On the Herbrand content of LK

We present a structural representation of the Herbrand content of LK-proofs with cuts of complexity prenex Sigma-2/Pi-2. The representation takes the form of a typed non-deterministic tree grammar of order 2 which generates a finite language of first-order terms that appear in the Herbrand expansions obtained through cut-elimination. In particular, for every Gentzen-style reduction between LK-proofs we study the induced grammars and classify the cases in which language equality and inclusion hold.Comment: In Proceedings CL&C 2016, arXiv:1606.0582

On closure ordinals for the modal mu-calculus

The closure ordinal of a formula of modal mu-calculus mu X phi is the least ordinal kappa, if it exists, such that the denotation of the formula and the kappa-th iteration of the monotone operator induced by phi coincide across all transition systems (finite and infinite). It is known that for every alpha < omega^2 there is a formula phi of modal logic such that mu X phi has closure ordinal alpha (Czarnecki 2010). We prove that the closure ordinals arising from the alternation-free fragment of modal mu-calculus (the syntactic class capturing Sigma_2 cap Pi_2) are bounded by omega^2. In this logic satisfaction can be characterised in terms of the existence of tableaux, trees generated by systematically breaking down formulae into their constituents according to the semantics of the calculus. To obtain optimal upper bounds we utilise the connection between closure ordinals of formulae and embedded order-types of the corresponding tableaux

On closure ordinals for the modal mu-calculus

The closure ordinal of a formula of modal mu-calculus mu X phi is the least ordinal kappa, if it exists, such that the denotation of the formula and the kappa-th iteration of the monotone operator induced by phi coincide across all transition systems (finite and infinite). It is known that for every alpha < omega^2 there is a formula phi of modal logic such that mu X phi has closure ordinal alpha (Czarnecki 2010). We prove that the closure ordinals arising from the alternation-free fragment of modal mu-calculus (the syntactic class capturing Sigma_2 cap Pi_2) are bounded by omega^2. In this logic satisfaction can be characterised in terms of the existence of tableaux, trees generated by systematically breaking down formulae into their constituents according to the semantics of the calculus. To obtain optimal upper bounds we utilise the connection between closure ordinals of formulae and embedded order-types of the corresponding tableaux

Finitary proof systems for Kozen’s μ.

We present three finitary cut-free sequent calculi for the modal [my]-calculus. Two of these derive annotated sequents in the style of Stirling’s ‘tableau proof system with names’ (4236) and feature special inferences that discharge open assumptions. The third system is a variant of Kozen’s axiomatisation in which cut is replaced by a strengthening of the v-induction inference rule. Soundness and completeness for the three systems is proved by establishing a sequence of embeddings between the calculi, starting at Stirling’s tableau-proofs and ending at the original axiomatisation of the [my]-calculus due to Kozen. As a corollary we obtain a completeness proof for Kozen’s axiomatisation which avoids the usual detour through automata or games

A Cyclic Proof System for Full Computation Tree Logic

Full Computation Tree Logic, commonly denoted CTL*, is the extension of Linear Temporal Logic LTL by path quantification for reasoning about branching time. In contrast to traditional Computation Tree Logic CTL, the path quantifiers are not bound to specific linear modalities, resulting in a more expressive language. We present a sound and complete hypersequent calculus for CTL*. The proof system is cyclic in the sense that proofs are finite derivation trees with back-edges. A syntactic success condition on non-axiomatic leaves guarantees soundness. Completeness is established by relating cyclic proofs to a natural ill-founded sequent calculus for the logic

Demystifying μ\mu

We develop the theory of illfounded and cyclic proof systems in the context of the modal $\mu$-calculus. A fine analysis of provability and admissibility bridges the finitary, cyclic and illfounded notions of proof for this logic and re-enforces the subtlety of two important normal form theorems: guardedness and disjunctiveness

Gravity in the 3+1-Split Formalism I: Holography as an Initial Value Problem

We present a detailed analysis of the 3+1-split formalism of gravity in the presence of a cosmological constant. The formalism helps revealing the intimate connection between holography and the initial value formulation of gravity. We show that the various methods of holographic subtraction of divergences correspond just to different transformations of the canonical variables, such that the initial value problem is properly set up at the boundary. The renormalized boundary energy momentum tensor is a component of the Weyl tensor.Comment: 28 pages; v2: minor improvements, references adde

Asymptotic generators of fermionic charges and boundary conditions preserving supersymmetry

We use a covariant phase space formalism to give a general prescription for defining Hamiltonian generators of bosonic and fermionic symmetries in diffeomorphism invariant theories, such as supergravities. A simple and general criterion is derived for a choice of boundary condition to lead to conserved generators of the symmetries on the phase space. In particular, this provides a criterion for the preservation of supersymmetries. For bosonic symmetries corresponding to diffeomorphisms, our prescription coincides with the method of Wald et al. We then illustrate these methods in the case of certain supergravity theories in $d=4$. In minimal AdS supergravity, the boundary conditions such that the supercharges exist as Hamiltonian generators of supersymmetry transformations are unique within the usual framework in which the boundary metric is fixed. In extended ${\mathcal N}=4$ AdS supergravity, or more generally in the presence of chiral matter superfields, we find that there exist many boundary conditions preserving ${\mathcal N}=1$ supersymmetry for which corresponding generators exist. These choices are shown to correspond to a choice of certain arbitrary boundary ``superpotentials,'' for suitably defined ``boundary superfields.'' We also derive corresponding formulae for the conserved bosonic charges, such as energy, in those theories, and we argue that energy is always positive, for any supersymmetry-preserving boundary conditions. We finally comment on the relevance and interpretation of our results within the AdS-CFT correspondence.Comment: 45 pages, Latex, no figures, v2: extended discussion of positive energy theorem and explicit form of fermionic generators, references adde