arXiv subjects

Andrzej S. Murawski

Publications and source records attributed to Andrzej S. Murawski.

15 recordsLinked to original sources

arXiv2026

Contextual MetaML: Syntax and Full Abstraction

MetaML-style metaprogramming languages allow programmers to construct, manipulate and run code. In the presence of higher-order references for code, ensuring type safety is challenging, as free variables can escape their binders. In this paper, we present Contextual MetaML, \textit{the first metaprogramming language that supports storing and running open code under a strong type safety guarantee}. The type system utilises contextual modal types to track and reason about free variables in code explicitly. A crucial concern in metaprogramming-based program optimisations is whether the optimised program preserves the meaning of the original program. Addressing this question requires a notion of program equivalence and techniques to reason about it. In this paper, we provide a semantic model that captures contextual equivalence for Contextual MetaML, establishing \textit{the first full abstraction result for an imperative MetaML-style language}. Our model is based on traces derived via operational game semantics, where the meaning of a program is modelled by its possible interactions with the environment. We also establish a novel closed instances of use theorem that accounts for both call-by-value and call-by-name closing substitutions.

cs.PL↗

arXiv2025

Bisimilarity in fresh-register automata

Register automata are a basic model of computation over infinite alphabets. Fresh-register automata extend register automata with the capability to generate fresh symbols in order to model computational scenarios involving name creation. This paper investigates the complexity of the bisimilarity problem for classes of register and fresh-register automata. We examine all main disciplines that have appeared in the literature: general register assignments; assignments where duplicate register values are disallowed; and assignments without duplicates in which registers cannot be empty. In the general case, we show that the problem is EXPTIME-complete. However, the absence of duplicate values in registers enables us to identify inherent symmetries inside the associated bisimulation relations, which can be used to establish a polynomial bound on the depth of Attacker-winning strategies. Furthermore, they enable a highly succinct representation of the corresponding bisimulations. By exploiting results from group theory and computational group theory, we can then show solvability in PSPACE and NP respectively for the latter two register disciplines. In each case, we find that freshness does not affect the complexity class of the problem. The results allow us to close a complexity gap for language equivalence of deterministic register automata. We show that deterministic language inequivalence for the no-duplicates fragment is NP-complete, which disproves an old conjecture of Sakamoto. Finally, we discover that, unlike in the finite-alphabet case, the addition of pushdown store makes bisimilarity undecidable, even in the case of visibly pushdown storage.

cs.LO↗

arXiv2024

Guaranteed Bounds on Posterior Distributions of Discrete Probabilistic Programs with Loops

We study the problem of bounding the posterior distribution of discrete probabilistic programs with unbounded support, loops, and conditioning. Loops pose the main difficulty in this setting: even if exact Bayesian inference is possible, the state of the art requires user-provided loop invariant templates. By contrast, we aim to find guaranteed bounds, which sandwich the true distribution. They are fully automated, applicable to more programs and provide more provable guarantees than approximate sampling-based inference. Since lower bounds can be obtained by unrolling loops, the main challenge is upper bounds, and we attack it in two ways. The first is called residual mass semantics, which is a flat bound based on the residual probability mass of a loop. The approach is simple, efficient, and has provable guarantees. The main novelty of our work is the second approach, called geometric bound semantics. It operates on a novel family of distributions, called eventually geometric distributions (EGDs), and can bound the distribution of loops with a new form of loop invariants called contraction invariants. The invariant synthesis problem reduces to a system of polynomial inequality constraints, which is a decidable problem with automated solvers. If a solution exists, it yields an exponentially decreasing bound on the whole distribution, and can therefore bound moments and tail asymptotics as well, not just probabilities as in the first approach. Both semantics enjoy desirable theoretical properties. In particular, we prove soundness and convergence, i.e. the bounds converge to the exact posterior as loops are unrolled further. On the practical side, we describe Diabolo, a fully-automated implementation of both semantics, and evaluate them on a variety of benchmarks from the literature, demonstrating their general applicability and the utility of the resulting bounds.

cs.PL↗

arXiv2023

Saturating automata for game semantics

Saturation is a fundamental game-semantic property satisfied by strategies that interpret higher-order concurrent programs. It states that the strategy must be closed under certain rearrangements of moves, and corresponds to the intuition that program moves (P-moves) may depend only on moves made by the environment (O-moves). We propose an automata model over an infinite alphabet, called saturating automata, for which all accepted languages are guaranteed to satisfy a closure property mimicking saturation. We show how to translate the finitary fragment of Idealized Concurrent Algol (FICA) into saturating automata, confirming their suitability for modelling higher-order concurrency. Moreover, we find that, for terms in normal form, the resultant automaton has linearly many transitions and states with respect to term size, and can be constructed in polynomial time. This is in contrast to earlier attempts at finding automata-theoretic models of FICA, which did not guarantee saturation and involved an exponential blow-up during translation, even for normal forms.

