This paper investigates the usage of generating functions (GFs) encoding measures over the program variables for reasoning about discrete probabilistic programs. To that end, we define a denotational GF-transformer semantics for probabilistic while-programs, and show that it instantiates Kozen's seminal distribution transformer semantics. We then study the effective usage of GFs for program analysis. We show that finitely expressible GFs enable checking super-invariants by means of computer algebra tools, and that they can be used to determine termination probabilities. The paper concludes by characterizing a class of -- possibly infinite-state -- programs whose semantics is a rational GF encoding a discrete phase-type distribution.
We are concerned with the average case runtime complexity analysis of a prototypical imperative language endowed with primitives for sampling and probabilistic choice. Taking inspiration from known approaches from to the modular resource analysis of non-probabilistic programs, we investigate how a modular runtime analysis is obtained for probabilistic programs.
In this work, we consider the almost-sure termination problem for probabilistic programs that asks whether a given probabilistic program terminates with probability 1. Scalable approaches for program analysis often rely on modularity as their theoretical basis. In non-probabilistic programs, the classical variant rule (V-rule) of Floyd-Hoare logic provides the foundation for modular analysis. Extension of this rule to almost-sure termination of probabilistic programs is quite tricky, and a probabilistic variant was proposed in [Fioriti and Hermanns 2015]. While the proposed probabilistic varia…
This paper presents a wp-style calculus for obtaining bounds on the expected run-time of probabilistic programs. Its application includes determining the (possibly infinite) expected termination time of a probabilistic program and proving positive almost-sure termination - does a program terminate with probability one in finite expected time? We provide several proof rules for bounding the run-time of loops, and prove the soundness of the approach with respect to a simple operational model. We show that our approach is a conservative extension of Nielson's approach for reasoning about the run-…
The weakest pre-expectation calculus has been proved to be a mature theory to analyze quantitative properties of probabilistic and nondeterministic programs. We present an automatic method for proving quantitative linear properties on any denumerable state space using iterative backwards fixed point calculation in the general framework of abstract interpretation. In order to accomplish this task we present the technique of random variable abstraction (RVA) and we also postulate a sufficient condition to achieve exact fixed point computation in the abstract domain. The feasibility of our approa…
Research on deductive verification of probabilistic programs has considered expectation-based logics, where pre- and post-conditions are real-valued functions on states, and assertion-based logics, where pre- and post-conditions are boolean predicates on state distributions. Both approaches have developed over nearly four decades, but they have different standings today. Expectation-based systems have managed to formalize many sophisticated case studies, while assertion-based systems today have more limited expressivity and have targeted simpler examples. We present Ellora, a sound and relativ…
Program sensitivity, also known as Lipschitz continuity, describes how small changes in a program's input lead to bounded changes in the output. We propose an average notion of program sensitivity for probabilistic programs---expected sensitivity---that averages a distance function over a probabilistic coupling of two output distributions from two similar inputs. By varying the distance, expected sensitivity recovers useful notions of probabilistic function sensitivity, including stability of machine learning algorithms and convergence of Markov chains. Furthermore, expected sensitivity satisf…
We study a syntax for specifying quantitative "assertions" - functions mapping program states to numbers - for probabilistic program verification. We prove that our syntax is expressive in the following sense: Given any probabilistic program $C$, if a function $f$ is expressible in our syntax, then the function mapping each initial state $σ$ to the expected value of $f$ evaluated in the final states reached after termination of $C$ on $σ$ (also called the weakest preexpectation $\textit{wp} [C](f)$) is also expressible in our syntax. As a consequence, we obtain a relatively complete verificati…
Probabilistic programs are typically normal-looking programs describing posterior probability distributions. They intrinsically code up randomized algorithms and have long been at the heart of modern machine learning and approximate computing. We explore the theory of generating functions [19] and investigate its usage in the exact quantitative reasoning of probabilistic programs. Important topics include the exact representation of program semantics [13], proving exact program equivalence [5], and -- as our main focus in this extended abstract -- exact probabilistic inference. In probabilisti…
University College London（伦敦大学学院）；Cornell University（康奈尔大学）
We present a denotational semantics for higher-order probabilistic programs in terms of linear operators between Banach spaces. Our semantics is rooted in the classical theory of Banach spaces and their tensor products, but bears similarities with the well-known Scott semantics of higher-order programs through the use ordered Banach spaces which allow definitions in terms of fixed points. Being based on a monoidal rather than cartesian closed structure, our semantics effectively treats randomness as a resource.
