Normal Forms of Conditional Knowledge Bases Respecting Entailments and Renamings

2020 ◽  
pp. 22-41
Author(s):  
Christoph Beierle ◽  
Jonas Haldimann
Keyword(s):  
Normal Forms ◽  
Knowledge Bases ◽  
Conditional Knowledge

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.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document