From coinductive proofs to exact real arithmetic: theory and applications

Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. We discuss several examples including the extraction of programs for polynomials up to degree two and the definite integral of continuous maps

Minlog - A Tool for Program Extraction Supporting Algebras and Coalgebras

Minlog is an interactive proof system which implements prooftheoreticmethods and applies them to verication and program extraction.We give an overview of the system and demonstrate how it can beused to exploit the computational content in (co)algebraic proofs and todevelop correct and ecient programs

Proofs, Programs, Processes

The objective of this paper is to provide a theoretical foundation for program extraction from proofs. We give a realizability interpretation for first-order proofs involving inductive and coinductive definitions and discuss its application to the synthesis of provably correct programs. We show that realizers, although per-se untyped, can be assigned polymorphic recursive types and hence represent valid programs in a lazy functional programming language such as Haskell. Programs extracted from proofs using coinduction can be understood as perpetual processes producing infinite streams of data. Typical applications of such processes are computations in exact real arithmetic. As an example we show how to extract a program computing the average of two real numbers w.r.t.\ to the binary signed digit representation

From Coinductive Proofs to Exact Real Arithmetic

Abstract. We give a coinductive characterisation of the set of continuous functions defined on a compact real interval, and extract certified programs that construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. This is a pilot study in using proof-theoretic methods for obtaining certified algorithms in exact real arithmetic.

Putting the P in Ptilotus: a phosphorus-accumulating herb native to Australia

Background and Aims Ptilotus polystachyus (green mulla mulla; ptilotus) is a short-lived perennial herb that occurs widely in Australia in arid and semi-arid regions with nutrient poor soils. As this species shows potential for domestication, its response to addition of phosphorus (P) and nitrogen (N) was compared to a variety of the domesticated exotic perennial pasture herb Cichorium intybus (chicory), ‘Puna’. Methods Pots were filled with 3 kg of an extremely nutrient-deficient sterilized field soil that contained 3 mg kg−1 mineral N and 2 mg kg−1 bicarbonate-extractable P. The growth and P and N accumulation of ptilotus and chicory in response to seven rates of readily available phosphorus (0–300 mg P pot−1) and nitrogen (N) (0–270 mg N pot−1) was examined. Key Results Ptilotus grew extremely well under low P conditions: shoot dry weights were 23, 6 and 1·7 times greater than for chicory at the three lowest levels of P addition, 0, 15 and 30 mg P pot−1, respectively. Ptilotus could not downregulate P uptake. Concentrations of P in shoots approached 4 % of dry weight and cryo-scanning electron microscopy and X-ray microanalysis showed 35–196 mm of P in cell vacuoles in a range of tissues from young leaves. Ptilotus had a remarkable tolerance of high P concentrations in shoots. While chicory exhibited symptoms of P toxicity at the highest rate of P addition (300 mg P pot−1), no symptoms were present for ptilotus. The two species responded in a similar manner to addition of N. Conclusions In comparison to chicory, ptilotus demonstrated an impressive ability to grow well under conditions of low and high P availability. Further study of the mechanisms of P uptake and tolerance in ptilotus is warranted