arXiv subjects

Florian Zuleger

Publications and source records attributed to Florian Zuleger.

At least 19 recordsLinked to original sources

arXiv2026

Robust Algebraic Theories of Triangle Graphs

Triangle graphs are graphs of tree-width at most three in which every edge belongs to a triangle. This class encompasses well-known graph families such as Apollonian networks. We also consider fan graphs, a subclass of triangle graphs closely related to the 3-connected triangle graphs. Our main result is an algebraic characterization of both classes. We introduce two graph algebras based on parallel composition and a ternary serial composition, and show that they generate exactly the triangle and fan graphs, respectively. These algebras provide a natural extension of the classical algebra of series-parallel graphs from tree-width two to tree-width three. Building on these characterizations, we investigate context-free, recognizable, and logically-definable graph languages. We show that counting monadic second-order logic (CMSO) is decidable over the context-free sets of triangle and fan graphs. Moreover, we prove that recognizable graph languages coincide with languages definable in CMSO for both algebras.

cs.FL↗

arXiv2026

A Cost-Aware Probability Monad for Liquid Haskell

Probabilistic algorithms and data structures are widely used to obtain favourable expected performance guarantees. While their mathematical analysis is often well understood, mechanising expected-cost analyses remains challenging, requiring reasoning about probability distributions, expectations, and recursive stochastic behaviour. Existing formal approaches frequently require substantial manual proof effort, since expected costs are often encoded separately from probabilistic computations and must therefore be propagated explicitly throughout proofs. In this paper, we present a cost-aware probability monad for \LH/ that supports reasoning about probabilistic programs together with their expected costs. Our approach combines executable probabilistic programs with refinement-type-based verification and SMT-supported automation. The monad intrinsically tracks probability mass, expected values, and expected costs through refinement types, enabling many quantitative properties of probabilistic computations to be inferred compositionally from program structure. We evaluate our approach on several classical probabilistic algorithms and data structures, including meldable heaps, randomised quicksort and quickselect, randomised splay trees, random permutations, and the hiring problem. The case studies demonstrate different points along the spectrum between automated and interactive verification.

cs.LO↗

arXiv2026

Automated Amortised Analysis of Skew Heaps and Leftist Heaps (Extended Version)

We study the fully automated amortised analysis of purely functional data structures like skew heaps, as well as weight- and rank-biased leftist heaps. For that we generalise earlier works on automated amortised resource analysis by developing a type inference based approach with a generic type system. This allows for modular reasoning and the inference of precise and optimal cost bounds. More specifically, we extend the work on the ATLAS system by Leutgeb et al. which was developed to cover the analysis of splay trees and some closely related data structures. To enable the analysis of skew heaps, however, and the even more challenging (amortised) analysis of leftist heaps, we have developed a range of new techniques for type-based automated analysis. By introducing a generic type system we allow for arbitrary (classes of) potential functions, compared to the use of hard-coded potential functions in ATLAS, which we have implemented in Haskell in an entirely modular way. We have also greatly enhanced the existing type inference algorithm by extensions in multiple directions, including path-sensitive reasoning, data structure invariants, and template parameters for piecewise defined potential functions. We show how our newly developed system supports the use of all known potential functions for analysing skew heaps and leftist heaps, confirming the known bounds.

cs.PL↗

arXiv2026

Regular Grammars as Effective Representations of Recognizable Sets of Series-Parallel Graphs

Series-parallel (SP) graphs are binary edge-labeled graphs with a designated source and target vertex, built using serial and parallel composition. A set of graphs is recognizable if membership depends only on its image under a homomorphism into a finite algebra. For SP-graphs, and more generally, for graphs of bounded tree-width, recognizability coincides with definability in Counting Monadic Second-Order (CMSO) logic. Despite this strong logical characterization, the conciseness and algorithmic effectiveness of syntactic representations of recognizable sets of SP (and bounded-tree-width) graphs remain poorly understood. Building on previously introduced regular grammars for SP-graphs, we show that recognizable sets admit concise and effective syntactic representations. The main contribution is an improved construction of finite recognizer algebras whose size is singly-exponential in the size of a regular grammar, improving upon the previously known double-exponential bound. As a consequence, the problems of intersection and language inclusion for sets represented by regular grammars are shown to be EXPTIME-complete, thus improving on a previously known 2EXPTIME upper bound.

