Weak MSO: Automata and Expressiveness Modulo Bisimilarity

We prove that the bisimulation-invariant fragment of weak monadic second-order logic (WMSO) is equivalent to the fragment of the modal $\mu$-calculus where the application of the least fixpoint operator $\mu p.\varphi$ is restricted to formulas $\varphi$ that are continuous in $p$. Our proof is automata-theoretic in nature; in particular, we introduce a class of automata characterizing the expressive power of WMSO over tree models of arbitrary branching degree. The transition map of these automata is defined in terms of a logic $\mathrm{FOE}_1^\infty$ that is the extension of first-order logic with a generalized quantifier $\exists^\infty$, where $\exists^\infty x. \phi$ means that there are infinitely many objects satisfying $\phi$. An important part of our work consists of a model-theoretic analysis of $\mathrm{FOE}_1^\infty$.Comment: Technical Report, 57 page

Scaling limit for subsystems and Doplicher-Roberts reconstruction

Given an inclusion $B \subset F$ of (graded) local nets, we analyse the structure of the corresponding inclusion of scaling limit nets $B_0 \subset F_0$, giving conditions, fulfilled in free field theory, under which the unicity of the scaling limit of $F$ implies that of the scaling limit of $B$. As a byproduct, we compute explicitly the (unique) scaling limit of the fixpoint nets of scalar free field theories. In the particular case of an inclusion $A \subset B$ of local nets with the same canonical field net $F$, we find sufficient conditions which entail the equality of the canonical field nets of $A_0$ and $B_0$.Comment: 31 page

On Soliton Automorphisms in Massive and Conformal Theories

For massive and conformal quantum field theories in 1+1 dimensions with a global gauge group we consider soliton automorphisms, viz. automorphisms of the quasilocal algebra which act like two different global symmetry transformations on the left and right spacelike complements of a bounded region. We give a unified treatment by providing a necessary and sufficient condition for the existence and Poincare' covariance of soliton automorphisms which is applicable to a large class of theories. In particular, our construction applies to the QFT models with the local Fock property -- in which case the latter property is the only input from constructive QFT we need -- and to holomorphic conformal field theories. In conformal QFT soliton representations appear as twisted sectors, and in a subsequent paper our results will be used to give a rigorous analysis of the superselection structure of orbifolds of holomorphic theories.Comment: latex2e, 20 pages. Proof of Thm. 3.14 corrected, 2 references added. Final version as to appear in Rev. Math. Phy

Unguarded Recursion on Coinductive Resumptions

We study a model of side-effecting processes obtained by starting from a monad modelling base effects and adjoining free operations using a cofree coalgebra construction; one thus arrives at what one may think of as types of non-wellfounded side-effecting trees, generalizing the infinite resumption monad. Correspondingly, the arising monad transformer has been termed the coinductive generalized resumption transformer. Monads of this kind have received some attention in the recent literature; in particular, it has been shown that they admit guarded iteration. Here, we show that they also admit unguarded iteration, i.e. form complete Elgot monads, provided that the underlying base effect supports unguarded iteration. Moreover, we provide a universal characterization of the coinductive resumption monad transformer in terms of coproducts of complete Elgot monads.Comment: 47 pages, extended version of http://www.sciencedirect.com/science/article/pii/S157106611500079

Fair Testing

In this paper we present a solution to the long-standing problem of characterising the coarsest liveness-preserving pre-congruence with respect to a full (TCSP-inspired) process algebra. In fact, we present two distinct characterisations, which give rise to the same relation: an operational one based on a De Nicola-Hennessy-like testing modality which we call should-testing, and a denotational one based on a refined notion of failures. One of the distinguishing characteristics of the should-testing pre-congruence is that it abstracts from divergences in the same way as Milner¿s observation congruence, and as a consequence is strictly coarser than observation congruence. In other words, should-testing has a built-in fairness assumption. This is in itself a property long sought-after; it is in notable contrast to the well-known must-testing of De Nicola and Hennessy (denotationally characterised by a combination of failures and divergences), which treats divergence as catrastrophic and hence is incompatible with observation congruence. Due to these characteristics, should-testing supports modular reasoning and allows to use the proof techniques of observation congruence, but also supports additional laws and techniques. Moreover, we show decidability of should-testing (on the basis of the denotational characterisation). Finally, we demonstrate its advantages by the application to a number of examples, including a scheduling problem, a version of the Alternating Bit-protocol, and fair lossy communication channel

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.Comment: To appear in SIAM J. Computin