From Nonstandard Analysis to various flavours of Computability Theory

As suggested by the title, it has recently become clear that theorems of Nonstandard Analysis (NSA) give rise to theorems in computability theory (no longer involving NSA). Now, the aforementioned discipline divides into classical and higher-order computability theory, where the former (resp. the latter) sub-discipline deals with objects of type zero and one (resp. of all types). The aforementioned results regarding NSA deal exclusively with the higher-order case; we show in this paper that theorems of NSA also give rise to theorems in classical computability theory by considering so-called textbook proofs.Comment: To appear in the proceedings of TAMC2017 (http://tamc2017.unibe.ch/

Completeness for a First-order Abstract Separation Logic

Existing work on theorem proving for the assertion language of separation logic (SL) either focuses on abstract semantics which are not readily available in most applications of program verification, or on concrete models for which completeness is not possible. An important element in concrete SL is the points-to predicate which denotes a singleton heap. SL with the points-to predicate has been shown to be non-recursively enumerable. In this paper, we develop a first-order SL, called FOASL, with an abstracted version of the points-to predicate. We prove that FOASL is sound and complete with respect to an abstract semantics, of which the standard SL semantics is an instance. We also show that some reasoning principles involving the points-to predicate can be approximated as FOASL theories, thus allowing our logic to be used for reasoning about concrete program verification problems. We give some example theories that are sound with respect to different variants of separation logics from the literature, including those that are incompatible with Reynolds's semantics. In the experiment we demonstrate our FOASL based theorem prover which is able to handle a large fragment of separation logic with heap semantics as well as non-standard semantics.Comment: This is an extended version of the APLAS 2016 paper with the same titl

Soundness and completeness proofs by coinductive methods

We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems for various flavors of first-order logic. Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isabelle/HOL’s recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL’s new coinductive specification language such as nesting through non-free types and mixed recursion–corecursion

UVSSA and USP7, a new couple in transcription-coupled DNA repair

Transcription-coupled nucleotide excision repair (TC-NER) specifically removes transcription-blocking lesions from our genome. Defects in this pathway are associated with two human disorders: Cockayne syndrome (CS) and UV-sensitive syndrome (UVSS). Despite a similar cellular defect in the UV DNA damage response, patients with these syndromes exhibit strikingly distinct symptoms; CS patients display severe developmental, neurological, and premature aging features, whereas the phenotype of UVSS patients is mostly restricted to UV hypersensitivity. The exact molecular mechanism behind these clinical differences is still unknown; however, they might be explained by additional functions of CS proteins beyond TC-NER. A short overview of the current hypotheses addressing possible molecular mechanisms and the proteins involved are presented in this review. In addition, we will focus on two new players involved in TC-NER which were recently identified: UV-stimulated scaffold protein A (UVSSA) and ubiquitin-specific protease 7 (USP7). UVSSA has been found to be the causative gene for UVSS and, together with USP7, is implicated in regulating TC-NER activity. We will discuss the function of UVSSA and USP7 and how the discovery of these proteins contributes to a better understanding of the molecular mechanisms underlying the clinical differences between UVSS and the more severe CS

Constructivism in mathematics

Studies in Logic and the Foundations of Mathematics, Volume 123: Constructivism in Mathematics: An Introduction, Vol. II focuses on various studies in mathematics and logic, including metric spaces, polynomial rings, and Heyting algebras.The publication first takes a look at the topology of metric spaces, algebra, and finite-type arithmetic and theories of operators. Discussions focus on intuitionistic finite-type arithmetic, theories of operators and classes, rings and modules, linear algebra, polynomial rings, fields and local rings, complete separable metric spaces, and located sets. The t

Constructivism in mathematics: an introduction

Constructivism in Mathematics Vol.