cs.FL↗

arXiv2026

Characterizations of Monadic Second Order Definable Context-Free Sets of Graphs

We give a characterization of the sets of graphs that are both definable in Counting Monadic Second Order Logic (CMSO) and context-free, i.e., least solutions of Hyperedge-Replacement (HR) grammars introduced by Courcelle and Engelfriet. We prove the equivalence of these sets with: (a) recognizable sets (in the algebra of graphs with HR-operations) of bounded tree-width; we refine this condition further and show equivalence with recognizability in a finitely generated subalgebra of the HR-algebra of graphs; (b) parsable sets, for which there is a definable transduction from graphs to a set of derivation trees labelled by HR operations, such that the set of graphs is the image of the set of derivation trees under the canonical evaluation of the HR operations; (c) images of recognizable unranked sets of trees under a definable transduction, whose inverse is also definable. We rely on a novel connection between two seminal results, a logical characterization of context-free graph languages in terms of tree-to-graph definable transductions, by Courcelle and Engelfriet and a proof that an optimal-width tree decomposition of a graph can be built by an definable transduction, by Bojanczyk and Pilipczuk.

cs.FL↗

arXiv2025

Iterating Non-Aggregative Structure Compositions

An aggregative composition is a binary operation obeying the principle that the whole is determined by the sum of its parts. The development of graph algebras, on which the theory of formal graph languages is built, relies on aggregative compositions that behave like disjoint union, except for a set of well-marked interface vertices from both sides, that are joined. The same style of composition has been considered in the context of relational structures, that generalize graphs and use constant symbols to label the interface. In this paper, we study a non-aggregative composition operation, called \emph{fusion}, that joins non-deterministically chosen elements from disjoint structures. The sets of structures obtained by iteratively applying fusion do not always have bounded tree-width, even when starting from a tree-width bounded set. First, we prove that the problem of the existence of a bound on the tree-width of the closure of a given set under fusion is decidable, when the input set is described inductively by a finite \emph{hyperedge-replacement} (HR) grammar, written using the operations of aggregative composition, forgetting and renaming of constants. Such sets are usually called \emph{context-free}. Second, assuming that the closure under fusion of a context-free set has bounded tree-width, we show that it is the language of an effectively constructible HR grammar. A possible application of the latter result is the possiblity of checking whether all structures from a non-aggregatively closed set having bounded tree-width satisfy a given monadic second order logic formula.

cs.FL↗

arXiv2025

To Zip Through the Cost Analysis of Probabilistic Programs

Probabilistic programming and the formal analysis of probabilistic algorithms are active areas of research, driven by the widespread use of randomness to improve performance. While functional correctness has seen substantial progress, automated reasoning about expected runtime remains comparatively limited. In this work, we address this challenge by introducing a refinement-typed probability monad in Liquid Haskell. Our monad enables automated reasoning about expected values and costs by encoding probabilistic behaviour directly in types. Initially defined for discrete distributions over finite support, it is extended to support infinite distributions via an axiomatic approach. By leveraging Liquid Haskell's SMT-based refinement type checking, our framework provides a high degree of automation. We evaluate our approach through four case studies: meldable heaps, coupon collector, randomised quicksort, and zip trees. The first two demonstrate automation with minimal annotation overhead. The latter two showcase how our monad integrates with interactive proofs, including the first formal verification of the expected runtime of zip trees.

cs.LO↗

arXiv2025

Parameterized Model-checking of Discrete-Timed Networks and Symmetric-Broadcast Systems

We study the complexity of the model-checking problem for parameterized discrete-timed systems with arbitrarily many anonymous and identical processes, with and without a distinguished "controller", and communicating via synchronous rendezvous. Our framework extends the seminal work from German and Sistla on untimed systems by adding discrete-time clocks to processes. For the case without a controller, we show that the systems can be efficiently simulated -- and vice versa -- by systems of untimed processes that communicate via rendezvous and symmetric broadcast, which we call "RB-systems". Symmetric broadcast is a novel communication primitive that allows all processes to synchronize at once; however, it does not distinguish between sending and receiving processes. We show that the parameterized model-checking problem for safety specifications is pspace-complete, and for liveness specifications it is decidable in exptime. The latter result is proved using automata theory, rational linear programming, and geometric reasoning for solving certain reachability questions in a new variant of vector addition systems called "vector rendezvous systems". We believe these proof techniques are of independent interest and will be useful in solving related problems. For the case with a controller, we show that the parameterized model-checking problems for RB-systems and systems with asymmetric broadcast as a primitive are inter-reducible. This allows us to prove that for discrete timed-networks with a controller the parameterized model-checking problem is undecidable for liveness specifications. Our work exploits the intimate connection between parameterized discrete-timed systems and systems of processes communicating via broadcast, providing a rare and surprising decidability result for liveness properties of parameterized timed-systems, as well as extend work from untimed systems to timed systems.

