Roberto Sebastiani

Giuseppe Spallitta, Roberto Sebastiani, Armin Biere

A basic algorithm for enumerating disjoint propositional models (disjoint AllSAT) is based on adding blocking clauses incrementally, ruling out previously found models. On the one hand, blocking clauses have the potential to reduce the number of generated models exponentially, as they can handle partial models. On the other hand, the introduction of a large number of blocking clauses affects memory consumption and drastically slows down unit propagation. We propose a new approach that allows for enumerating disjoint partial models with no need for blocking clauses by integrating: Conflict-Driven Clause-Learning (CDCL), Chronological Backtracking (CB), and methods for shrinking models (Implicant Shrinking). Experiments clearly show the benefits of our novel approach.

Giuseppe Spallitta, Gabriele Masina, Paolo Morettin et al.

The development of efficient exact and approximate algorithms for probabilistic inference is a long-standing goal of artificial intelligence research. Whereas substantial progress has been made in dealing with purely discrete or purely continuous domains, adapting the developed solutions to tackle hybrid domains, characterised by discrete and continuous variables and their relationships, is highly non-trivial. Weighted Model Integration (WMI) recently emerged as a unifying formalism for probabilistic inference in hybrid domains. Despite a considerable amount of recent work, allowing WMI algorithms to scale with the complexity of the hybrid problem is still a challenge. In this paper we highlight some substantial limitations of existing state-of-the-art solutions, and develop an algorithm that combines SMT-based enumeration, an efficient technique in formal verification, with an effective encoding of the problem structure. This allows our algorithm to avoid generating redundant models, resulting in drastic computational savings. Additionally, we show how SMT-based approaches can seamlessly deal with different integration techniques, both exact and approximate, significantly expanding the set of problems that can be tackled by WMI technology. An extensive experimental evaluation on both synthetic and real-world datasets confirms the substantial advantage of the proposed solution over existing alternatives. The application potential of this technology is further showcased on a prototypical task aimed at verifying the fairness of probabilistic programs.

Giuseppe Spallitta, Gabriele Masina, Paolo Morettin et al.

Weighted Model Integration (WMI) is a popular formalism aimed at unifying approaches for probabilistic inference in hybrid domains, involving logical and algebraic constraints. Despite a considerable amount of recent work, allowing WMI algorithms to scale with the complexity of the hybrid problem is still a challenge. In this paper we highlight some substantial limitations of existing state-of-the-art solutions, and develop an algorithm that combines SMT-based enumeration, an efficient technique in formal verification, with an effective encoding of the problem structure. This allows our algorithm to avoid generating redundant models, resulting in substantial computational savings. An extensive experimental evaluation on both synthetic and real-world datasets confirms the advantage of the proposed solution over existing alternatives.

Giuseppe Spallitta, Roberto Sebastiani, Armin Biere

All-Solution Satisfiability (AllSAT) and its extension, All-Solution Satisfiability Modulo Theories (AllSMT), have become more relevant in recent years, mainly in formal verification and artificial intelligence applications. The goal of these problems is the enumeration of all satisfying assignments of a formula (for SAT and SMT problems, respectively), making them useful for test generation, model checking, and probabilistic inference. Nevertheless, traditional AllSAT algorithms face significant computational challenges due to the exponential growth of the search space and inefficiencies caused by blocking clauses, which cause memory blowups and degrade unit propagation performances in the long term. This paper presents two novel solvers: tabularAllSAT, a projected AllSAT solver, and tabularAllSMT, a projected AllSMT solver. Both solvers combine Conflict-Driven Clause Learning (CDCL) with chronological backtracking to improve efficiency while ensuring disjoint enumeration. To retrieve compact partial assignments we propose a novel aggressive implicant shrinking algorithm, compatible with chronological backtracking, to minimize the number of partial assignments, reducing overall search complexity. Furthermore, we extend the solver framework to handle projected enumeration and SMT formulas effectively and efficiently, adapting the baseline framework to integrate theory reasoning and the distinction between important and non-important variables. An extensive experimental evaluation demonstrates the superiority of our approach compared to state-of-the-art solvers, particularly in scenarios requiring projection and SMT-based reasoning.

Massimo Michelutti, Gabriele Masina, Giuseppe Spallitta et al.

