Completeness of a Hypersequent Calculus for Some First-order Gödel Logics with Delta

All first-order Gödel logics G_V with globalization operator based on truth value sets V C [0,1] where 0 and 1 lie in the perfect kernel of V are axiomatized by Ciabattoni’s hypersequent calculus HGIF

Foundation of Computer (Algebra) ANALYSIS Systems: Semantics, Logic, Programming, Verification

We propose a semantics of operating on real numbers that is sound, Turing-complete, and practical. It modifies the intuitive but super-recursive Blum-Shub-Smale model (formalizing Computer ALGEBRA Systems), to coincide in power with the realistic but inconvenient Type-2 Turing machine underlying Computable Analysis: reconciling both as foundation to a Computer ANALYSIS System. Several examples illustrate the elegance of rigorous numerical coding in this framework, formalized as a simple imperative programming language ERC with denotational semantics for REALIZING a real function $f$: arguments $x$ are given as exact real numbers, while values $y=f(x)$ suffice to be returned approximately up to absolute error $2^p$ with respect to an additionally given integer parameter $p\to-\infty$. Real comparison (necessarily) becomes partial, possibly 'returning' the lazy Kleenean value UNKNOWN (subtly different from $\bot$ for classically undefined expressions like 1/0). This asserts closure under composition, and in fact 'Turing-completeness over the reals': All and only functions computable in the sense of Computable Analysis can be realized in ERC. Programs thus operate on a many-sorted structure involving real numbers and integers, the latter connected via the 'error' embedding $Z\ni p\mapsto 2^p\in R$, whose first-order theory is proven decidable and model-complete. This logic serves for formally specifying and formally verifying correctness of ERC programs

First-order Goedel logics

First-order Goedel logics are a family of infinite-valued logics where the sets of truth values V are closed subsets of [0, 1] containing both 0 and 1. Different such sets V in general determine different Goedel logics G_V (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that G_V is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Goedel logics are also characterized.Comment: 37 page

Deciding logics of linear Kripke frames with scattered end pieces

We show that logics based on linear Kripke frames – with or without constant domains – that have a scattered end piece are not recursively enumerable.This is done by reduction to validity in all finite classical models

Introduction to R and Time Series (Study of Mathematical Software and Its Effective Use for Mathematics Education)

We give a short introduction to R., and discuss how to deal with tim series based data using R by providing examples from network data analysis