On Rational Entailment for Propositional Typicality Logic

in Artificial Intelligence and Law (in press)

Propositional Typicality Logic (PTL) is a recently proposed logic, ob- tained by enriching classical propositional logic with a typicality opera- tor capturing the most typical (alias normal or ... [more ▼]

Propositional Typicality Logic (PTL) is a recently proposed logic, ob- tained by enriching classical propositional logic with a typicality opera- tor capturing the most typical (alias normal or conventional) situations in which a given sentence holds. The semantics of PTL is in terms of ranked models as studied in the well-known KLM approach to preferen- tial reasoning and therefore KLM-style rational consequence relations can be embedded in PTL. In spite of the non-monotonic features introduced by the semantics adopted for the typicality operator, the obvious Tarskian definition of entailment for PTL remains monotonic and is therefore not appropriate in many contexts. Our first important result is an impossibil- ity theorem showing that a set of proposed postulates that at first all seem appropriate for a notion of entailment with regard to typicality cannot be satisfied simultaneously. Closer inspection reveals that this result is best interpreted as an argument for advocating the development of more than one type of PTL entailment. In the spirit of this interpretation, we in- vestigate three different (semantic) versions of entailment for PTL, each one based on the definition of rational closure as introduced by Lehmann and Magidor for KLM-style conditionals, and constructed using different notions of minimality. [less ▲]

Taking Defeasible Entailment beyond Rational Closure

in Proceedings of the 16th European Conference on Logics in Artificial Intelligence (JELIA-19) (2019)

We present a systematic approach for extending the KLM framework for defeasible entailment. We first present a class of basic defeasible entailment relations, characterise it in three distinct ways and ... [more ▼]

We present a systematic approach for extending the KLM framework for defeasible entailment. We first present a class of basic defeasible entailment relations, characterise it in three distinct ways and provide a high-level algorithm for computing it. This framework is then refined, with the refined version being characterised in a similar manner. We show that the two well-known forms of defeasible entailment, rational closure and lexicographic closure, fall within our refined framework, that rational closure is the most conservative of the defeasible entailment relations within the framework (with respect to subset inclusion), but that there are forms of defeasible entailment within our framework that are more “adventurous” than lexicographic closure. [less ▲]

A KLM Perspective on Defeasible Reasoning for Description Logics

in Turhan, Anni-Yasmin; Wolter, Frank; Lutz, Carsten (Eds.) et al Description Logic, Theory Combination, and All That - Essays Dedicated to Franz Baader on the Occasion of His 60th Birthday (2019)

In this paper we present an approach to defeasible reasoning for the description logic ALC. The results discussed here are based on work done by Kraus, Lehmann and Magidor (KLM) on defeasible conditionals ... [more ▼]

In this paper we present an approach to defeasible reasoning for the description logic ALC. The results discussed here are based on work done by Kraus, Lehmann and Magidor (KLM) on defeasible conditionals in the propositional case. We consider versions of a preferential semantics for two forms of defeasible subsumption, and link these semantic constructions formally to KLM-style syntactic properties via representation results. In addition to showing that the semantics is appropriate, these results pave the way for more effective decision procedures for defeasible reasoning in description logics. With the semantics of the defeasible version of ALC in place, we turn to the investigation of an appropriate form of defeasible entailment for this enriched version of ALC. This investigation includes an algorithm for the computation of a form of defeasible entailment known as rational closure in the propositional case. Importantly, the algorithm relies completely on classical entailment checks and shows that the computational complexity of reasoning over defeasible ontologies is no worse than that of the underlying classical ALC. Before concluding, we take a brief tour of some existing work on defeasible extensions of ALC that go beyond defeasible subsumption. [less ▲]

A Semantic Perspective on Belief Change in a Preferential Non-Monotonic Framework

in Proceedings of the Sixteenth International Conference on Principles of Knowledge Representation and Reasoning (KR 2018) (2018)

Belief change and non-monotonic reasoning are usually viewed as two sides of the same coin, with results showing that one can formally be defined in terms of the other. In this paper we investigate the ... [more ▼]

Belief change and non-monotonic reasoning are usually viewed as two sides of the same coin, with results showing that one can formally be defined in terms of the other. In this paper we investigate the integration of the two formalisms by studying belief change for a (preferential) non-monotonic framework. We show that the standard AGM approach to be- lief change can be transferred to a preferential non-monotonic framework in the sense that change operations can be defined on conditional knowledge bases. We take as a point of depar- ture the results presented by Casini and Meyer (2017), and we develop and extend such results with characterisations based on semantics and entrenchment relations, showing how some of the constructions defined for propositional logic can be lifted to our preferential non-monotonic framework. [less ▲]

On Rational Entailment for Propositional Typicality Logic

Report (2018)

Propositional Typicality Logic (PTL) is a recently proposed logic, obtained by enriching classical propositional logic with a typicality operator capturing the most typical (alias normal or conventional ... [more ▼]

