Tanaka et al. proposed a type system for verifying functional correctness properties of programs that use arrays and pointer arithmetic. Their system extends ConSORT - a type system combining fractional ownership and refinement types for imperative program verification - with support for pointer arithmetic. Their idea was to extend fractional ownership so that it can depend on an array index. Their formulation, however, does not handle nested arrays, which are essential for representing practical data structures such as matrices. We extend Tanaka et al.’s type system to support nested arrays by generalizing the notion of ownership to be able to refer to the indices of the outer arrays and prove the soundness of the extended type system. We have implemented a verifier based on the proposed type system and demonstrated that it can verify the correctness of programs that manipulate nested arrays, which were beyond the reach of Tanaka et al.

Neural encoding and decoding provide perspectives for understanding neural representations of sensory inputs. Recent functional magnetic resonance imaging (fMRI) studies have succeeded in building prediction models for encoding and decoding numerous stimuli by representing a complex stimulus as a combination of simple elements. While arbitrary visual images were reconstructed using a modular model that combined the outputs of decoder modules for multiscale local image bases (elements), the shapes of the image bases were heuristically determined. In this work, we propose a method to establish mappings between the stimulus and the brain by automatically extracting modules from measured data. We develop a model based on Bayesian canonical correlation analysis, in which each module is modeled by a latent variable that relates a set of pixels in a visual image to a set of voxels in an fMRI activity pattern. The estimated mapping from a latent variable to pixels can be regarded as an image basis. We show that the model estimates a modular representation with spatially localized multiscale image bases. Further, using the estimated mappings, we derive encoding and decoding models that produce accurate predictions for brain activity and stimulus images. Our approach thus provides a novel means of revealing neural representations of stimuli by automatically extracting modules, which can be used to generate effective prediction models for encoding and decoding.

Image representation based on image bases provides a framework for understanding neural representation of visual perception. A recent fMRI study has shown that arbitrary contrast-defined visual images can be reconstructed from fMRI activity patterns using a combination of multi-scale local image bases. In the reconstruction model, the mapping from an fMRI activity pattern to the contrasts of the image bases was learned from measured fMRI responses to visual images. But the shapes of the images bases were fixed, and thus may not be optimal for reconstruction. Here, we propose a method to build a reconstruction model in which image bases are automatically extracted from the measured data. We constructed a probabilistic model that relates the fMRI activity space to the visual image space via a set of latent variables. The mapping from the latent variables to the visual image space can be regarded as a set of image bases. We found that spatially localized, multi-scale image bases were estimated near the fovea, and that the model using the estimated image bases was able to accurately reconstruct novel visual images. The proposed method provides a means to discover a novel functional mapping between stimuli and brain activity patterns.