arXiv subjects
Publications and source records attributed to Alexandra Silva.
We propose Weighted Guarded Kleene Algebra with Tests (wGKAT), an uninterpreted weighted programming language equipped with branching, conditionals, and loops. We provide an operational semantics for wGKAT using a variant of weighted automata and introduce a sound and complete axiomatization. We also provide a polynomial time decision procedure for bisimulation equivalence.
Accurately identifying spatial patterns of species distribution is crucial for scientific insight and societal benefit, aiding our understanding of species fluctuations. The increasing quantity and quality of ecological datasets present heightened statistical challenges, complicating spatial species dynamics comprehension. Addressing the complex task of integrating multiple data sources to enhance spatial fish distribution understanding in marine ecology, this study introduces a pioneering five-layer Joint model. The model adeptly integrates fishery-independent and fishery-dependent data, accommodating zero-inflated data and distinct sampling processes. A comprehensive simulation study evaluates the model performance across various preferential sampling scenarios and sample sizes, elucidating its advantages and challenges. Our findings highlight the model's robustness in estimating preferential parameters, emphasizing differentiation between presence-absence and biomass observations. Evaluation of estimation of spatial covariance and prediction performance underscores the model's reliability. Augmenting sample sizes reduces parameter estimation variability, aligning with the principle that increased information enhances certainty. Assessing the contribution of each data source reveals successful integration, providing a comprehensive representation of biomass patterns. Empirical validation within a real-world context further solidifies the model's efficacy in capturing species' spatial distribution. This research advances methodologies for integrating diverse datasets with different sampling natures further contributing to a more informed understanding of spatial dynamics of marine species.
Various categories have been proposed as targets for the denotational semantics of higher-order probabilistic programming languages. One such proposal involves joint probability distributions (couplings) used in Bayesian statistical models with conditioning. In previous treatments, composition of joint measures was performed by disintegrating to obtain Markov kernels, composing the kernels, then reintegrating to obtain a joint measure. Disintegrations exist only under certain restrictions on the underlying spaces. In this paper we propose a category whose morphisms are joint finite measures in which composition is defined without reference to disintegration, allowing its application to a broader class of spaces. The category is symmetric monoidal with a pleasing symmetry in which the dagger structure is a simple transpose.
Programs increasingly rely on randomization in applications such as cryptography and machine learning. Analyzing randomized programs has been a fruitful research direction, but there is a gap when programs also exploit nondeterminism (for concurrency, efficiency, or algorithmic design). In this paper, we introduce Demonic Outcome Logic for reasoning about programs that exploit both randomization and nondeterminism. The logic includes several novel features, such as reasoning about multiple executions in tandem and manipulating pre- and postconditions using familiar equational laws -- including the distributive law of probabilistic choices over nondeterministic ones. We also give rules for loops that both establish termination and quantify the distribution of final outcomes from a single premise. We illustrate the reasoning capabilities of Demonic Outcome Logic through several case studies, including the Monty Hall problem, an adversarial protocol for simulating fair coins, and a heuristic based probabilistic SAT solver.
We develop new data structures and algorithms for checking verification queries in NetKAT, a domain-specific language for specifying the behavior of network data planes. Our results extend the techniques obtained in prior work on symbolic automata and provide a framework for building efficient and scalable verification tools. We present KATch, an implementation of these ideas in Scala, featuring an extended set of NetKAT operators that are useful for expressing network-wide specifications, and a verification engine that constructs a bisimulation or generates a counter-example showing that none exists. We evaluate the performance of our implementation on real-world and synthetic benchmarks, verifying properties such as reachability and slice isolation, typically returning a result in well under a second, which is orders of magnitude faster than previous approaches. Our advancements underscore NetKAT's potential as a practical, declarative language for network specification and verification.
Guarded Kleene Algebra with Tests (GKAT) is a fragment of Kleene Algebra with Tests (KAT) that was recently introduced to reason efficiently about imperative programs. In contrast to KAT, GKAT does not have an algebraic axiomatization, but relies on an analogue of Salomaa's axiomatization of Kleene Algebra. In this paper, we present an algebraic axiomatization and prove two completeness results for a large fragment of GKAT consisting of skip-free programs.
