Erratum: Indistinguishability of independent single photons [Phys. Rev. A 79, 013824 (2009)]

The number of cotyledons in angiosperm monocots and dicots is tightly constrained. But in the gymnosperm Pinaceae, including conifers, cotyledon number (n c ) can vary widely, commonly between 2 to 12. Conifer cotyledons form in whorled rings, on a domed embryo geometry. We measured embryo diameters and counted cotyledons to determine the radial positioning of the whorl and the circumferential spacing between cotyledons. Results were similar between Douglas fir (Pseudotsuga), Sitka spruce (Picea) and larch (Larix), indicating a common mechanism for cotyledon positioning in conifers. Disrupting transport of the growth regulator auxin (with NPA) led to cup-shaped embryos, indicating that whorl (ring) formation is separable from cotyledon patterning within the ring. NPA inhibits cotyledon outgrowth, but not the spacing (distance) between cotyledons. The NPA effect is direct; it does not operate indirectly on embryo size. These results support a hierarchical model for cotyledon positioning in conifers, in which a first stage (not requiring auxin transport) sets the whorl position, constraining the second stage (which requires auxin transport) to form cotyledons within this whorl. Similarly, recent studies in Arabidopsis have shown that different components of complex developmental patterns can have different transport properties; this aspect of patterning may be shared across plants.

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Current static verification techniques do not provide good support for incrementality, making it difficult for developers to focus on specifying and verifying the properties and components that are most important. Dynamic verification approaches support incrementality, but cannot provide static guarantees. To bridge this gap, prior work proposed gradual verification, which supports incrementality by allowing every assertion to be complete, partial, or omitted, and provides sound verification that smoothly scales from dynamic to static checking. The prior approach to gradual verification, however, was limited to programs without recursive data structures. This paper extends gradual verification to programs that manipulate recursive, mutable data structures on the heap. We address several technical challenges, such as semantically connecting iso-and equi-recursive interpretations of abstract predicates, and supporting gradual verification of heap ownership. This work thus lays the foundation for future tools that work on realistic programs and support verification within an engineering process in which cost-benefit trade-offs can be made.

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