This paper presents a wp-style calculus for obtaining expectations on the outcomes of (mutually) recursive probabilistic programs. We provide several proof rules to derive one-- and two--sided bounds for such expectations, and show the soundness of our wp-calculus with respect to a probabilistic pushdown automaton semantics. We also give a wp-style calculus for obtaining bounds on the expected runtime of recursive programs that can be used to determine the (possibly infinite) time until termination of such programs.
In probabilistic reasoning, the traditionally discrete domain has been elevated to the hybrid domain encompassing additionally continuous random variables. Inference in the hybrid domain, however, usually necessitates to condone trade-offs on either the inference on discrete or continuous random variables. We introduce a novel approach based on weighted model integration and algebraic model counting that circumvents these trade-offs. We then show how it supports knowledge compilation and exact probabilistic inference. Moreover, we introduce the hybrid probabilistic logic programming language H…
We present a novel modular approach to infer upper bounds on the expected runtime of probabilistic integer programs automatically. To this end, it computes bounds on the runtime of program parts and on the sizes of their variables in an alternating way. To evaluate its power, we implemented our approach in a new version of our open-source tool KoAT.
Probabilistic logic programming is a major part of statistical relational artificial intelligence, where approaches from logic and probability are brought together to reason about and learn from relational domains in a setting of uncertainty. However, the behaviour of statistical relational representations across variable domain sizes is complex, and scaling inference and learning to large domains remains a significant challenge. In recent years, connections have emerged between domain size dependence, lifted inference and learning from sampled subpopulations. The asymptotic behaviour of stati…
The semantics of probabilistic languages has been extensively studied, but specification languages for their properties have received little attention. This paper introduces the probabilistic dynamic logic pDL, a specification logic for programs in the probabilistic guarded command language (pGCL) of McIver and Morgan. The proposed logic pDL can express both first-order state properties and probabilistic reachability properties, addressing both the non-deterministic and probabilistic choice operators of pGCL. In order to precisely explain the meaning of specifications, we formally define the s…
This paper considers the computational hardness of computing expected outcomes and deciding almost-sure termination of probabilistic programs. We show that deciding almost-sure termination and deciding whether the expected outcome of a program equals a given rational value is $Π^0_2$-complete. Computing lower and upper bounds on the expected outcome is shown to be recursively enumerable and $Σ^0_2$-complete, respectively.
Essential tasks for the verification of probabilistic programs include bounding expected outcomes and proving termination in finite expected runtime. We contribute a simple yet effective inductive synthesis approach for proving such quantitative reachability properties by generating inductive invariants on source-code level. Our implementation shows promise: It finds invariants for (in)finite-state programs, can beat state-of-the-art probabilistic model checkers, and is competitive with modern tools dedicated to invariant synthesis and expected runtime reasoning.
We describe a mathematical structure that can give extensional denotational semantics to higher-order probabilistic programs. It is not limited to discrete probabilities, and it is compatible with integration in a way the models that have been proposed before are not. It is organised as a model of propositional linear logic in which all the connectives have intuitive probabilistic interpretations. In addition, it has least fixed points for all maps, so it can interpret recursion.
There are numerous applications where we have to deal with temporal uncertainty associated with events. The Temporal Probabilistic (TP) Logic Programs should provide support for valid-time indeterminacy of events, by proposing the concept of an indeterminate instant, that is, an interval of time-points (event's time-window) with an associated, lower and upper, probability distribution. In particular, we propose the new semantics, for the TP Logic Programs of Dekhtyar and Subrahmanian. Our semantics, based on the possible world semantics is a generalization of the possible world semantics for (…
Proving programs terminating is a fundamental computer science challenge. Recent research has produced powerful tools that can check a wide range of programs for termination. The analog for probabilistic programs, namely termination with probability one ("almost-sure termination"), is an equally important property for randomized algorithms and probabilistic protocols. We suggest a novel algorithm for proving almost-sure termination of probabilistic programs. Our algorithm exploits the power of state-of-the-art model checkers and termination provers for nonprobabilistic programs: it calls such …
Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these combined features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus i…
We study the almost-sure termination problem for probabilistic programs. First, we show that supermartingales with lower bounds on conditional absolute difference provide a sound approach for the almost-sure termination problem. Moreover, using this approach we can obtain explicit optimal bounds on tail probabilities of non-termination within a given number of steps. Second, we present a new approach based on Central Limit Theorem for the almost-sure termination problem, and show that this approach can establish almost-sure termination of programs which none of the existing approaches can hand…
We present iSAPP (Imperative Static Analyser for Probabilistic Polynomial Time), a complexity verifier tool that is sound and extensionally complete for the Probabilistic Polynomial Time (PP) complexity class. iSAPP works on an imperative programming language for stack machines. The certificate of polynomiality can be built in polytime, with respect to the number of stacks used.
