Ecumenical modal logic

The discussion about how to put together Gentzen's systems for classical and intuitionistic logic in a single unified system is back in fashion. Indeed, recently Prawitz and others have been discussing the so called Ecumenical Systems, where connectives from these logics can co-exist in peace. In Prawitz' system, the classical logician and the intuitionistic logician would share the universal quantifier, conjunction, negation, and the constant for the absurd, but they would each have their own existential quantifier, disjunction, and implication, with different meanings. Prawitz' main idea is that these different meanings are given by a semantical framework that can be accepted by both parties. In a recent work, Ecumenical sequent calculi and a nested system were presented, and some very interesting proof theoretical properties of the systems were established. In this work we extend Prawitz' Ecumenical idea to alethic K-modalities

Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers

We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., $(x^n \in \mathbb{Q})$ and $(x^n \in \mathbb{Z})$. Our decision procedure combines computation over real algebraic cells with the rational root theorem and witness construction via algebraic number density arguments.Comment: To appear in CADE-25: 25th International Conference on Automated Deduction, 2015. Proceedings to be published by Springer-Verla

The Epsilon Calculus and Herbrand Complexity

Hilbert's epsilon-calculus is based on an extension of the language of predicate logic by a term-forming operator $\epsilon_{x}$. Two fundamental results about the epsilon-calculus, the first and second epsilon theorem, play a role similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.Comment: 23 p

Introduction to Milestones in Interactive Theorem Proving

On March 8, 2018, Tobias Nipkow celebrated his sixtieth birthday. In anticipation of the occasion, in January 2016, two of his former students, Gerwin Klein and Jasmin Blanchette, and one of his former postdocs, Andrei Popescu, approached the editorial board of the Journal of Automated Reasoning with a proposal to publish a surprise Festschrift issue in his honor. The e-mail was sent to twenty-six members of the board, leaving out one, for reasons that will become clear in a moment. It is a sign of the love and respect that Tobias commands from his colleagues that within two days every recipient of the e-mail had responded favorably and enthusiastically to the proposal

On the computational content of Zorn's lemma

We give a computational interpretation to an abstract instance of Zorn's lemma formulated as a wellfoundedness principle in the language of arithmetic in all finite types. This is achieved through G\"odel's functional interpretation, and requires the introduction of a novel form of recursion over non-wellfounded partial orders whose existence in the model of total continuous functionals is proven using domain theoretic techniques. We show that a realizer for the functional interpretation of open induction over the lexicographic ordering on sequences follows as a simple application of our main results

The New Deal for Communities experience: a final assessment - The New Deal for Communities evaluation: Final report – Volume 7

We show that for all integers $t\geq 8$ and arbitrarily small $\epsilon>0$, there exists a graph property $\Pi$ (which depends on $\epsilon$) such that $\epsilon$-testing $\Pi$ has non-adaptive query complexity $Q=\~{\Theta}(q^{2-2/t})$, where $q=\~{\Theta}(\epsilon^{-1})$ is the adaptive query complexity. This resolves the question of how beneficial adaptivity is, in the context of proximity-dependent properties (\cite{benefits-of-adaptivity}). This also gives evidence that the canonical transformation of Goldreich and Trevisan (\cite{canonical-testers}) is essentially optimal when converting an adaptive property tester to a non-adaptive property tester. To do so, we provide optimal adaptive and non-adaptive testers for the combined property of having maximum degree $O(\epsilon N)$ and being a \emph{blow-up collection} of an arbitrary base graph $H$.Comment: Keywords: Sublinear-Time Algorithms, Property Testing, Dense-Graph Model, Adaptive vs Nonadaptive Queries, Hierarchy Theore

From ductile to brittle: evolution and localization of deformation below a crustal detachment (Tinos, Cyclades, Greece)

International audienceThe Cycladic Oligo-Miocene detachment of Tinos island is an example of a flat-lying extensional shear zone evolving into a low-angle brittle detachment. A clear continuum of extensional strain from ductile to brittle regime is observed in the footwall. The main brittle structures marking extension are shallow- and steeply dipping normal faults associated with subvertical extensional joints and veins. The earliest brittle structures are lowangle normal faults which commonly superimpose on, and reactivate, earlier (precursory) ductile shear bands, but newly formed low-angle normal faults could also be observed. Low-angle normal faults are cut by late steeply dipping normal faults. The inversion of fault slip data collected within, and away from, the main detachment zone shows that the direction of the minimum stress axis is strictly parallel to the NE-SW stretching lineation and that the maximum principal stress axis remained subvertical during the whole brittle evolution, in agreement with the subvertical attitude of veins throughout the island. The high angle of s1 to the main detachment suggests that the detachment was weak. This observation, together with the presence of a thick layer of cataclasites below the main detachment and the kinematic continuum from ductile to brittle, leads us to propose a kinematic model for the formation of the detachment. Boudinage at the crustal scale induces formation, near the brittle-ductile transition, of ductile shear zones near the edges of boudins. Shear zones are progressively exhumed and replaced by shallowdipping cataclastic shear zones when they reached the brittle field. Most of the displacement is achieved through cataclastic flow in the upper crust and only the last increment of strain gives rise to the formation of brittle faults. The formation of the low-angle brittle detachment is thus ''prepared'' by the ductile shear zone and the cataclasites and favored by the circulation of surface-derived fluids in the shear zone

A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint

A cytosolic invertase is required for normal growth and cell development in the model legume, Lotus japonicus

Neutral/alkaline invertases are a subgroup, confined to plants and cyanobacteria, of a diverse family of enzymes. A family of seven closely-related genes, LjINV1–LjINV7, is described here and their expression in the model legume, Lotus japonicus, is examined. LjINV1 previously identified as encoding a nodule-enhanced isoform is the predominant isoform present in all parts of the plant. Mutants for two isoforms, LjINV1 and LjINV2, were isolated using TILLING. A premature stop codon allele of LjINV2 had no effect on enzyme activity nor did it show a visible phenotype. For LjINV1, premature stop codon and missense mutations were obtained and the phenotype of the mutants examined. Recovery of homozygous mutants was problematic, but their phenotype showed a severe reduction in growth of the root and the shoot, a change in cellular development, and impaired flowering. The cellular organization of both roots and leaves was altered; leaves were smaller and thicker with extra layers of cells and roots showed an extended and broader zone of cell division. Moreover, anthers contained no pollen. Both heterozygotes and homozygous mutants showed decreased amounts of enzyme activity in nodules and shoot tips. Shoot tips also contained up to a 9-fold increased level of sucrose. However, mutants were capable of forming functional root nodules. LjINV1 is therefore crucial to whole plant development, but is clearly not essential for nodule formation or function