cs.PL↗

arXiv2023

Exact Bayesian Inference on Discrete Models via Probability Generating Functions: A Probabilistic Programming Approach

We present an exact Bayesian inference method for discrete statistical models, which can find exact solutions to a large class of discrete inference problems, even with infinite support and continuous priors. To express such models, we introduce a probabilistic programming language that supports discrete and continuous sampling, discrete observations, affine functions, (stochastic) branching, and conditioning on discrete events. Our key tool is probability generating functions: they provide a compact closed-form representation of distributions that are definable by programs, thus enabling the exact computation of posterior probabilities, expectation, variance, and higher moments. Our inference method is provably correct and fully automated in a tool called Genfer, which uses automatic differentiation (specifically, Taylor polynomials), but does not require computer algebra. Our experiments show that Genfer is often faster than the existing exact inference tools PSI, Dice, and Prodigy. On a range of real-world inference problems that none of these exact tools can solve, Genfer's performance is competitive with approximate Monte Carlo methods, while avoiding approximation errors.

cs.PL↗

arXiv2022

The Big-O Problem

Given two weighted automata, we consider the problem of whether one is big-O of the other, i.e., if the weight of every finite word in the first is not greater than some constant multiple of the weight in the second. We show that the problem is undecidable, even for the instantiation of weighted automata as labelled Markov chains. Moreover, even when it is known that one weighted automaton is big-O of another, the problem of finding or approximating the associated constant is also undecidable. Our positive results show that the big-O problem is polynomial-time solvable for unambiguous automata, coNP-complete for unlabelled weighted automata (i.e., when the alphabet is a single character) and decidable, subject to Schanuel's conjecture, when the language is bounded (i.e., a subset of $w_1^*\dots w_m^*$ for some finite words $w_1,\dots,w_m$) or when the automaton has finite ambiguity. On labelled Markov chains, the problem can be restated as a ratio total variation distance, which, instead of finding the maximum difference between the probabilities of any two events, finds the maximum ratio between the probabilities of any two events. The problem is related to $\varepsilon$-differential privacy, for which the optimal constant of the big-O notation is exactly $\exp(\varepsilon)$.

cs.FL↗

arXiv2021

Complete trace models of state and control

We consider a hierarchy of four typed call-by-value languages with either higher-order or ground-type references and with either callcc or no control operator.Our first result is a fully abstract trace model for the most expressive setting, featuring both higher-order references and callcc, constructed in the spirit of operational game semantics. Next we examine the impact of suppressing higher-order references and callcc in contexts and provide an operational explanation for the game-semantic conditions known as visibility and bracketing respectively.This allows us to refine the original model to provide fully abstract trace models of interaction with contexts that need not use higher-order references or callcc. Along the way, we discuss the relationship between error- and termination-based contextual testing in each case, and relate the two to trace and complete trace equivalence respectively.Overall, the paper provides a systematic development of operational game semantics for all four cases, which represent the state-based face of the so-called semantic cube.

cs.LO↗

arXiv2021

Leafy Automata for Higher-Order Concurrency

Finitary Idealized Concurrent Algol (FICA) is a prototypical programming language combining functional, imperative, and concurrent computation. There exists a fully abstract game model of FICA, which in principle can be used to prove equivalence and safety of FICA programs. Unfortunately, the problems are undecidable for the whole language, and only very rudimentary decidable sub-languages are known. We propose leafy automata as a dedicated automata-theoretic formalism for representing the game semantics of FICA. The automata use an infinite alphabet with a tree structure. We show that the game semantics of any FICA term can be represented by traces of a leafy automaton. Conversely, the traces of any leafy automaton can be represented by a FICA term. Because of the close match with FICA, we view leafy automata as a promising starting point for finding decidable subclasses of the language and, more generally, to provide a new perspective on models of higher-order concurrent computation. Moreover, we identify a fragment of FICA that is amenable to verification by translation into a particular class of leafy automata. Using a locality property of the latter class, where communication between levels is restricted and every other level is bounded, we show that their emptiness problem is decidable by reduction to Petri net reachability.

