The finite embeddability property for residuated groupoids

Models of complex systems are widely used in the physical and social sciences, and the concept of layering, typically building upon graph-theoretic structure, is a common feature. We describe an intuitionistic substructural logic that gives an account of layering. As in other bunched systems, the logic includes the usual intuitionistic connectives, together with a non-commutative, non-associative conjunction (used to capture layering) and its associated implications. We give a soundness and completeness theorem for a labelled tableaux system with respect to a Kripke semantics on graphs. To demonstrate the utility of the logic, we show how to represent systems and security examples, illuminating the relationship between services/policies and the infrastructures/architectures to which they are applied.

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“…The rest of the proof now proceeds as in [26]. Define operations λ, σ : A → A by λ(a) = C {c ∈ C | a ≤ A c} and σ(a) = C {c ∈ C | c ≤ A a}.…”

Section: Finite Model Property and Decidabilitymentioning

confidence: 99%

Intuitionistic Layered Graph Logic

Docherty

Pym

2017

Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence

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“…Decidability of the distributive non-associative full Lambek calculus can be also proved using algebraic methods via finite embeddability property for residuated grupoids, as was done by Horčík and Haniková in [11], or using proof-theoretical methods as was done by Farulewski in [10]. The final model property for the distributive full Lambek calculus was proved by Kozak in [12].…”

Section: Remark 410mentioning

confidence: 99%

Epistemic logics for sceptical agents

Bílková

Majer

Peliš

2015

J Logic Computation

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In this article, we introduce an epistemic modal operator modelling knowledge over distributive non-associative full Lambek calculus with a negation. Our approach is based on the relational semantics for substructural logics: we interpret the elements of a relational frame as information states consisting of collections of data. The principal epistemic relation between the states is the one of being a reliable source of information, on the basis of which we explicate the notion of knowledge as information confirmed by a reliable source. From this point of view it is natural to define the epistemic operator formally as the backward-looking diamond modality. The framework is a generalization and extension of the system of relevant epistemic logic proposed by Majer and Peliš (2009, college Publications, 123-135) and developed by Bílková et al. (2010, college Publications, 22-38). The system is modular in the sense that the axiomatization of the epistemic operator is sound and complete with respect to a wide class of background logics, which makes the system potentially applicable to a wide class of epistemic contexts. Our system admits a weak form of logical omniscience (the monotonicity rule), but avoids stronger ones (a necessitation rule and a K-axiom) as well as some closure properties discussed in normal epistemic logics (like positive and negative introspection). For these properties we provide characteristic frame conditions, so that they can be present in the system if they are considered to be appropriate for some specific epistemic context. We also prove decidability of the weakest epistemic logic we consider, using a filtration method. Finally, we outline further extensions of our framework to a multiagent system.

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“…Therefore we prove that their word problem is undecidable. This solves negatively Problem 7.1 in [11]. Moreover, the encoding of the halting problem for 2-tag systems can be used directly to obtain a similar result for their {·, ∨}-reduct-joinsemilattices expanded by a groupoid operation (product) where product distributes over join.…”

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confidence: 94%

“…However, we show that the distributivity of product over join is sufficient. It is worth noting that the consequence relation in the distributive FNL, the distributive laws hold for {∧, ∨}, is decidable, see [4,7,11].…”

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confidence: 99%