Deciding KAT and Hoare Logic with Derivatives

Kleene algebra with tests (KAT) is an equational system for program verification, which is the combination of Boolean algebra (BA) and Kleene algebra (KA), the algebra of regular expressions. In particular, KAT subsumes the propositional fragment of Hoare logic (PHL) which is a formal system for the specification and verification of programs, and that is currently the base of most tools for checking program correctness. Both the equational theory of KAT and the encoding of PHL in KAT are known to be decidable. In this paper we present a new decision procedure for the equivalence of two KAT expressions based on the notion of partial derivatives. We also introduce the notion of derivative modulo particular sets of equations. With this we extend the previous procedure for deciding PHL. Some experimental results are also presented.Comment: In Proceedings GandALF 2012, arXiv:1210.202

Modularizing the Elimination of r=0 in Kleene Algebra

Given a universal Horn formula of Kleene algebra with hypotheses of the form r = 0, it is already known that we can efficiently construct an equation which is valid if and only if the Horn formula is valid. This is an example of elimination of hypotheses, which is useful because the equational theory of Kleene algebra is decidable while the universal Horn theory is not. We show that hypotheses of the form r = 0 can still be eliminated in the presence of other hypotheses. This lets us extend any technique for eliminating hypotheses to include hypotheses of the form r = 0

Boolean like algebras

Using Vaggione’s concept of central element in a double pointed algebra, we introduce the notion of Boolean like variety as a generalization of Boolean algebras to an arbitrary similarity type. Appropriately relaxing the requirement that every element be central in any member of the variety, we obtain the more general class of semi-Boolean like varieties, which still retain many of the pleasing properties of Boolean algebras. We prove that a double pointed variety is discriminator i↵ it is semi-Boolean like, idempotent, and 0-regular. This theorem yields a new Maltsev-style characterization of double pointed discriminator varieties. Moreover, we show that every idempotent semi-Boolean-like variety is term equivalent to a variety of noncommutative Boolean algebras with additional regular operations

Synthesis of Strategies Using the Hoare Logic of Angelic and Demonic Nondeterminism

We study a propositional variant of Hoare logic that can be used for reasoning about programs that exhibit both angelic and demonic nondeterminism. We work in an uninterpreted setting, where the meaning of the atomic actions is specified axiomatically using hypotheses of a certain form. Our logical formalism is entirely compositional and it subsumes the non-compositional formalism of safety games on finite graphs. We present sound and complete Hoare-style calculi that are useful for establishing partial-correctness assertions, as well as for synthesizing implementations. The computational complexity of the Hoare theory of dual nondeterminism is investigated using operational models, and it is shown that the theory is complete for exponential time

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

On kleene algebras for weighted computation

Kleene algebra with tests (KAT) was introduced as an alge- braic structure to model and reason about classic imperative programs, i.e. sequences of discrete actions guarded by Boolean tests. This paper introduces two generalisations of this structure able to ex- press programs as weighted transitions and tests with outcomes in a not necessary bivalent truth space, namely graded Kleene algebra with tests (GKAT) and Heyting Kleene algebra with tests (HKAT). On these contexts, in analogy to Kozen's encoding of Propositional Hoare Logic (PHL) in KAT [10], we discuss the encoding of a graded PHL in HKAT and of its while-free fragment in GKAT.This work is financed by the ERDF - European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT - Fundacao para a Ciencia e a Tecnologia, within projects POCI-01-0145-FEDER-016692 and UID/MAT/04106/2013. The second author is also supported by the individual grant SFRH/BPD/103004/2014

A Coalgebraic Approach to Kleene Algebra with Tests

Kleene algebra with tests is an extension of Kleene algebra, the algebra of regular expressions, which can be used to reason about programs. We develop a coalgebraic theory of Kleene algebra with tests, along the lines of the coalgebraic theory of regular expressions based on deterministic automata. Since the known automata-theoretic presentation of Kleene algebra with tests does not lend itself to a coalgebraic theory, we define a new interpretation of Kleene algebra with tests expressions and a corresponding automata-theoretic presentation. One outcome of the theory is a coinductive proof principle, that can be used to establish equivalence of our Kleene algebra with tests expressions.Comment: 21 pages, 1 figure; preliminary version appeared in Proc. Workshop on Coalgebraic Methods in Computer Science (CMCS'03

Embedding Kozen-Tiuryn Logic into Residuated One-Sorted Kleene Algebra with Tests

Kozen and Tiuryn have introduced the substructural logic $\mathsf{S}$ for reasoning about correctness of while programs (ACM TOCL, 2003). The logic $\mathsf{S}$ distinguishes between tests and partial correctness assertions, representing the latter by special implicational formulas. Kozen and Tiuryn's logic extends Kleene altebra with tests, where partial correctness assertions are represented by equations, not terms. Kleene algebra with codomain, $\mathsf{KAC}$, is a one-sorted alternative to Kleene algebra with tests that expands Kleene algebra with an operator that allows to construct a Boolean subalgebra of tests. In this paper we show that Kozen and Tiuryn's logic embeds into the equational theory of the expansion of $\mathsf{KAC}$ with residuals of Kleene algebra multiplication and the upper adjoint of the codomain operator