We study termination time and recurrence time in programs with unbounded recursion, which are either randomized or operate on some statistically quantified inputs. As the underlying formal model for such programs we use probabilistic pushdown automata (pPDA) which are equivalent to probabilistic recursive state machines. We obtain tail bounds for the distribution of termination time for pPDA. We also study the recurrence time for probabilistic recursive programs that are not supposed to terminate (such as system daemons, network servers, etc.). Typically, such programs react to certain request…
The Credal semantics is a probabilistic extension of the answer set semantics which can be applied to programs that may or may not be stratified. It assigns to atoms a set of acceptable probability distributions characterised by its lower and upper bounds. Performing exact probabilistic inference in the Credal semantics is computationally intractable. This paper presents a first solver, based on sampling, for probabilistic inference under the Credal semantics called PASOCS (Parallel Approximate SOlver for the Credal Semantics). PASOCS performs both exact and approximate inference for queries g…
We present a new proof rule for verifying lower bounds on quantities of probabilistic programs. Our proof rule is not confined to almost-surely terminating programs -- as is the case for existing rules -- and can be used to establish non-trivial lower bounds on, e.g., termination probabilities and expected values, for possibly divergent probabilistic loops, e.g., the well-known three-dimensional random walk on a lattice.
We consider a programming language that can manipulate both classical and quantum information. Our language is type-safe and designed for variational quantum programming, which is a hybrid classical-quantum computational paradigm. The classical subsystem of the language is the Probabilistic FixPoint Calculus (PFPC), which is a lambda calculus with mixed-variance recursive types, term recursion and probabilistic choice. The quantum subsystem is a first-order linear type system that can manipulate quantum information. The two subsystems are related by mixed classical/quantum terms that specify h…
Hybrid probabilistic logic programs can represent several scenarios thanks to the expressivity of Logic Programming extended with facts representing discrete and continuous distributions. The semantics for this type of programs is crucial since it ensures that a probability can be assigned to every query. Here, following one recent semantics proposal, we illustrate a concrete syntax, and we analyse the syntactic requirements needed to preserve the well-definedness.
We define a probabilistic programming language for Gaussian random variables with a first-class exact conditioning construct. We give operational, denotational and equational semantics for this language, establishing convenient properties like exchangeability of conditions. Conditioning on equality of continuous random variables is nontrivial, as the exact observation may have probability zero; this is Borel's paradox. Using categorical formulations of conditional probability, we show that the good properties of our language are not particular to Gaussians, but can be derived from universal pr…
We initiate the study of probabilistic parallel programs with dynamic process creation and synchronisation. To this end, we introduce probabilistic split-join systems (pSJSs), a model for parallel programs, generalising both probabilistic pushdown systems (a model for sequential probabilistic procedural programs which is equivalent to recursive Markov chains) and stochastic branching processes (a classical mathematical model with applications in various areas such as biology, physics, and language processing). Our pSJS model allows for a possibly recursive spawning of parallel processes; the s…
We study discrete probabilistic programs with potentially unbounded looping behaviors over an infinite state space. We present, to the best of our knowledge, the first decidability result for the problem of determining whether such a program generates exactly a specified distribution over its outputs (provided the program terminates almost surely). The class of distributions that can be specified in our formalism consists of standard distributions (geometric, uniform, etc.) and finite convolutions thereof. Our method relies on representing these (possibly infinite-support) distributions as pro…
We introduce a formal meta-language for probabilistic programming, capable of expressing both programs and the type systems in which they are embedded. We are motivated here by the desire to allow an AGI to learn not only relevant knowledge (programs/proofs), but also appropriate ways of reasoning (logics/type systems). We draw on the frameworks of cubical type theory and dependent typed metagraphs to formalize our approach. In doing so, we show that specific constructions within the meta-language can be related via bisimulation (implying path equivalence) to the type systems they correspond. …
Probabilistic logic programs are logic programs in which some of the facts are annotated with probabilities. This paper investigates how classical inference and learning tasks known from the graphical model community can be tackled for probabilistic logic programs. Several such tasks such as computing the marginals given evidence and learning from (partial) interpretations have not really been addressed for probabilistic logic programs before. The first contribution of this paper is a suite of efficient algorithms for various inference tasks. It is based on a conversion of the program and the …