cs.LO↗

arXiv2025

Regular Grammars for Sets of Graphs of Tree-Width 2

Regular word grammars are restricted context-free grammars that define all the recognizable languages of words. This paper generalizes regular grammars from words to certain classes of graphs, by defining regular grammars for unordered unranked trees and graphs of tree-width 2 at most. The qualifier ``regular'' is justified because these grammars define precisely the recognizable (equivalently, CMSO-definable) sets of the respective graph classes. The proof of equivalence between regular and recognizable sets of graphs relies on the effective construction of a recognizer algebra of size doubly-exponential in the size of the grammar. This sets a 2EXPTIME upper bound on the (EXPTIME-hard) problem of inclusion of a context-free language in a regular language, for graphs of tree-width 2 at most. A further syntactic restriction of regular grammars suffices to capture precisely the MSO-definable sets of graphs of tree-width 2 at most, i.e., the sets defined by CMSO formulae without cardinality constraints. Moreover, we show that MSO-definability coincides with recognizability by algebras having an aperiodic parallel composition semigroup, for each class of graphs defined by a bound on the tree-width.

cs.FL↗

arXiv2025

Compositional Shape Analysis with Shared Abduction and Biabductive Loop Acceleration (Extended Version)

Biabduction-based shape analysis is a compositional verification and analysis technique that can prove memory safety in the presence of complex, linked data structures. Despite its usefulness, several open problems persist for this kind of analysis; two of which we address in this paper. On the one hand, the original analysis is path-sensitive but cannot combine safety requirements for related branches. This causes the analysis to require additional soundness checks and decreases the analysis' precision. We extend the underlying symbolic execution and propose a framework for shared abduction where a common pre-condition is maintained for related computation branches. On the other hand, prior implementations lift loop acceleration methods from forward analysis to biabduction analysis by applying them separately on the pre- and post-condition, which can lead to imprecise or even unsound acceleration results that do not form a loop invariant. In contrast, we propose biabductive loop acceleration, which explicitly constructs and checks candidate loop invariants. For this, we also introduce a novel heuristic called shape extrapolation. This heuristic takes advantage of locality in the handling of list-like data structures (which are the most common data structures found in low-level code) and jointly accelerates pre- and post-conditions by extrapolating the related shapes. In addition to making the analysis more precise, our techniques also make biabductive analysis more efficient since they are sound in just one analysis phase. In contrast, prior techniques always require two phases (as the first phase can produce contracts that are unsound and must hence be verified). We experimentally confirm that our techniques improve on prior techniques; both in terms of precision and runtime of the analysis.

cs.LO↗

arXiv2024

Automated Expected Amortised Cost Analysis of Probabilistic Data Structures

In this paper, we present the first fully-automated expected amortised cost analysis of self-adjusting data structures, that is, of randomised splay trees, randomised splay heaps and randomised meldable heaps, which so far have only (semi-) manually been analysed in the literature. Our analysis is stated as a type-and-effect system for a first-order functional programming language with support for sampling over discrete distributions, non-deterministic choice and a ticking operator. The latter allows for the specification of fine-grained cost models. We state two soundness theorems based on two different -- but strongly related -- typing rules of ticking, which account differently for the cost of non-terminating computations. Finally we provide a prototype implementation able to fully automatically analyse the aforementioned case studies.

cs.LO↗

arXiv2024

The Treewidth Boundedness Problem for an Inductive Separation Logic of Relations

The treewidth boundedness problem for a logic asks for the existence of an upper bound on the treewidth of the models of a given formula in that logic. This problem is found to be undecidable for first order logic. We consider a generalization of Separation Logic over relational signatures, interpreted over standard relational structures, and describe an algorithm for the treewidth boundedness problem in the context of this logic.

cs.LO↗

arXiv2024

Deciding Boolean Separation Logic via Small Models (Technical Report)

We present a novel decision procedure for a fragment of separation logic (SL) with arbitrary nesting of separating conjunctions with boolean conjunctions, disjunctions, and guarded negations together with a support for the most common variants of linked lists. Our method is based on a model-based translation to SMT for which we introduce several optimisations$\unicode{x2013}$the most important of them is based on bounding the size of predicate instantiations within models of larger formulae, which leads to a much more efficient translation of SL formulae to SMT. Through a series of experiments, we show that, on the frequently used symbolic heap fragment, our decision procedure is competitive with other existing approaches, and it can outperform them outside the symbolic heap fragment. Moreover, our decision procedure can also handle some formulae for which no decision procedure has been implemented so far.