We introduce a sound and complete equational theory capturing equivalence of discrete probabilistic programs, that is, programs extended with primitives for Bernoulli distributions and conditioning, to model distributions over finite sets of events. To do so, we translate these programs into a graphical syntax of probabilistic circuits, formalised as string diagrams, the two-dimensional syntax of symmetric monoidal categories. We then prove a first completeness result for the equational theory of the conditioning-free fragment of our syntax. Finally, we extend this result to a complete equational theory for the entire language. Note these developments are also of interest for the development of probability theory in Markov categories: our first result gives a presentation by generators and equations of the category of Markov kernels, restricted to objects that are powers of the two-elements set.
We introduce Probabilistic Regular Expressions (PRE), a probabilistic analogue of regular expressions denoting probabilistic languages in which every word is assigned a probability of being generated. We present and prove the completeness of an inference system for reasoning about probabilistic language equivalence of PRE based on Salomaa's axiomatisation of Kleene Algebra.
Guarded Kleene Algebra with Tests (GKAT for short) is an efficient fragment of Kleene Algebra with Tests, suitable for reasoning about simple imperative while-programs. Following earlier work by Das and Pous on Kleene Algebra, we study GKAT from a proof-theoretical perspective. The deterministic nature of GKAT allows for a non-well-founded sequent system whose set of regular proofs is complete with respect to the guarded language model. This is unlike the situation with Kleene Algebra, where hypersequents are required. Moreover, the decision procedure induced by proof search runs in NLOGSPACE, whereas that of Kleene Algebra is in PSPACE.
We present a novel \emph{weakest pre calculus} for \emph{reasoning about quantitative hyperproperties} over \emph{nondeterministic and probabilistic} programs. Whereas existing calculi allow reasoning about the expected value that a quantity assumes after program termination from a \emph{single initial state}, we do so for \emph{initial sets of states} or \emph{initial probability distributions}. We thus (i)~obtain a weakest pre calculus for hyper Hoare logic and (ii)~enable reasoning about so-called \emph{hyperquantities} which include expected values but also quantities (e.g. variance) out of scope of previous work. As a byproduct, we obtain a novel strongest post for weighted programs that extends both existing strongest and strongest liberal post calculi. Our framework reveals novel dualities between forward and backward transformers, correctness and incorrectness, as well as nontermination and unreachability.
Separation logic's compositionality and local reasoning properties have led to significant advances in scalable static analysis. But program analysis has new challenges -- many programs display computational effects and, orthogonally, static analyzers must handle incorrectness too. We present Outcome Separation Logic (OSL), a program logic that is sound for both correctness and incorrectness reasoning in programs with varying effects. OSL has a frame rule -- just like separation logic -- but uses different underlying assumptions that open up local reasoning to a larger class of properties than can be handled by any single existing logic. Building on this foundational theory, we also define symbolic execution algorithms that use bi-abduction to derive specifications for programs with effects. This involves a new tri-abduction procedure to analyze programs whose execution branches due to effects such as nondeterministic or probabilistic choice. This work furthers the compositionality promised by separation logic by opening up the possibility for greater reuse of analysis tools across two dimensions: bug-finding vs verification in programs with varying effects.
We give a lightweight alternative construction of Jacobs's distributive law for multisets and distributions that does not involve any combinatorics. We first give a distributive law for lists and distributions, then apply a general theorem on 2-categories that allows properties of lists to be transferred automatically to multisets. The theorem states that equations between 2-cells are preserved by epic 2-natural transformations. In our application, the appropriate epic 2-natural transformation is defined in terms of the Parikh map, familiar from formal language theory, that takes a list to its multiset of elements.