Propositional Typicality Logic (PTL) is a recently proposed logic, obtained by enriching classical propositional logic with a typicality operator capturing the most typical (alias normal or conventional) situations in which a given sentence holds. The semantics of PTL is in terms of ranked models as studied in the well-known KLM approach to preferential reasoning and therefore KLM-style rational consequence relations can be embedded in PTL. In spite of the non-monotonic features introduced by the semantics adopted for the typicality operator, the obvious Tarskian definition of entailment for PTL remains monotonic and is therefore not appropriate in many contexts. Our first important result is an impossibility theorem showing that a set of proposed postulates that at first all seem appropriate for a notion of entailment with regard to typicality cannot be satisfied simultaneously. Closer inspection reveals that this result is best interpreted as an argument for advocating the development of more than one type of PTL entailment. In the spirit of this interpretation, we investigate three different (semantic) versions of entailment for PTL, each one based on the definition of rational closure as introduced by Lehmann and Magidor for KLM-style conditionals, and constructed using different notions of minimality. [less ▲]

Belief Change in a Preferential Non-Monotonic Framework

in Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (2017, August)

Belief change and non-monotonic reasoning are usually viewed as two sides of the same coin, with results showing that one can formally be defined in terms of the other. In this paper we show that we can ... [more ▼]

Belief change and non-monotonic reasoning are usually viewed as two sides of the same coin, with results showing that one can formally be defined in terms of the other. In this paper we show that we can also integrate the two formalisms by studying belief change within a (preferential) non-monotonic framework. This integration relies heavily on the identification of the monotonic core of a non-monotonic framework. We consider belief change operators in a non-monotonic propositional setting with a view towards preserving consistency. These results can also be applied to the preservation of coherence—an important notion within the field of logic-based ontologies. We show that the standard AGM approach to belief change can be adapted to a preferential non-monotonic framework, with the definition of expansion, contraction, and revision operators, and corresponding representation results. Surprisingly, preferential AGM belief change, as defined here, can be obtained in terms of classical AGM belief change. [less ▲]

Extending Typicality for Description Logics

Report (2017)

Recent extensions of description logics for dealing with different forms of non-monotonic reasoning don’t take us beyond the case of defeasible subsumption. In this paper we enrich the DL EL⊥ with a ... [more ▼]

Recent extensions of description logics for dealing with different forms of non-monotonic reasoning don’t take us beyond the case of defeasible subsumption. In this paper we enrich the DL EL⊥ with a (constrained version of) a typicality operator •, the intuition of which is to capture the most typical members of a class, providing us with the DL EL•⊥. We argue that EL•⊥ is the smallest step one can take to increase the expressivity beyond the case of defeasible subsumption for DLs, while still retaining all the rationality properties an appropriate notion of defeasible subsumption is required to satisfy, and investigate what an appropriate notion of non-monotonic entailment for EL•⊥ should look like. [less ▲]

Using Defeasible Information to Obtain Coherence

in Baral, Chitta; Delgrande, James; Wolter, Frank (Eds.) Proceedings of the 15th International Conference on Principle of Knowledge Representation and Reasoning (KR-16) (2016, April)

We consider the problem of obtaining coherence in a propositional knowledge base using techniques from Belief Change. Our motivation comes from the field of formal ontologies where coherence is ... [more ▼]

We consider the problem of obtaining coherence in a propositional knowledge base using techniques from Belief Change. Our motivation comes from the field of formal ontologies where coherence is interpreted to mean that a concept name has to be satisfiable. In the propositional case we consider here, this translates to a propositional formula being satisfiable. We define belief change operators in a framework of nonmonotonic preferential reasoning.We show how the introduction of defeasible information using contraction operators can be an effective means for obtaining coherence. [less ▲]

Using Defeasible Information to Obtain Coherence

in BNAIC 2016 - Proceedings of the 28th Benelux Conference on Artificial intelligence (2016)

In this paper we consider the problem of obtaining coherence in a propositional knowledge base using techniques from Belief Change. Our motivation comes from the field of formal ontologies where coherence ... [more ▼]

In this paper we consider the problem of obtaining coherence in a propositional knowledge base using techniques from Belief Change. Our motivation comes from the field of formal ontologies where coherence is interpreted to mean that a concept name has to be satisfiable. [less ▲]

On Revision of Partially Specified Convex Probabilistic Belief Bases

in Proceedings of the 22nd European Conference on Artificial Intelligence (ECAI-16) (2016)

We propose a method for an agent to revise its incomplete probabilistic beliefs when a new piece of propositional information is observed. In this work, an agent’s beliefs are represented by a set of ... [more ▼]

We propose a method for an agent to revise its incomplete probabilistic beliefs when a new piece of propositional information is observed. In this work, an agent’s beliefs are represented by a set of probabilistic formulae – a belief base. The method involves determining a representative set of ‘boundary’ probability distributions consistent with the current belief base, revising each of these probability distributions and then translating the revised information into a new belief base. We use a version of Lewis Imaging as the revision operation. The correctness of the approach is proved. An analysis of the approach is done against six rationality postulates. The expressivity of the belief bases under consideration are rather restricted, but has some applications. We also discuss methods of belief base revision employing the notion of optimum entropy, and point out some of the benefits and difficulties in those methods. Both the boundary distribution method and the optimum entropy methods are reasonable, yet yield different results. [less ▲]