Canonical proof nets for classical logic

Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cut-elimination procedure which preserves correctness. Previous attempts to give proof-net-like objects for propositional classical logic have failed at least one of the above conditions. In Richard McKinley (2010) [22], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK∗ in this paper, is a novel one-sided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a self-contained extended version of Richard McKinley (2010) [22]), we give a full proof of (c) for expansion nets with respect to LK∗, and in addition give a cut-elimination procedure internal to expansion nets – this makes expansion nets the first notion of proof-net for classical logic satisfying all four criteria.

Two-dimensional Lattice Gauge Theories with Superconducting Quantum Circuits

A quantum simulator of U(1) lattice gauge theories can be implemented with superconducting circuits. This allows the investigation of confined and deconfined phases in quantum link models, and of valence bond solid and spin liquid phases in quantum dimer models. Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well. Here we show how state-of-the-art superconducting technology allows us to simulate these phenomena in relatively small circuit lattices. By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions. We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories. The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.

Proof internalization in generalized Frege systems for classical logic

We present a general method for inserting proofs in Frege systems for classical logic that produces systems that can internalize their own proofs.

Censors for Boolean Description Logic

Protecting different kinds of information has become an important area of research. One aspect is to provide effective means to avoid that secrets can be deduced from the answers of legitimate queries. In the context of atomic propositional databases several methods have been developed to achieve this goal. However, in those databases it is not possible to formalize structural information. Also they are quite restrictive with respect to the specification of secrets. In this paper we extend those methods to match the much greater expressive power of Boolean description logics. In addition to the formal framework, we provide a discussion of various kinds of censors and establish different levels of security they can provide.

Neural circuits of emotion regulation: a comparison of mindfulness-based and cognitive reappraisal strategies

Dealing with one's emotions is a core skill in everyday life. Effective cognitive control strategies have been shown to be neurobiologically represented in prefrontal structures regulating limbic regions. In addition to cognitive strategies, mindfulness-associated methods are increasingly applied in psychotherapy. We compared the neurobiological mechanisms of these two strategies, i.e. cognitive reappraisal and mindfulness, during both the cued expectation and perception of negative and potentially negative emotional pictures. Fifty-three healthy participants were examined with functional magnetic resonance imaging (47 participants included in analysis). Twenty-four subjects applied mindfulness, 23 used cognitive reappraisal. On the neurofunctional level, both strategies were associated with comparable activity of the medial prefrontal cortex and the amygdala. When expecting negative versus neutral stimuli, the mindfulness group showed stronger activations in ventro- and dorsolateral prefrontal cortex, supramarginal gyrus as well as in the left insula. During the perception of negative versus neutral stimuli, the two groups only differed in an increased activity in the caudate in the cognitive group. Altogether, both strategies recruited overlapping brain regions known to be involved in emotion regulation. This result suggests that common neural circuits are involved in the emotion regulation by mindfulness-based and cognitive reappraisal strategies. Identifying differential activations being associated with the two strategies in this study might be one step towards a better understanding of differential mechanisms of change underlying frequently used psychotherapeutic interventions.