Approximating Holant problems by winding

We give an FPRAS for Holant problems with parity constraints and not-all-equal constraints, a generalisation of the problem of counting sink-free-orientations. The approach combines a sampler for near-assignments of "windable" functions -- using the cycle-unwinding canonical paths technique of Jerrum and Sinclair -- with a bound on the weight of near-assignments. The proof generalises to a larger class of Holant problems; we characterise this class and show that it cannot be extended by expressibility reductions. We then ask whether windability is equivalent to expressibility by matchings circuits (an analogue of matchgates), and give a positive answer for functions of arity three

Interval Slopes as Numerical Abstract Domain for Floating-Point Variables

The design of embedded control systems is mainly done with model-based tools such as Matlab/Simulink. Numerical simulation is the central technique of development and verification of such tools. Floating-point arithmetic, that is well-known to only provide approximated results, is omnipresent in this activity. In order to validate the behaviors of numerical simulations using abstract interpretation-based static analysis, we present, theoretically and with experiments, a new partially relational abstract domain dedicated to floating-point variables. It comes from interval expansion of non-linear functions using slopes and it is able to mimic all the behaviors of the floating-point arithmetic. Hence it is adapted to prove the absence of run-time errors or to analyze the numerical precision of embedded control systems

Optimization in SMT with LA(Q) Cost Functions

In the contexts of automated reasoning and formal verification, important decision problems are effectively encoded into Satisfiability Modulo Theories (SMT). In the last decade efficient SMT solvers have been developed for several theories of practical interest (e.g., linear arithmetic, arrays, bit-vectors). Surprisingly, very few work has been done to extend SMT to deal with optimization problems; in particular, we are not aware of any work on SMT solvers able to produce solutions which minimize cost functions over arithmetical variables. This is unfortunate, since some problems of interest require this functionality. In this paper we start filling this gap. We present and discuss two general procedures for leveraging SMT to handle the minimization of LA(Q) cost functions, combining SMT with standard minimization techniques. We have implemented the proposed approach within the MathSAT SMT solver. Due to the lack of competitors in AR and SMT domains, we experimentally evaluated our implementation against state-of-the-art tools for the domain of linear generalized disjunctive programming (LGDP), which is closest in spirit to our domain, on sets of problems which have been previously proposed as benchmarks for the latter tools. The results show that our tool is very competitive with, and often outperforms, these tools on these problems, clearly demonstrating the potential of the approach.Comment: A shorter version is currently under submissio

Programming with Specifications

This thesis explores the use of specifications for the construction of correct programs. We go beyond their standard use as run-time assertions, and present algorithms, techniques and implementations for the tasks of 1) program verification, 2) declarative programming and 3) software synthesis. These results are made possible by our advances in the domains of decision procedure design and implementation. In the first part of this thesis, we present a decidability result for a class of logics that support user-defined recursive function definitions. Constraints in this class can encode expressive properties of recursive data structures, such as sortedness of a list, or balancing of a search tree. As a result, complex verification conditions can be stated concisely and solved entirely automatically. We also present a new decision procedure for a logic to reason about sets and constraints over their cardinalities. The key insight lies in a technique to decompose con- straints according to mutual dependencies. Compared to previous techniques, our algorithm brings significant improvements in running times, and for the first time integrates reasoning about cardinalities within the popular DPLL(T ) setting. We integrated our algorithmic ad- vances into Leon, a static analyzer for functional programs. Leon can reason about constraints involving arbitrary recursive function definitions, and has the desirable theoretical property that it will always find counter-examples to assertions that do not hold. We illustrate the flexibility and efficiency of Leon through experimental evaluation, where we used it to prove detailed correctness properties of data structure implementations. We then illustrate how program specifications can be used as a high-level programming construct ; we present Kaplan, an extension of Scala with first-class logical constraints. Kaplan allows programmers to create, manipulate and combine constraints as they would any other data structure. Our implementation of Kaplan illustrates how declarative programming can be incorporated into an existing mainstream programming language. Moreover, we examine techniques to transform, at compile-time, program specifications into efficient executable code. This approach of software synthesis combines the correctness benefits of declarative programming with the efficiency of imperative or functional programming

An Abstract Domain to Infer Types over Zones in Spreadsheets

International audienceSpreadsheet languages are very commonly used, by large user bases, yet they are error prone. However, many semantic issues and errors could be avoided by enforcing a stricter type discipline. As declaring and specifying type information would represent a prohibitive amount of work for users, we propose an abstract interpretation based static analysis for spreadsheet programs that infers type constraints over zones of spreadsheets, viewed as two-dimensional arrays. Our abstract domain consists in a cardinal power from a numerical abstraction describing zones in a spreadsheet to an abstraction of cell values, including type properties. We formalize this abstract domain and its operators (transfer functions, join, widening and reduction) as well as a static analysis for a simplified spreadsheet language. Last, we propose a representation for abstract values and present an implementation of our analysis

Programmiersprachen und Rechenkonzepte

Seit 1984 veranstaltet die GI-Fachgruppe "Programmiersprachen und Rechenkonzepte", die aus den ehemaligen Fachgruppen 2.1.3 "Implementierung von Programmiersprachen" und 2.1.4 "Alternative Konzepte für Sprachen und Rechner" hervorgegangen ist, regelmäßig im Frühjahr einen Workshop im Physikzentrum Bad Honnef. Das Treffen dient in erster Linie dem gegenseitigen Kennenlernen, dem Erfahrungsaustausch, der Diskussion und der Vertiefung gegenseitiger Kontakte

Controller Synthesis for Timeline-based Games

In the timeline-based approach to planning, the evolution over time of a set of state variables (the timelines) is governed by a set of temporal constraints. Traditional timeline-based planning systems excel at the integration of planning with execution by handling temporal uncertainty. In order to handle general nondeterminism as well, the concept of timeline-based games has been recently introduced. It has been proved that finding whether a winning strategy exists for such games is 2EXPTIME-complete. However, a concrete approach to synthesize controllers implementing such strategies is missing. This paper fills this gap, by providing an effective and computationally optimal approach to controller synthesis for timeline-based games.Comment: arXiv admin note: substantial text overlap with arXiv:2209.1031