Strong Klee-And\^o Theorems through an Open Mapping Theorem for cone-valued multi-functions

A version of the classical Klee-And\^o Theorem states the following: For every Banach space $X$, ordered by a closed generating cone $C\subseteq X$, there exists some $\alpha>0$ so that, for every $x\in X$, there exist $x^{\pm}\in C$ so that $x=x^{+}-x^{-}$ and $\|x^{+}\|+\|x^{-}\|\leq\alpha\|x\|$. The conclusion of the Klee-And\^o Theorem is what is known as a conormality property. We prove stronger and somewhat more general versions of the Klee-And\^o Theorem for both conormality and coadditivity (a property that is intimately related to conormality). A corollary to our result shows that the functions $x\mapsto x^{\pm}$, as above, may be chosen to be bounded, continuous, and positively homogeneous, with a similar conclusion yielded for coadditivity. Furthermore, we show that the Klee-And\^o Theorem generalizes beyond ordered Banach spaces to Banach spaces endowed with arbitrary collections of cones. Proofs of our Klee-And\^o Theorems are achieved through an Open Mapping Theorem for cone-valued multi-functions/correspondences. We very briefly discuss a potential further strengthening of The Klee-And\^o Theorem beyond what is proven in this paper, and motivate a conjecture that there exists a Banach space $X$, ordered by a closed generating cone $C\subseteq X$, for which there exist no Lipschitz functions $(\cdot)^{\pm}:X\to C$ satisfying $x=x^{+}-x^{-}$ for all $x\in X$.Comment: Major rewrite. Large parts were removed which a referee pointed out can be proven through much easier method

The Use of Symbolic Execution for Testing of Real-Time Safety-Related Software

Tato práce zkoumá vhodnost nástroje symbolické exekuce KLEE pro verifikaci real-time bezpečnostně kritických systémů. Real-time bezpečnostně kritické systémy jsou ty, u kterých selhání může vést ke ztrátě lidských životů anebo ke škodám na životním prostředí. V této práci modifikuji dva real-time bezpečnostně kritické systémy, abych je mohl otestovat s KLEE a posoudit, jak komplikovaná jejich modifikace byla a zda má smysl začít používat KLEE k verifikaci real-time bezpečnostně kritických systémů. Došel jsem k závěru, že KLEE může být cenný nástroj pro verifikaci real-time bezpečnostně kritických systémů, ale daný systém musí být navržen s ohledem na KLEE.This thesis investigates fitness of symbolic execution tool KLEE for verification of real-time safety-critical systems. Real-time safety-critical systems are those systems, whose malfunction might result in loss of life and/or environmental damage. In this thesis I modify two pieces of real-time safety-critical software to test them with KLEE and to evaluate how complex the modification was and whether using KLEE for verification of real-time safety-critical systems is viable going forward. I conclude that KLEE can be a valuable tool for verifying real-time safety-critical software, but the software has to be designed with KLEE in mind

Note on Bessaga-Klee classification

We collect several variants of the proof of the third case of the Bessaga-Klee relative classification of closed convex bodies in topological vector spaces. We were motivated by the fact that we have not found anywhere in the literature a complete correct proof. In particular, we point out an error in the proof given in the book of C.~Bessaga and A.~Pe\l czy\'nski (1975). We further provide a simplified version of T.~Dobrowolski's proof of the smooth classification of smooth convex bodies in Banach spaces which works simultaneously in the topological case.Comment: 14 pages; we made few corrections, added one reference and precised the abstrac