The logic of Dependence and Independence Bunched Implications (DIBI) is a logic to reason about conditional independence (CI); for instance, DIBI formulas can characterise CI in probability distributions and relational databases, using the probabilistic and relational DIBI models, respectively. Despite the similarity of the probabilistic and relational models, a uniform, more abstract account remains unsolved. The laborious case-by-case verification of the frame conditions required for constructing new models also calls for such a treatment. In this paper, we develop an abstract framework for systematically constructing DIBI models, using category theory as the unifying mathematical language. In particular, we use string diagrams -- a graphical presentation of monoidal categories -- to give a uniform definition of the parallel composition and subkernel relation in DIBI models. Our approach not only generalises known models, but also yields new models of interest and reduces properties of DIBI models to structures in the underlying categories. Furthermore, our categorical framework enables a logical notion of CI, in terms of the satisfaction of specific DIBI formulas. We compare it with string diagrammatic approaches to CI and show that it is an extension of string diagrammatic CI under reasonable conditions.
Active automata learning algorithms cannot easily handle conflict in the observation data (different outputs observed for the same inputs). This inherent inability to recover after a conflict impairs their effective applicability in scenarios where noise is present or the system under learning is mutating. We propose the Conflict-Aware Active Automata Learning (C3AL) framework to enable handling conflicting information during the learning process. The core idea is to consider the so-called observation tree as a first-class citizen in the learning process. Though this idea is explored in recent work, we take it to its full effect by enabling its use with any existing learner and minimizing the number of tests performed on the system under learning, specially in the face of conflicts. We evaluate C3AL in a large set of benchmarks, covering over 30 different realistic targets, and over 18,000 different scenarios. The results of the evaluation show that C3AL is a suitable alternative framework for closed-box learning that can better handle noise and mutations.
Active automata learning algorithms cannot easily handle conflict in the observation data (different outputs observed for the same inputs). This inherent inability to recover after a conflict impairs their effective applicability in scenarios where noise is present or the system under learning is mutating. We propose the Conflict-Aware Active Automata Learning (C3AL) framework to enable handling conflicting information during the learning process. The core idea is to consider the so-called observation tree as a first-class citizen in the learning process. Though this idea is explored in recent work, we take it to its full effect by enabling its use with any existing learner and minimizing the number of tests performed on the system under learning, specially in the face of conflicts. We evaluate C3AL in a large set of benchmarks, covering over 30 different realistic targets, and over 18,000 different scenarios. The results of the evaluation show that C3AL is a suitable alternative framework for closed-box learning that can better handle noise and mutations.
We present a new symbolic execution semantics of probabilistic programs that include observe statements and sampling from continuous distributions. Building on Kozen's seminal work, this symbolic semantics consists of a countable collection of measurable functions, along with a partition of the state space. We use the new semantics to provide a full correctness proof of symbolic execution for probabilistic programs. We also implement this semantics in the tool symProb, and illustrate its use on examples.
It is well-known that every regular language admits a unique minimal deterministic acceptor. Establishing an analogous result for non-deterministic acceptors is significantly more difficult, but nonetheless of great practical importance. To tackle this issue, a number of sub-classes of non-deterministic automata have been identified, all admitting canonical minimal representatives. In previous work, we have shown that such representatives can be recovered categorically in two steps. First, one constructs the minimal bialgebra accepting a given regular language, by closing the minimal coalgebra with additional algebraic structure over a monad. Second, one identifies canonical generators for the algebraic part of the bialgebra, to derive an equivalent coalgebra with side effects in a monad. In this paper, we further develop the general theory underlying these two steps. On the one hand, we show that deriving a minimal bialgebra from a minimal coalgebra can be realized by applying a monad on an appropriate category of subobjects. On the other hand, we explore the abstract theory of generators and bases for algebras over a monad.
We introduce Probabilistic Guarded Kleene Algebra with Tests (ProbGKAT), an extension of GKAT that allows reasoning about uninterpreted imperative programs with probabilistic branching. We give its operational semantics in terms of special class of probabilistic automata. We give a sound and complete Salomaa-style axiomatisation of bisimilarity of ProbGKAT expressions. Finally, we show that bisimilarity of ProbGKAT expressions can be decided in $O(n^3 \log n)$ time via a generic partition refinement algorithm.