cs.FL↗

arXiv2020

Collapsible Pushdown Parity Games

This paper studies a large class of two-player perfect-information turn-based parity games on infinite graphs, namely those generated by collapsible pushdown automata. The main motivation for studying these games comes from the connections from collapsible pushdown automata and higher-order recursion schemes, both models being equi-expressive for generating infinite trees. Our main result is to establish the decidability of such games and to provide an effective representation of the winning region as well as of a winning strategy. Thus, the results obtained here provide all necessary tools for an in-depth study of logical properties of trees generated by collapsible pushdown automata/recursion schemes.

cs.FL↗

arXiv2018

Bisimilarity Distances for Approximate Differential Privacy

Differential privacy is a widely studied notion of privacy for various models of computation. Technically, it is based on measuring differences between probability distributions. We study $ε,δ$-differential privacy in the setting of labelled Markov chains. While the exact differences relevant to $ε,δ$-differential privacy are not computable in this framework, we propose a computable bisimilarity distance that yields a sound technique for measuring $δ$, the parameter that quantifies deviation from pure differential privacy. We show this bisimilarity distance is always rational, the associated threshold problem is in NP, and the distance can be computed exactly with polynomially many calls to an NP oracle.

cs.FL↗

arXiv2016

Higher-Order Linearisability

Linearisability is a central notion for verifying concurrent libraries: a given library is proven safe if its operational history can be rearranged into a new sequential one which, in addition, satisfies a given specification. Linearisability has been examined for libraries in which method arguments and method results are of ground type, including libraries parameterised with such methods. In this paper we extend linearisability to the general higher-order setting: methods can be passed as arguments and returned as values. A library may also depend on abstract methods of any order. We use this generalised notion to show correctness of several higher-order example libraries.

cs.PL↗

arXiv2016

Block structure vs scope extrusion: between innocence and omniscience

We study the semantic meaning of block structure using game semantics. To that end, we introduce the notion of block-innocent strategies and characterise call-by-value computation with block-allocated storage through soundness, finite definability and universality results. This puts us in a good position to conduct a comparative study of purely functional computation, computation with block storage as well as that with dynamic memory allocation. For example, we can show that dynamic variable allocation can be replaced with block-allocated variables exactly when the term involved (open or closed) is of base type and that block-allocated storage can be replaced with purely functional computation when types of order two are involved. To illustrate the restrictive nature of block structure further, we prove a decidability result for a finitary fragment of call-by-value Idealized Algol for which it is known that allowing for dynamic memory allocation leads to undecidability.

cs.PL↗

arXiv2015

Fragments of ML Decidable by Nested Data Class Memory Automata

The call-by-value language RML may be viewed as a canonical restriction of Standard ML to ground-type references, augmented by a "bad variable" construct in the sense of Reynolds. We consider the fragment of (finitary) RML terms of order at most 1 with free variables of order at most 2, and identify two subfragments of this for which we show observational equivalence to be decidable. The first subfragment consists of those terms in which the P-pointers in the game semantic representation are determined by the underlying sequence of moves. The second subfragment consists of terms in which the O-pointers of moves corresponding to free variables in the game semantic representation are determined by the underlying moves. These results are shown using a reduction to a form of automata over data words in which the data values have a tree-structure, reflecting the tree-structure of the threads in the game semantic plays. In addition we show that observational equivalence is undecidable at every third- or higher-order type, every second-order type which takes at least two first-order arguments, and every second-order type (of arity greater than one) that has a first-order argument which is not the final argument.

cs.PL↗

arXiv2012

Bisimilarity of Pushdown Systems is Nonelementary

Given two pushdown systems, the bisimilarity problem asks whether they are bisimilar. While this problem is known to be decidable our main result states that it is nonelementary, improving EXPTIME-hardness, which was the previously best known lower bound for this problem. Our lower bound result holds for normed pushdown systems as well.

cs.FL↗

arXiv2012

On the Complexity of the Equivalence Problem for Probabilistic Automata

Checking two probabilistic automata for equivalence has been shown to be a key problem for efficiently establishing various behavioural and anonymity properties of probabilistic systems. In recent experiments a randomised equivalence test based on polynomial identity testing outperformed deterministic algorithms. In this paper we show that polynomial identity testing yields efficient algorithms for various generalisations of the equivalence problem. First, we provide a randomized NC procedure that also outputs a counterexample trace in case of inequivalence. Second, we show how to check for equivalence two probabilistic automata with (cumulative) rewards. Our algorithm runs in deterministic polynomial time, if the number of reward counters is fixed. Finally we show that the equivalence problem for probabilistic visibly pushdown automata is logspace equivalent to the Arithmetic Circuit Identity Testing problem, which is to decide whether a polynomial represented by an arithmetic circuit is identically zero.

cs.FL↗