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...