Completeness and Incompleteness of Synchronous Kleene Algebra

Synchronous Kleene algebra (SKA), an extension of Kleene algebra (KA), was proposed by Prisacariu as a tool for reasoning about programs that may execute synchronously, i.e., in lock-step. We provide a countermodel witnessing that the axioms of SKA are incomplete w.r.t. its language semantics, by exploiting a lack of interaction between the synchronous product operator and the Kleene star. We then propose an alternative set of axioms for SKA, based on Salomaa's axiomatisation of regular languages, and show that these provide a sound and complete characterisation w.r.t. the original language semantics.Comment: Accepted at MPC 201

Certification of Compiler Optimizations using Kleene Algebra with Tests

We use Kleene algebra with tests to verify a wide assortment of common compiler optimizations, including dead code elimination, common subexpression elimination, copy propagation, loop hoisting, induction variable elimination, instruction scheduling, algebraic simplification, loop unrolling, elimination of redundant instructions, array bounds check elimination, and introduction of sentinels. In each of these cases, we give a formal equational proof of the correctness of the optimizing transformation

Compositional closure for Bayes Risk in probabilistic noninterference

We give a sequential model for noninterference security including probability (but not demonic choice), thus supporting reasoning about the likelihood that high-security values might be revealed by observations of low-security activity. Our novel methodological contribution is the definition of a refinement order and its use to compare security measures between specifications and (their supposed) implementations. This contrasts with the more common practice of evaluating the security of individual programs in isolation. The appropriateness of our model and order is supported by our showing that our refinement order is the greatest compositional relation --the compositional closure-- with respect to our semantics and an "elementary" order based on Bayes Risk --- a security measure already in widespread use. We also relate refinement to other measures such as Shannon Entropy. By applying the approach to a non-trivial example, the anonymous-majority Three-Judges protocol, we demonstrate by example that correctness arguments can be simplified by the sort of layered developments --through levels of increasing detail-- that are allowed and encouraged by compositional semantics

Testing the Equivalence of Regular Languages

The minimal deterministic finite automaton is generally used to determine regular languages equality. Antimirov and Mosses proposed a rewrite system for deciding regular expressions equivalence of which Almeida et al. presented an improved variant. Hopcroft and Karp proposed an almost linear algorithm for testing the equivalence of two deterministic finite automata that avoids minimisation. In this paper we improve the best-case running time, present an extension of this algorithm to non-deterministic finite automata, and establish a relationship between this algorithm and the one proposed in Almeida et al. We also present some experimental comparative results. All these algorithms are closely related with the recent coalgebraic approach to automata proposed by Rutten

Differential Hoare Logics and Refinement Calculi for Hybrid Systems with Isabelle/HOL

We present simple new Hoare logics and refinement calculi for hybrid systems in the style of differential dynamic logic. (Refinement) Kleene algebra with tests is used for reasoning about the program structure and generating verification conditions at this level. Lenses capture hybrid program stores in a generic algebraic way. The approach has been formalised with the Isabelle/HOL proof assistant. A number of examples explains the workflow with the resulting verification components

Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions

Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here, we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chromatic polynomial and the zero-temperature partition function of the Potts antiferromagnet on the same graph. Implementing this bottom-up algorithm using appropriate computer algebra, the new method outperforms standard top-down methods by several orders of magnitude, already for moderately sized graphs. As a first application, we compute chromatic polynomials of samples of the simple cubic lattice, for the first time computationally accessing three-dimensional lattices of physical relevance. The method offers straightforward generalizations to several other counting problems.Comment: 7 pages, 4 figure

Respostas do cafeeiro à calagem

Field experiments were conducted in a stablished coffee plantation for 8 years (1975-82) on Brazilian Oxisols to investigate the effects of liming these soils (0.0,2.5, 5 and 10 tons/ha) on the soil chemical properties, production and mineral nutrition of coffee (Coffea arabica L.) trees. The soil pH, cation exchange capacity (CEC) and exchangeable Ca and Mg increased, while exchangeable AI and K decreased whith increasing dolomitic lime rates. The lime effects were limited to the top soil only (0-30 cm). Liming significantly increased leaf Ca and Mg, reduced leaf Mn (eliminated the toxic effects), Zn and K, and had no effect on leaf N, P, and Cu in the leaves. The yields of coffee were increased by reducing the exchangeable Al and by adjusting the Ca-K, Ca-Mg, and Mg-K ratios to 13:1, 4:1 and 3:1, respectively. The best coffee yields were obtained with the lowest lime rate (2.5 tons/ha). Higher lime rates (5 and 10 tons/ha) resulted in decreased yields.Experimentos de campo foram conduzidos em dois dos principais solos da região cafeeira do Paraná (LRd e LEd), por um período de 8 anos (1975-82), com o objetivo de estudar os efeitos de doses crescentes de calcário dolomítico (0, 2,5, 5,0 e 10,0 t/ha) nas propriedades químicas do solo, produção e estado nutricional do cafeeiro (Coffea arábica L.). O pH do solo, capacidade de troca de cátions (CTC) e Ca e Mg trocáveis aumentaram, enquanto que o Al e K trocáveis diminuíram com o aumento das doses de calcário. Estes efeitos foram evidentes apenas na superfície do solo (0-30 cm). Os efeitos da calagem no estado nutricional do cafeeiro foram pronunciados, em virtude do aumento nas concentrações de Ca e Mg, redução nas de Mn (eliminou os efeitos tóxicos), Zn e K e inalteração nas de N, P e Cu nos tecidos foliares. A neutralização do Al tóxico e o ajustamento das relações entre Ca-K, Ca-Mg e Mg-K para 13:1, 4:1 e 3:1, respectivamente, aumentaram a produção do cafeeiro. As melhores produções de café foram associadas com a mais baixa dose de calcário (2,5 t/ha), sendo que as mais elevadas (5 e 10 t/ha) diminuíram sistematicamente a produção do cafeeiro