Some new results on decidability for elementary algebra and geometry

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the AE fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbert's 10th problem show that the EA fragments for metric and normed spaces and the AE fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3

Double Negation Semantics for Generalisations of Heyting Algebras

This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko. We view these as variant semantics and present a semantic formulation of Troelstra’s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we demonstrate that failure via specific finite examples. Using these, we prove that the syntactic version of these translations will fail to satisfy Troelstra’s criteria in the corresponding substructural logical setting

A General Framework for Sound and Complete Floyd-Hoare Logics

This paper presents an abstraction of Hoare logic to traced symmetric monoidal categories, a very general framework for the theory of systems. Our abstraction is based on a traced monoidal functor from an arbitrary traced monoidal category into the category of pre-orders and monotone relations. We give several examples of how our theory generalises usual Hoare logics (partial correctness of while programs, partial correctness of pointer programs), and provide some case studies on how it can be used to develop new Hoare logics (run-time analysis of while programs and stream circuits).Comment: 27 page

Self-Formalisation of Higher-Order Logic: Semantics, Soundness, and a Verified Implementation

This is the final version of the article. It first appeared from Springer via http://dx.doi.org/10.1007/s10817-015-9357-xWe present a mechanised semantics for higher-order logic (HOL), and a proof of soundness for the inference system, including the rules for making definitions, implemented by the kernel of the HOL Light theorem prover. Our work extends Harrison’s verification of the inference system without definitions. Soundness of the logic extends to soundness of a theorem prover, because we also show that a synthesised implementation of the kernel in CakeML refines the inference system. Apart from adding support for definitions and synthesising an implementation, we improve on Harrison’s work by making our model of HOL parametric on the universe of sets, and we prove soundness for an improved principle of constant specification in the hope of encouraging its adoption. Our semantics supports defined constants directly via a context, and we find this approach cleaner than our previous work formalising Wiedijk’s Stateless HOL.The first author was supported by the Gates Cambridge Trust. The second author was funded in part by the EPSRC (grant number EP/K503769/1). The third author was partially supported by the Royal Society UK and the Swedish Research Council

Heteropogon ‐ Themeda grasses evolve to occupy either tropical grassland or wetland biomes

Species of the Heteropogon‐Themeda clade are ecologically important grasses distributed across the tropics, including widespread species, such as the pantropical Heteropogon contortus and Themeda triandra, and range‐restricted species such as Heteropogon ritchiei and Themeda anathera. Here, we examine habitat preferences of the grassland/savanna and wetland species by describing bioclimatic niche characteristics, characterizing functional traits, and investigating the evolution of functional traits of 31 species in the Heteropogon‐Themeda clade in relation to precipitation and temperature. The climatic limits of the clade are linked to mean annual precipitation and seasonality that also distinguish seven wetland species from 24 grassland/savanna species. Tests of niche equivalency highlighted the unique bioclimatic niche of the wetland species. However, climatic factors do not fully explain species geographic range, and other factors are likely to contribute to their distribution ranges. Trait analyses demonstrated that the wetland and grassland/savanna species were separated by culm height, leaf length, leaf area, awn length, and awn types. Phylogenetic analyses showed that the wetland species had tall stature with long and large leaves and lack of hygroscopic awns, which suggest selective pressures in the shift between savanna/grassland and wetland. The two most widespread species, H. contortus and T. triandra, have significantly different bioclimatic niches, but we also found that climatic niche alone does not explain the current geographic distributions of H. contortus and T. triandra. Our study provides a new understanding of the biogeography and evolutionary history of an ecologically important clade of C4 tropical grasses