Decision diagrams (DDs) are powerful tools to represent effectively propositional formulas, which are largely used in many domains, in particular in formal verification and in knowledge compilation. Some forms of DDs (e.g., OBDDs, SDDs) are canonical, that is, (under given conditions on the atom list) they univocally represent equivalence classes of formulas. Given the limited expressiveness of propositional logic, a few attempts to leverage DDs to SMT level have been presented in the literature. Unfortunately, these techniques still suffer from some limitations: most procedures are theory-specific; some produce theory DDs (T-DDs) which do not univocally represent T-valid formulas or T-inconsistent formulas; none of these techniques provably produces theory-canonical T-DDs, which (under given conditions on the T-atom list) univocally represent T-equivalence classes of formulas. Also, these procedures are not easy to implement, and very few implementations are actually available. In this paper, we present a novel very-general technique to leverage DDs to SMT level, which has several advantages: it is very easy to implement on top of an AllSMT solver and a DD package, which are used as blackboxes; it works for every form of DDs and every theory, or combination thereof, supported by the AllSMT solver; it produces theory-canonical T-DDs if the propositional DD is canonical. We have implemented a prototype tool for both T-OBDDs and T-SDDs on top of OBDD and SDD packages and the MathSAT SMT solver. Some preliminary empirical evaluation supports the effectiveness of the approach.

Roberto Sebastiani

Many procedures for SAT-related problems, in particular for those requiring the complete enumeration of satisfying truth assignments, rely their efficiency and effectiveness on the detection of (possibly small) partial assignments satisfying an input formula. Surprisingly, there seems to be no unique universally-agreed definition of formula satisfaction by a partial assignment in the literature. In this paper we analyze in deep the issue of satisfaction by partial assignments, raising a flag about some ambiguities and subtleties of this concept, and investigating their practical consequences. We identify two alternative notions that are implicitly used in the literature, namely verification and entailment, which coincide if applied to CNF formulas but differ and present complementary properties if applied to non-CNF or to existentially-quantified formulas. We show that, although the former is easier to check and as such is implicitly used by most current search procedures, the latter has better theoretical properties, and can improve the efficiency and effectiveness of enumeration procedures.

Paolo Morettin, Andrea Passerini, Roberto Sebastiani

In machine learning (ML) verification, the majority of procedures are non-quantitative and therefore cannot be used for verifying probabilistic models, or be applied in domains where hard guarantees are practically unachievable. The probabilistic formal verification (PFV) of ML models is in its infancy, with the existing approaches limited to specific ML models, properties, or both. This contrasts with standard formal methods techniques, whose successful adoption in real-world scenarios is also due to their support for a wide range of properties and diverse systems. We propose a unifying framework for the PFV of ML systems based on Weighted Model Integration (WMI), a relatively recent formalism for probabilistic inference with algebraic and logical constraints. Crucially, reducing the PFV of ML models to WMI enables the verification of many properties of interest over a wide range of systems, addressing multiple limitations of deterministic verification and ad-hoc algorithms. We substantiate the generality of the approach on prototypical tasks involving the verification of group fairness, monotonicity, robustness to noise, probabilistic local robustness and equivalence among predictors. We characterize the challenges related to the scalability of the approach and, through our WMI-based perspective, we show how successful scaling techniques in the ML verification literature can be generalized beyond their original scope.

Roberto Sebastiani

Many procedures for SAT and SAT-related problems -- in particular for those requiring the complete enumeration of satisfying truth assignments -- rely their efficiency on the detection of partial assignments satisfying an input formula. In this paper we analyze the notion of partial-assignment satisfiability -- in particular when dealing with non-CNF and existentially-quantified formulas -- raising a flag about the ambiguities and subtleties of this concept, and investigating their practical consequences. This may drive the development of more effective assignment-enumeration algorithms.

Chi Mai Nguyen, Roberto Sebastiani, Paolo Giorgini et al.

We are interested in supporting software evolution caused by changing requirements and/or environmental settings. For example, users of a system may require new functionality (changing requirements), or performance enhancements to cope with growing user population. Specifically, we propose to use goal models to capture such changes, and exploit reasoning techniques that derive optimal new specifications for a system whose requirements and/or environment have changed. Moreover, we are interested in discovering new classes of evolution requirements, for example, that give preference to evolutions that minimize implementation effort for the implementation of the evolution. To address both of these problems, we exploit Constraint Goal Models (CGMs) an expressive language for modelling goals that comes with scalable solvers that can solve hybrid constraint and optimization problems using a combination of Satisfiability Modulo Theories (SMT) and Optimization Modulo Theories (OMT) solvers. We evaluate our proposal by modeling and reasoning with a goal model for meeting scheduling.

