Learning definite Horn formulas from closure queries

Arias, Marta; Balcázar, José L.; Tîrnźucź, Cristina

doi:10.1016/j.tcs.2015.12.019

Cited by 7 publications

(3 citation statements)

References 21 publications

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“…The choice of implication logic as a foundational support for explanations benefits both from FCA's results as well as from the approximate nature of Horn's clauses in the face of counterfactual issues asked by explainee [80], which we will consider in a future work. However, our proposal is only a first proof of concept towards a formalization of the notions involved in AI-assisted explaining of events (or justification of decisions) of CS (always keeping in mind that a number of sophisticated AI-based systems are actually CS).…”

Section: Future Workmentioning

confidence: 99%

Knowledge representation for explainable artificial intelligence

Borrego-Díaz

Galán-Páez

2022

Complex Intell. Syst.

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Alongside the particular need to explain the behavior of black box artificial intelligence (AI) systems, there is a general need to explain the behavior of any type of AI-based system (the explainable AI, XAI) or complex system that integrates this type of technology, due to the importance of its economic, political or industrial rights impact. The unstoppable development of AI-based applications in sensitive areas has led to what could be seen, from a formal and philosophical point of view, as some sort of crisis in the foundations, for which it is necessary both to provide models of the fundamentals of explainability as well as to discuss the advantages and disadvantages of different proposals. The need for foundations is also linked to the permanent challenge that the notion of explainability represents in Philosophy of Science. The paper aims to elaborate a general theoretical framework to discuss foundational characteristics of explaining, as well as how solutions (events) would be justified (explained). The approach, epistemological in nature, is based on the phenomenological-based approach to complex systems reconstruction (which encompasses complex AI-based systems). The formalized perspective is close to ideas from argumentation and induction (as learning). The soundness and limitations of the approach are addressed from Knowledge representation and reasoning paradigm and, in particular, from Computational Logic point of view. With regard to the latter, the proposal is intertwined with several related notions of explanation coming from the Philosophy of Science.

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Section: Future Workmentioning

confidence: 99%

Knowledge representation for explainable artificial intelligence

Borrego-Díaz

Galán-Páez

2022

Complex Intell. Syst.

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“…Although this bound is weaker than Theorem 4.4, its proof demonstrates that this property of irredundant PC formulas is a consequence of the known properties of closure operators and their representation using Horn formulas. See Guigues and Duquenne (1986), Arias, Balcázar, and Tîrnăucă (2017) for more information on the relationship between closure operators (or closure systems) and Horn formulas. In particular, a formula is a PC formula, if it represents in the sense specified below the semantic closure operator on the sets of literals which are subsets of a set of polynomial size.…”

Section: Implicational Dual Rail Encoding Of Pc Formulasmentioning

confidence: 99%

Bounds on the size of PC and URC formulas

Kučera

Savický

2020

jair

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In this paper, we investigate CNF encodings, for which unit propagation is strong enough to derive a contradiction if the encoding is not consistent with a partial assignment of the variables (unit refutation complete or URC encoding) or additionally to derive all implied literals if the encoding is consistent with the partial assignment (propagation complete or PC encoding). We prove an exponential separation between the sizes of PC and URC encodings without auxiliary variables and strengthen the known results on their relationship to the PC and URC encodings that can use auxiliary variables. Besides of this, we prove that the sizes of any two irredundant PC formulas representing the same function differ at most by a factor polynomial in the number of the variables and present an example of a function demonstrating that a similar statement is not true for URC formulas. One of the separations above implies that a q-Horn formula may require an exponential number of additional clauses to become a URC formula. On the other hand, for every q-Horn formula, we present a polynomial size URC encoding of the same function using auxiliary variables. This encoding is not q-Horn in general.

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“…Let Ĥ be a Horn envelope of a set V ⊆ 2 Φ . For computing Ĥ, we need the membership query be answered relative to V. Such a query can be simulated by several implication queries relative to V. One well-known (see, e.g., [11]) method to do this is presented in Theorem 1.…”

Section: Simulating Membership Queriesmentioning

confidence: 99%

Probably approximately correct learning of Horn envelopes from queries

Borchmann¹,

Hanika²,

Obiedkov³

2020

Discrete Applied Mathematics

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We propose an algorithm for learning the Horn envelope of an arbitrary domain using an expert, or an oracle, capable of answering certain types of queries about this domain. Attribute exploration from formal concept analysis is a procedure that solves this problem, but the number of queries it may ask is exponential in the size of the resulting Horn formula in the worst case. We recall a well-known polynomial-time algorithm for learning Horn formulas with membership and equivalence queries and modify it to obtain a polynomial-time probably approximately correct algorithm for learning the Horn envelope of an arbitrary domain.

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Learning definite Horn formulas from closure queries

Cited by 7 publications

References 21 publications

Knowledge representation for explainable artificial intelligence

Knowledge representation for explainable artificial intelligence

Bounds on the size of PC and URC formulas

Probably approximately correct learning of Horn envelopes from queries

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