397 research outputs found
Convolution, Separation and Concurrency
A notion of convolution is presented in the context of formal power series
together with lifting constructions characterising algebras of such series,
which usually are quantales. A number of examples underpin the universality of
these constructions, the most prominent ones being separation logics, where
convolution is separating conjunction in an assertion quantale; interval
logics, where convolution is the chop operation; and stream interval functions,
where convolution is used for analysing the trajectories of dynamical or
real-time systems. A Hoare logic is constructed in a generic fashion on the
power series quantale, which applies to each of these examples. In many cases,
commutative notions of convolution have natural interpretations as concurrency
operations.Comment: 39 page
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Automated verification of refinement laws
Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs
Probabilistic Rely-guarantee Calculus
Jones' rely-guarantee calculus for shared variable concurrency is extended to
include probabilistic behaviours. We use an algebraic approach which combines
and adapts probabilistic Kleene algebras with concurrent Kleene algebra.
Soundness of the algebra is shown relative to a general probabilistic event
structure semantics. The main contribution of this paper is a collection of
rely-guarantee rules built on top of that semantics. In particular, we show how
to obtain bounds on probabilities by deriving rely-guarantee rules within the
true-concurrent denotational semantics. The use of these rules is illustrated
by a detailed verification of a simple probabilistic concurrent program: a
faulty Eratosthenes sieve.Comment: Preprint submitted to TCS-QAP
Stone-Type Dualities for Separation Logics
Stone-type duality theorems, which relate algebraic and
relational/topological models, are important tools in logic because -- in
addition to elegant abstraction -- they strengthen soundness and completeness
to a categorical equivalence, yielding a framework through which both algebraic
and topological methods can be brought to bear on a logic. We give a systematic
treatment of Stone-type duality for the structures that interpret bunched
logics, starting with the weakest systems, recovering the familiar BI and
Boolean BI (BBI), and extending to both classical and intuitionistic Separation
Logic. We demonstrate the uniformity and modularity of this analysis by
additionally capturing the bunched logics obtained by extending BI and BBI with
modalities and multiplicative connectives corresponding to disjunction,
negation and falsum. This includes the logic of separating modalities (LSM), De
Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics
extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as
corollaries soundness and completeness theorems for the specific Kripke-style
models of these logics as presented in the literature: for DMBI, the
sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene
BI (connecting our work to Concurrent Separation Logic), this is the first time
soundness and completeness theorems have been proved. We thus obtain a
comprehensive semantic account of the multiplicative variants of all standard
propositional connectives in the bunched logic setting. This approach
synthesises a variety of techniques from modal, substructural and categorical
logic and contextualizes the "resource semantics" interpretation underpinning
Separation Logic amongst them
On partial order semantics for SAT/SMT-based symbolic encodings of weak memory concurrency
Concurrent systems are notoriously difficult to analyze, and technological
advances such as weak memory architectures greatly compound this problem. This
has renewed interest in partial order semantics as a theoretical foundation for
formal verification techniques. Among these, symbolic techniques have been
shown to be particularly effective at finding concurrency-related bugs because
they can leverage highly optimized decision procedures such as SAT/SMT solvers.
This paper gives new fundamental results on partial order semantics for
SAT/SMT-based symbolic encodings of weak memory concurrency. In particular, we
give the theoretical basis for a decision procedure that can handle a fragment
of concurrent programs endowed with least fixed point operators. In addition,
we show that a certain partial order semantics of relaxed sequential
consistency is equivalent to the conjunction of three extensively studied weak
memory axioms by Alglave et al. An important consequence of this equivalence is
an asymptotically smaller symbolic encoding for bounded model checking which
has only a quadratic number of partial order constraints compared to the
state-of-the-art cubic-size encoding.Comment: 15 pages, 3 figure
Formal Analysis of Concurrent Programs
In this thesis, extensions of Kleene algebras are used to develop algebras for rely-guarantee style reasoning about concurrent programs. In addition to these algebras, detailed denotational models are implemented in the interactive theorem prover Isabelle/HOL. Formal soundness proofs link the algebras to their models. This follows a general algebraic approach for developing correct by construction verification tools within Isabelle. In this approach, algebras provide inference rules and abstract principles for reasoning about the control flow of programs, while the concrete models provide laws for reasoning about data flow. This yields a rapid, lightweight approach for the construction of verification and refinement tools. These tools are used to construct a popular example from the literature, via refinement, within the context of a general-purpose interactive theorem proving environment
An Algebraic Framework for Compositional Program Analysis
The purpose of a program analysis is to compute an abstract meaning for a
program which approximates its dynamic behaviour. A compositional program
analysis accomplishes this task with a divide-and-conquer strategy: the meaning
of a program is computed by dividing it into sub-programs, computing their
meaning, and then combining the results. Compositional program analyses are
desirable because they can yield scalable (and easily parallelizable) program
analyses.
This paper presents algebraic framework for designing, implementing, and
proving the correctness of compositional program analyses. A program analysis
in our framework defined by an algebraic structure equipped with sequencing,
choice, and iteration operations. From the analysis design perspective, a
particularly interesting consequence of this is that the meaning of a loop is
computed by applying the iteration operator to the loop body. This style of
compositional loop analysis can yield interesting ways of computing loop
invariants that cannot be defined iteratively. We identify a class of
algorithms, the so-called path-expression algorithms [Tarjan1981,Scholz2007],
which can be used to efficiently implement analyses in our framework. Lastly,
we develop a theory for proving the correctness of an analysis by establishing
an approximation relationship between an algebra defining a concrete semantics
and an algebra defining an analysis.Comment: 15 page
Kleene algebra with domain
We propose Kleene algebra with domain (KAD), an extension of Kleene algebra
with two equational axioms for a domain and a codomain operation, respectively.
KAD considerably augments the expressiveness of Kleene algebra, in particular
for the specification and analysis of state transition systems. We develop the
basic calculus, discuss some related theories and present the most important
models of KAD. We demonstrate applicability by two examples: First, an
algebraic reconstruction of Noethericity and well-foundedness; second, an
algebraic reconstruction of propositional Hoare logic.Comment: 40 page
Automated verification of reactive and concurrent programs by calculation
Reactive programs combine traditional sequential programming constructs with primitives to allow communication with other concurrent agents. They are ubiquitous in modern applications, ranging from components systems and web services, to cyber-physical systems and autonomous robots. In this paper, we present an algebraic verification strategy for concurrent reactive programs, with a large or infinite state space. We define novel operators to characterise interactions and state updates, and an associated equational theory. With this we can calculate a reactive program's denotational semantics, and thereby facilitate automated proof. Of note is our reasoning support for iterative programs with reactive invariants, based on Kleene algebra, and for parallel composition. We illustrate our strategy by verifying a reactive buffer. Our laws and strategy are mechanised in Isabelle/UTP, our implementation of Hoare and He's Unifying Theories of Programming (UTP) framework, to provide soundness guarantees and practical verification support
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