Stefano Teso, Roberto Sebastiani, Andrea Passerini

Modelling problems containing a mixture of Boolean and numerical variables is a long-standing interest of Artificial Intelligence. However, performing inference and learning in hybrid domains is a particularly daunting task. The ability to model this kind of domains is crucial in "learning to design" tasks, that is, learning applications where the goal is to learn from examples how to perform automatic {\em de novo} design of novel objects. In this paper we present Structured Learning Modulo Theories, a max-margin approach for learning in hybrid domains based on Satisfiability Modulo Theories, which allows to combine Boolean reasoning and optimization over continuous linear arithmetical constraints. The main idea is to leverage a state-of-the-art generalized Satisfiability Modulo Theory solver for implementing the inference and separation oracles of Structured Output SVMs. We validate our method on artificial and real world scenarios.

Stefano Teso, Roberto Sebastiani, Andrea Passerini

Generally speaking, the goal of constructive learning could be seen as, given an example set of structured objects, to generate novel objects with similar properties. From a statistical-relational learning (SRL) viewpoint, the task can be interpreted as a constraint satisfaction problem, i.e. the generated objects must obey a set of soft constraints, whose weights are estimated from the data. Traditional SRL approaches rely on (finite) First-Order Logic (FOL) as a description language, and on MAX-SAT solvers to perform inference. Alas, FOL is unsuited for con- structive problems where the objects contain a mixture of Boolean and numerical variables. It is in fact difficult to implement, e.g. linear arithmetic constraints within the language of FOL. In this paper we propose a novel class of hybrid SRL methods that rely on Satisfiability Modulo Theories, an alternative class of for- mal languages that allow to describe, and reason over, mixed Boolean-numerical objects and constraints. The resulting methods, which we call Learning Mod- ulo Theories, are formulated within the structured output SVM framework, and employ a weighted SMT solver as an optimization oracle to perform efficient in- ference and discriminative max margin weight learning. We also present a few examples of constructive learning applications enabled by our method.

Alessandro Cimatti, Alberto Griggio, Roberto Sebastiani

The problem of finding small unsatisfiable cores for SAT formulas has recently received a lot of interest, mostly for its applications in formal verification. However, propositional logic is often not expressive enough for representing many interesting verification problems, which can be more naturally addressed in the framework of Satisfiability Modulo Theories, SMT. Surprisingly, the problem of finding unsatisfiable cores in SMT has received very little attention in the literature. In this paper we present a novel approach to this problem, called the Lemma-Lifting approach. The main idea is to combine an SMT solver with an external propositional core extractor. The SMT solver produces the theory lemmas found during the search, dynamically lifting the suitable amount of theory information to the Boolean level. The core extractor is then called on the Boolean abstraction of the original SMT problem and of the theory lemmas. This results in an unsatisfiable core for the original SMT problem, once the remaining theory lemmas are removed. The approach is conceptually interesting, and has several advantages in practice. In fact, it is extremely simple to implement and to update, and it can be interfaced with every propositional core extractor in a plug-and-play manner, so as to benefit for free of all unsat-core reduction techniques which have been or will be made available. We have evaluated our algorithm with a very extensive empirical test on SMT-LIB benchmarks, which confirms the validity and potential of this approach.

Roberto Sebastiani, Michele Vescovi

In the last two decades, modal and description logics have been applied to numerous areas of computer science, including knowledge representation, formal verification, database theory, distributed computing and, more recently, semantic web and ontologies. For this reason, the problem of automated reasoning in modal and description logics has been thoroughly investigated. In particular, many approaches have been proposed for efficiently handling the satisfiability of the core normal modal logic K(m), and of its notational variant, the description logic ALC. Although simple in structure, K(m)/ALC is computationally very hard to reason on, its satisfiability being PSPACE-complete. In this paper we start exploring the idea of performing automated reasoning tasks in modal and description logics by encoding them into SAT, so that to be handled by state-of-the-art SAT tools; as with most previous approaches, we begin our investigation from the satisfiability in K(m). We propose an efficient encoding, and we test it on an extensive set of benchmarks, comparing the approach with the main state-of-the-art tools available. Although the encoding is necessarily worst-case exponential, from our experiments we notice that, in practice, this approach can handle most or all the problems which are at the reach of the other approaches, with performances which are comparable with, or even better than, those of the current state-of-the-art tools.

Roberto Sebastiani, Silvia Tomasi

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.