Failure of Cut-Elimination in the Cyclic Proof System of Bunched Logic with Inductive Propositions

Cyclic proof systems are sequent-calculus style proof systems that allow circular structures representing induction, and they are considered suitable for automated inductive reasoning. However, Kimura et al. have shown that the cyclic proof system for the symbolic heap separation logic does not satisfy the cut-elimination property, one of the most fundamental properties of proof systems. This paper proves that the cyclic proof system for the bunched logic with only nullary inductive predicates does not satisfy the cut-elimination property. It is hard to adapt the existing proof technique chasing contradictory paths in cyclic proofs since the bunched logic contains the structural rules. This paper proposes a new proof technique called proof unrolling. This technique can be adapted to the symbolic heap separation logic, and it shows that the cut-elimination fails even if we restrict the inductive predicates to nullary ones

Cut-Elimination for Cyclic Proof Systems with Inductively Defined Propositions (Theory and Applications of Proof and Computation)

Cyclic proof systems are extensions of the sequent-calculus style proof systems for logics with inductively defined predicates. In cyclic proof systems, inductive reasoning is realized as cyclic structures in proof trees. It has been already known that the cut-elimination property does not hold for the cyclic proof systems of some logics such as the first-order predicate logic and the separation logic. In this paper, we consider the cyclic proof systems with inductively defined propositions (that is, nullary predicates), and prove that the cut-elimination holds for the propositional logic, and it does not hold for the bunched logic

Cut elimination for propositional cyclic proof systems with fixed-point operators

Infinitary and cyclic proof systems are proof systems for logical formulas with fixed-point operators or inductive definitions. A cyclic proof system is a restriction of the corresponding infinitary proof system. Hence, these proof systems are generally not the same, as in the cyclic system may be weaker than the infinitary system. For several logics, the infinitary proof systems are shown to be cut-free complete. However, cyclic proof systems are characterized with many unknown problems on the (cut-free) completeness or the cut-elimination property. In this study, we show that the provability of infinitary and cyclic proof systems are the same for some propositional logics with fixed-point operators or inductive definitions and that the cyclic proof systems are cut-free complete

Suzaku Observations of PSR B1259-63: A New Manifestation of Relativistic Pulsar Wind

We observed PSR B1259-63, a young non-accreting pulsar orbiting around a Be star SS 2883, eight times with the Suzaku satellite in 2007, to characterize the X-ray emission arising from the interaction between a pulsar relativistic wind and Be star outflows. The X-ray spectra showed a featureless continuum in 0.6-10 keV, modeled by a power law with a wide range of photon index 1.3-1.8. When combined with the Suzaku PIN detector which allowed spectral analysis in the hard 15-50 keV band, X-ray spectra show a break at 5 keV in a certain epoch. Regarding the system as a compactified pulsar wind nebula, in which e+e- pairs are assumed to be accelerated at the inner shock front of the pulsar wind, we attribute the X-ray spectral break to the low-energy cutoff of the synchrotron radiation associated with the Lorentz factor of the relativistic pulsar wind gamma_1 = 4x10^5. Our result indicates that Comptonization of stellar photons by the unshocked pulsar wind will be accessible (or tightly constrained) by observations with the Fermi Gamma-ray Space Telescope during the next periastron passage. The PSR B1259-63 system allows us to probe the fundamental properties of the pulsar wind by a direct means, being complementary to the study of large-scale pulsar wind nebulae.Comment: 11 pages, 9 figures, accepted for publication in The Astrophysical Journa

Monadic translation of classical sequent calculus

International audienceWe study monadic translations of the call-by-name (cbn) and call-by-value (cbv) fragments of the classical sequent calculus ${\overline{\lambda}\mu\tilde{\mu}}$ due to Curien and Herbelin, and give modular and syntactic proofs of strong normalisation. The target of the translations is a new meta-language for classical logic, named monadic λμ. This language is a monadic reworking of Parigot's λμ-calculus, where the monadic binding is confined to commands, thus integrating the monad with the classical features. Also, its μ-reduction rule is replaced by a rule expressing the interaction between monadic binding and μ-abstraction.Our monadic translations produce very tight simulations of the respective fragments of ${\overline{\lambda}\mu\tilde{\mu}}$ within monadic λμ, with reduction steps of ${\overline{\lambda}\mu\tilde{\mu}}$ being translated in a 1–1 fashion, except for β steps, which require two steps. The monad of monadic λμ can be instantiated to the continuations monad so as to ensure strict simulation of monadic λμ within simply typed λ-calculus with β- and η-reduction. Through strict simulation, the strong normalisation of simply typed λ-calculus is inherited by monadic λμ, and then by cbn and cbv ${\overline{\lambda}\mu\tilde{\mu}}$, thus reproving strong normalisation in an elementary syntactical way for these fragments of ${\overline{\lambda}\mu\tilde{\mu}}$, and establishing it for our new calculus. These results extend to second-order logic, with polymorphic λ-calculus as the target, giving new strong normalisation results for classical second-order logic in sequent calculus style.CPS translations of cbn and cbv ${\overline{\lambda}\mu\tilde{\mu}}$ with the strict simulation property are obtained by composing our monadic translations with the continuations-monad instantiation. In an appendix to the paper, we investigate several refinements of the continuations-monad instantiation in order to obtain in a modular way improvements of the CPS translations enjoying extra properties like simulation by cbv β-reduction or reduction of administrative redexes at compile time

Thalidomide Prevents the Progression of Peritoneal Fibrosis in Mice

Thalidomide is clinically recognized as a therapeutic agent for multiple myeloma and has been known to exert anti-angiogenic actions. Recent studies have suggested the involvement of angiogenesis in the progression of peritoneal fibrosis. The present study investigated the effects of thalidomide on the development of peritoneal fibrosis induced by injection of chlorhexidine gluconate (CG) into the mouse peritoneal cavity every other day for 3 weeks. Thalidomide was given orally every day. Peritoneal tissues were dissected out 21 days after CG injection. Expression of CD31 (as a marker of endothelial cells), proliferating cell nuclear antigen (PCNA), vascular endothelial growth factor (VEGF), α-smooth muscle actin (as a marker of myofibroblasts), type III collagen and transforming growth factor (TGF)-β was examined using immunohistochemistry. CG group showed thickening of the submesothelial zone and increased numbers of vessels and myofibroblasts. Large numbers of VEGF-, PCNA-, and TGF-β-positive cells were observed in the submesothelial area. Thalidomide treatment significantly ameliorated submesothelial thickening and angiogenesis, and decreased numbers of PCNA- and VEGF-expressing cells, myofibroblasts, and TGF-β-positive cells. Moreover, thalidomide attenuated peritoneal permeability for creatinine, compared to the CG group. Our results indicate the potential utility of thalidomide for preventing peritoneal fibrosis