cs.LO↗

arXiv2024

Tree-Verifiable Graph Grammars

Hyperedge-Replacement grammars (HR) have been introduced by Courcelle in order to extend the notion of context-free sets from words and trees to graphs of bounded tree-width. While for words and trees the syntactic restrictions that guarantee that the associated languages of words resp. trees are regular - and hence, MSO-definable - are known, the situation is far more complicated for graphs. Here, Courcelle proposed the notion of regular graph grammars, a syntactic restriction of HR grammars that guarantees the definability of the associated languages of graphs in Counting Monadic Second Order Logic (CMSO). However, these grammars are not complete in the sense that not every CMSO-definable set of graphs of bounded tree-width can be generated by a regular graph grammar. In this paper, we introduce a new syntactic restriction of HR grammars, called tree-verifiable graph grammars, and a new notion of bounded tree-width, called embeddable bounded tree-width, where the later restricts the trees of a tree-decomposition to be a subgraph of the analyzed graph. The main property of tree-verifiable graph grammars is that their associated languages are CMSO-definable and that the have bounded embeddable tree-width. We show further that they strictly generalize the regular graph grammars of Courcelle. Finally, we establish a completeness result, showing that every language of graphs that is CMSO-definable and of bounded embeddable tree-width can be generated by a tree-verifiable graph grammar.

cs.FL↗

arXiv2024

Effective MSO-Definability for Tree-width Bounded Models of an Inductive Separation Logic of Relations

A class of graph languages is definable in Monadic Second-Order logic (MSO) if and only if it consists of sets of models of MSO formulæ. If, moreover, there is a computable bound on the tree-widths of the graphs in each such set, the satisfiability and entailment problems are decidable, by Courcelle's Theorem. This motivates the comparison of other graph logics to MSO. In this paper, we consider the MSO definability of a Separation Logic of Relations (SLR) that describes simple hyper-graphs, in which each sequence of vertices is attached to at most one edge with a given label. Our logic SLR uses inductive predicates whose recursive definitions consist of existentially quantified separated conjunctions of relation and predicate atoms. The main contribution of this paper is an expressive fragment of SLR that describes bounded tree-width sets of graphs which can, moreover, be effectively translated into MSO.

cs.LO↗

arXiv2023

Expressiveness Results for an Inductive Logic of Separated Relations

In this paper we study a Separation Logic of Relations (SLR) and compare its expressiveness to (Monadic)Second Order Logic (M)SO. SLR is based on the well-known Symbolic Heap fragment of Separation Logic, whose formulae are composed of points-to assertions, inductively defined predicates, with the separating conjunction as the only logical connective. SLR generalizes the Symbolic Heap fragment by supporting general relational atoms, instead of only points-to assertions. In this paper, we restrict ourselves to finite relational structures, and hence only consider Weak (M)SO, where quantification ranges over finite sets. Our main results are that SLR and MSO are incomparable on structures of unbounded treewidth, while SLR can be embedded in SO in general. Furthermore, MSO becomes a strict subset of SLR, when the treewidth of the models is bounded by a parameter and all vertices attached to some hyperedge belong to the interpretation of a fixed unary relation symbol. We also discuss the problem of identifying a fragment of SLR that is equivalent to MSO over models of bounded treewidth.

cs.LO↗

arXiv2022

On the Expressiveness of a Logic of Separated Relations

We compare the model-theoretic expressiveness of the existential fragment of Separation Logic over unrestricted relational signatures (SLR) -- with only separating conjunction as logical connective and higher-order inductive definitions, traditionally known as the symbolic heap fragment -- with the expressiveness of (Monadic) Second Order Logic ((M)SO). While SLR and MSO are incomparable on structures of unbounded treewidth, it turns out that SLR can be embedded in SO, in general, and that MSO becomes a strict subset of SLR, when the treewidth of the models is bounded by a parameter given as input. We also discuss the problem of defining a fragment of SLR that is equivalent to MSO over models of bounded treewidth. Such a fragment would then become the most general Separation Logic with a decidable entailment problem, a key ingredient of practical verification methods for self-adapting (reconfigurable) component-based and distributed systems.

cs.LO↗

arXiv2022

Low-Level Bi-Abduction

The paper proposes a new static analysis designed to handle open programs, i.e., fragments of programs, with dynamic pointer-linked data structures - in particular, various kinds of lists - that employ advanced low-level pointer operations. The goal is to allow such programs be analysed without a need of writing analysis harnesses that would first initialise the structures being handled. The approach builds on a special flavour of separation logic and the approach of bi-abduction. The code of interest is analyzed along the call tree, starting from its leaves, with each function analysed just once without any call context, leading to a set of contracts summarizing the behaviour of the analysed functions. In order to handle the considered programs, methods of abduction existing in the literature are significantly modified and extended in the paper. The proposed approach has been implemented in a tool prototype and successfully evaluated on not large but complex programs.

cs.LO↗