the worlds over the signature of $\mathcal {R}$ R . We show that system W exhibits further inference properties worthwhile for nonmonotonic reasoning, like satisfying the axioms of system P, respecting conditional indifference, and avoiding the drowning problem. The other main goal of this article is to provide results on our investigations, underlying the development of system W, of upper and lower bounds that can be used to restrict the set of c-representations that have to be taken into account for realizing c-inference. We show that the upper bound of n − 1 is sufficient for capturing c-inference with respect to $\mathcal {R}$ R having n conditionals if there is at least one world verifying all conditionals in $\mathcal {R}$ R . In contrast to the previous conjecture that the number of conditionals in $\mathcal {R}$ R is always sufficient, we prove that there are knowledge bases requiring an upper bound of 2n− 1, implying that there is no polynomial upper bound of the impacts assigned to the conditionals in $\mathcal {R}$ R for fully capturing c-inference.

scholarly journals Normal Form Near Orbit Segments of Convex Hamiltonian Systems

2021 ◽  
Author(s):  
Shahriar Aslani ◽  
Patrick Bernard
Keyword(s):  
Normal Form ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Vector Fields ◽  
Energy Surface ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
The Other ◽  
Finsler Metrics ◽  
The One

Abstract In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltonians by adding potentials (functions depending only on the base point). This led to the definition of Mañé genericity [ 8]: a property is generic if, given a Hamiltonian $H$, the set of potentials $g$ such that $H+g$ satisfies the property is generic. This notion is mostly used in the context of Hamiltonians that are convex in $p$, in the sense that $\partial ^2_{pp} H$ is positive definite at each point. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians $H+g$ and $H/(1-g)$ have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when $H$ is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be modified by a small perturbation. This kind of question depends strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, and Mañé perturbations of convex Hamiltonians. It is for example well known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka–Smale Theorem, see [ 5, 13] (the other part of the Kupka–Smale Theorem states that the stable and unstable manifolds intersect transversally; it has also been studied in the various settings mentioned above but will not be discussed here). In the context of Hamiltonian vector fields, the statement has to be weakened, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [ 11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [ 1, 2, 4]. The question was investigated only much more recently in the context of Mañé perturbations of convex Hamiltonians, see [ 9, 10]. It is proved in [ 10] that the same result holds: if $H$ is a convex Hamiltonian and $a$ is a regular value of $H$, then there exist arbitrarily small potentials $g$ such that all periodic orbits (including iterated ones) of $H+g$ at energy $a$ are non-degenerate. The proof given in [ 10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work in appropriate coordinates around an orbit segment. In the Riemannian case, one can use the so-called Fermi coordinates. In the Hamiltonian case, appropriate coordinates are considered in [ 10,Lemma 3.1] itself taken from [ 3, Lemma C.1]. However, as we shall detail below, the proof of this Lemma in [ 3], Appendix C, is incomplete, and the statement itself is actually wrong. Our goal in the present paper is to state and prove a corrected version of this normal form Lemma. Our proof is different from the one outlined in [ 3], Appendix C. In particular, it is purely Hamiltonian and does not rest on the results of [ 7] on Finsler metrics, as [ 3] did. Although our normal form is weaker than the one claimed in [ 10], it is actually sufficient to prove the main results of [ 6, 10], as we shall explain after the statement of Theorem 1, and probably also of the other works using [ 3, Lemma C.1].

Features of a Well-Working Civil Litigation System

2017 ◽  
Author(s):  
Steven P. Croley
Keyword(s):  
The Other ◽  
Civil Litigation ◽  
Civil Justice ◽  
Central Importance ◽  
Normative Framework ◽  
Other Hand ◽  
System P ◽  
The One

This chapter provides an analytical and normative framework for evaluating the civil litigation system as well as for understanding existing critiques of the system. It argues that civil justice requires, first, that courts are accessible to parties with valid legal claims and defenses and, second, that courts are capable of distinguishing between strong and weak claims and defenses, which the chapter defines as reliability. This chapter also explains the central importance of litigation costs, and notes that on the one hand litigation costs can impede access to the courts, while on the other hand some costs are crucial to the operation of the civil litigation system—in that distinguishing between strong and weak claims requires certain expenditures.

USING PRIORITIES TO COMBINE KNOWLEDGE BASES

1996 ◽  
Vol 05 (02n03) ◽  
pp. 333-364 ◽  
Author(s):  
SHEKHAR PRADHAN ◽  
JACK MINKER
Keyword(s):  
Knowledge Base ◽  
Knowledge Bases ◽  
The Other ◽  
The Third ◽  
Different Sources

Two or more companies, each with its own knowledge base, may merge. In that case one option is to merge the knowledge bases into one knowledge base. It can happen that some of the information contained in one or more knowledge bases may be in conflict with information in the other knowledge bases. There may be several such points of conflict and any information may be involved in several different such points of conflict. In that case, the integrator of the knowledge bases may prefer a certain claim to another in one conflict-point without necessarily preferring that claim in another conflict-point. Our work constructs a framework within which the consequences of a set of such preferences (expressed as priorities among sets of statements) can be computed. We give three types of semantics for priorities, two of which are shown to be equivalent to one another. The third type of semantics for priorities is shown to be more cautious than the other two. In terms of these semantics for priorities, we give a function for combining knowledge from different sources such that the combined knowledge is conflict-free and satisfies all the priorities.

An Efficient Technique to Ensure the Logical Consistency of Interacting Knowledge Bases

1997 ◽  
Vol 06 (01) ◽  
pp. 27-36 ◽  
Author(s):  
Bertrand Mazure ◽  
Lakhdar Saïs ◽  
Éric Grégoire
Keyword(s):  
Local Search ◽  
Fundamental Problem ◽  
Real Life ◽  
Knowledge Bases ◽  
The Other ◽  
Logical Consistency ◽  
Complete Problem ◽  
Np Complete ◽  
The One ◽  
Cooperative Knowledge

In this paper, we address a fundamental problem in the formalization and implementation of cooperative knowledge bases: the difficulty of preserving consistency while interacting or combining them. Indeed, knowledge bases that are individually consistent can exhibit global inconsistency. This stumbling-block problem is an even more serious drawback when knowledge and reasoning are expressed using logical terms. Indeed, on the one hand, two contradictory pieces of information lead to global inconsistency under complete standard rules of deduction: every assertion and its contrary can be deduced. On the other hand, checking the logical consistency of a propositional knowledge base is an NP-complete problem and is often out of reach for large real-life applications. In this paper, a new practical technique to locate inconsistent interacting pieces of information is presented in the context of cooperative logical knowledge bases. Based on a recently discovered heuristic about the work performed by local search techniques, it can be applied in the context of large interacting knowledge bases.

scholarly journals InfOCF-Web: An Online Tool for Nonmonotonic Reasoning with Conditionals and Ranking Functions

2021 ◽  
Author(s):  
Steven Kutsch ◽  
Christoph Beierle
Keyword(s):  
Nonmonotonic Reasoning ◽  
Minimal Models ◽  
Ranking Functions ◽  
Online Tool ◽  
Ranking Models ◽  
Conditional Knowledge ◽  
System P ◽  
Nonmonotonic Inference

InfOCF-Web provides implementations of system P and system Z inference, and of inference relations based on c-representation with respect to various inference modes and different classes of minimal models. It has an easy-to-use online interface for computing ranking models of a conditional knowledge R, and for answering queries and comparing inference results of nonmonotonic inference relations induced by R.

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