We study the differential properties of higher-order statistical probabilistic programs with recursion and conditioning. Our starting point is an open problem posed by Hongseok Yang: what class of statistical probabilistic programs have densities that are differentiable almost everywhere? To formalise the problem, we consider Statistical PCF (SPCF), an extension of call-by-value PCF with real numbers, and constructs for sampling and conditioning. We give SPCF a sampling-style operational semantics à la Borgström et al., and study the associated weight (commonly referred to as the density) function and value function on the set of possible execution traces.Our main result is that almost surely terminating SPCF programs, generated from a set of primitive functions (e.g. the set of analytic functions) satisfying mild closure properties, have weight and value functions that are almost everywhere differentiable. We use a stochastic form of symbolic execution to reason about almost everywhere differentiability. A by-product of this work is that almost surely terminating deterministic (S)PCF programs with real parameters denote functions that are almost everywhere differentiable.Our result is of practical interest, as almost everywhere differentiability of the density function is required to hold for the correctness of major gradient-based inference algorithms.

We present a simple resolution proof system for higher-order constrained Horn clauses (HoCHC)-a system of higher-order logic modulo theories-and prove its soundness and refutational completeness w.r.t. both standard and Henkin semantics. As corollaries, we obtain the compactness theorem and semi-decidability of HoCHC for semi-decidable background theories, and we prove that HoCHC satisfies a canonical model property. Moreover a variant of the well-known translation from higher-order to 1st-order logic is shown to be sound and complete for HoCHC in both semantics. We illustrate how to transfer decidability results for (fragments of) 1st-order logic modulo theories to our higher-order setting, using as example the Bernays-Schönfinkel-Ramsey fragment of HoCHC modulo a restricted form of Linear Integer Arithmetic.

This paper is a study of high-throughput filter structures such as block structures and their behavior in finite precision environments. Block structures achieve high throughput rates by using a large number of processors working in parallel. It has been believed that block structures which are relatively robust to round-off noise must also be robust to coefficient quantization errors. However, our research has shown that block structures, in fact, have high coefficient sensitivity. A potential problem that arises as a result of coefficient quantization is a periodically time-varying behavior exhibited by the realized filter. We will demonstrate how finite wordlength errors can change a nominally time-invariant filter into a time-varying system. We will identify the block structures that have low coefficient sensitivity, and develop high-speed structures that are immune to the time-varying problems caused by coefficient quantization.

An experimental .r-y measurement system is described that was designed for the high-speed. high-precision measurements required in integrated circuit manufacturing and for optical measurement applications in which a sufficiently large data base is required for statistical process analysis. The technology for this experimental system differs considerably from that of conventional optical measuring systems in current use and utilizes a computer for data acquisition. manipulation and evaluation. The system, utilizing the edge detection principle, presently operates at a measuring speed of 2.5 cm/s. An analysis gives both the short-term and the long-term precision of the system. The standard deviation for the short-term precision is 0.038 iuc:

We present initial limit Datalog, a new extensible class of constrained Horn clauses for which the satisfiability problem is decidable. The class may be viewed as a generalisation to higher-order logic (with a simple restriction on types) of the first-order language limit Datalog Z (a fragment of Datalog modulo linear integer arithmetic), but can be instantiated with any suitable background theory. For example, the fragment is decidable over any countable well-quasi-order with a decidable first-order theory, such as natural number vectors under componentwise linear arithmetic, and words of a bounded, context-free language ordered by the subword relation. Formulas of initial limit Datalog have the property that, under some assumptions on the background theory, their satisfiability can be witnessed by a new kind of term model which we call entwined structures. Whilst the set of all models is typically uncountable, the set of all entwined structures is recursively enumerable, and model checking is decidable.