Strong Completeness and Limited Canonicity for PDL

Lavalette, Gerard Renardel de; Kooi, Barteld; Verbrugge, Rineke

doi:10.1007/s10849-009-9083-z

Cited by 2 publications

(2 citation statements)

References 3 publications

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“…But about the second one we think that one possible way is to use infinitary proof systems. This method has been used in [39] to show an strong axiomatization of Λ(g, CFr, [0, 1] L ) (see Section 3.3), but even in the classical modal setting the method has been successfully considered (see [44]). In our opinion the addition of infinitary rules may help to obtain a completeness proof based on a canonical model construction.…”

Section: Discussionmentioning

confidence: 99%

On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

Bou

Esteva

Godo

et al. 2009

Journal of Logic and Computation

120

173

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This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones only evaluated in the idempotent elements and the ones evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truth-constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truth-constants in the language.

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Section: Discussionmentioning

confidence: 99%

On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

Bou

Esteva

Godo

et al. 2009

Journal of Logic and Computation

120

173

View full text Add to dashboard Cite

show abstract

“…Proof systems for PDL itself can broadly be characterised as one of two sorts. Falling into the first category are a multitude of infinitary systems [33,22,14] employing either infinitely-wide ω-proof rules, or (equivalently) allowing countably infinite contexts. In the other category are tableau-based algorithms for deciding PDL-satisfiability [16,18].…”

Section: Introductionmentioning

confidence: 99%

A Non-wellfounded, Labelled Proof System for Propositional Dynamic Logic

Docherty

Rowe

2019

Lecture Notes in Computer Science

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We define an infinitary labelled sequent calculus for PDL, G3PDL ∞ . A finitarily representable cyclic system, G3PDL ω , is then given. We show that both are sound and complete with respect to standard models of PDL and, further, that G3PDL ∞ is cut-free complete. We additionally investigate proof-search strategies in the cyclic system for the fragment of PDL without tests.where • denotes relational composition, R n denotes the composition of R with itself n times, Π 1 returns a set by projecting the first component of each tuple in a relation, and Id(X) denotes the identity relation over the set X.We write m, s |= ϕ to mean s ∈ I m (ϕ), and m |= ϕ to mean that m, s |= ϕ for all states s ∈ S. A PDL formula ϕ is valid when m |= ϕ for all models m.We now define a sequent calculus for deriving theorems (i.e. valid formulas) of PDL. This proof system has two important features. The first is that it is a labelled proof system. Thus sequents contain assertions about the structure of the underlying Kripke models and formulas are labelled with atoms denoting specific states in which they should be interpreted. Secondly, we allow proofs of infinite height.We assume a countable set L of labels (ranged over by x, y, z) that we will use to denote particular states. A relational atom is an expression of the form x R a y, where x and y are labels and a is an atomic program. A labelled formula is an expression of the form x : ϕ, where x is a label and ϕ is a formula. We define a label substitution operation by z{x/y} = y when x = z, and z{x/y} = z otherwise. We lift this to relational atoms and labelled formulas by:Sequents are expressions of the form Γ ⇒ ∆, where Γ and ∆ are finite sets of relational atoms and labelled formulas. We denote an arbitrary member of such a set using A, B, etc. As usual, Γ, A and A, Γ both denote the set {A} ∪ Γ , and Γ {z/y} denotes the application of the (label) substitution {x/y} to all the elements in Γ . We denote by [α]Γ the set of formulas obtained from Γ by prepending the modality [α] to every labelled formula. That is, we define [α]Γ = {x R a y | x R a y ∈ Γ } ∪ {x : [α]ϕ | x : ϕ ∈ Γ }. labs(Γ ) denotes the set of all labels ocurring in the relational atoms and labelled formulas in Γ .We interpret sequents with respect to PDL models using label valuations v, which are functions from labels to states. We write m, v |= x R a y to mean that (v(x), v(y)) ∈ I m (a). We write m, v |= x : ϕ to mean m, v(x) |= ϕ. For a sequent Γ ⇒ ∆, denoted by S, we write m, v |= S to mean that m, v |= B for some B ∈ ∆ whenever m, v |= A for all A ∈ Γ . We write m, v |= S whenever this is not the case, i.e. when m, v |= A for all A ∈ Γ and m, v |= B for all B ∈ ∆. We say S is valid, and write |= S, when m, v |= S for all models m and valuations v that map each label to some state of m.The sequent calculus G3PDL ∞ is defined by the inference rules in fig. 1. A pre-proof is a possibly infinite derivation tree built from these inference rules.Definition 3 (Pre-proof ). A pre-proof is a possibly infinite (i.e. non-wel...

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Strong Completeness and Limited Canonicity for PDL

Cited by 2 publications

References 3 publications

On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

A Non-wellfounded, Labelled Proof System for Propositional Dynamic Logic

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