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.

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.

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 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.

Possibilistic and Probabilistic Logic under Coherence: Default Reasoning and System P

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
Giulianella Coletti ◽  
Romano Scozzafava ◽  
Barbara Vantaggi
Keyword(s):  
Conditional Probability ◽  
Nonmonotonic Reasoning ◽  
Default Reasoning ◽  
Probabilistic Logic ◽  
System P

AbstractSome results on coherence in probabilistic and in possibilistic frameworks are presented in order to deal with nonmonotonic reasoning. Moreover, we extend these results to conditional decomposable measures. We deal with entailment and prove that it satisfies the axiomatization of System P by referring to conditional necessities or to specific conditional decomposable measures (which include conditional probability). Finally, we study some aspects concerning a notion of irrelevance.

scholarly journals Preferred extensions as stable models

2008 ◽  
Vol 8 (04) ◽  
pp. 527-543 ◽  
Author(s):  
JUAN CARLOS NIEVES ◽  
ULISES CORTÉS ◽  
MAURICIO OSORIO
Keyword(s):  
Direct Relationship ◽  
Logic Program ◽  
Nonmonotonic Reasoning ◽  
Mapping Function ◽  
Propositional Formula ◽  
Stable Model ◽  
Minimal Models ◽  
Argumentation Framework ◽  
Polynomial Size ◽  
Stable Models

AbstractGiven an argumentation frameworkAF, we introduce a mapping function that constructs a disjunctive logic programP, such that the preferred extensions ofAFcorrespond to the stable models ofP, after intersecting each stable model with the relevant atoms. The given mapping function is of polynomial size w.r.t.AF.In particular, we identify that there is a direct relationship between the minimal models of a propositional formula and the preferred extensions of an argumentation framework by working on representing the defeated arguments. Then we show how to infer the preferred extensions of an argumentation framework by using UNSAT algorithms and disjunctive stable model solvers. The relevance of this result is that we define a direct relationship between one of the most satisfactory argumentation semantics and one of the most successful approach of nonmonotonic reasoning i.e., logic programming with the stable model semantics.

scholarly journals Nonmonotonic Reasoning, Expectations Orderings, and Conceptual Spaces

2021 ◽  
Author(s):  
Matías Osta-Vélez ◽  
Peter Gärdenfors
Keyword(s):  
Knowledge Representation ◽  
Nonmonotonic Reasoning ◽  
Knowledge Representation And Reasoning ◽  
Conceptual Spaces ◽  
Reasoning Under Uncertainty ◽  
Consequence Relation ◽  
Nonmonotonic Inference ◽  
Default Logics

AbstractIn Gärdenfors and Makinson (Artif Intell 65(2):197–245, 1994) and Gärdenfors (Knowledge representation and reasoning under uncertainty, Springer-Verlag, 1992) it was shown that it is possible to model nonmonotonic inference using a classical consequence relation plus an expectation-based ordering of formulas. In this article, we argue that this framework can be significantly enriched by adopting a conceptual spaces-based analysis of the role of expectations in reasoning. In particular, we show that this can solve various epistemological issues that surround nonmonotonic and default logics. We propose some formal criteria for constructing and updating expectation orderings based on conceptual spaces, and we explain how to apply them to nonmonotonic reasoning about objects and properties.

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