Giuseppe Spallitta

Emanuele Civini, Gabriele Masina, Giuseppe Spallitta et al.

Lifting Boolean-reasoning techniques to the SMT level most often requires producing theory lemmas that rule out theory-inconsistent truth assignments. With standard SMT solving, it is common to "lazily" generate such lemmas on demand during the search; with some harder SMT-level tasks -- such as unsat-core extraction, MaxSMT, T-OBDD or T-SDD compilation -- it may be beneficial or even necessary to "eagerly" pre-compute all the needed theory lemmas upfront. Whereas in principle "classic" eager SMT encodings could do the job, they are specific for very few and easy theories, they do not comply with theory combination, and may produce lots of unnecessary lemmas. In this paper, we present theory-agnostic methods for enumerating complete sets of theory lemmas tailored to a given formula. Starting from AllSMT as a baseline approach, we propose improved lemma-enumeration techniques, including divide&conquer, projected enumeration, and theory-driven partitioning, which are highly parallelizable and which may drastically improve scalability. An experimental evaluation demonstrates that these techniques significantly enhance efficiency and enable the method to scale to substantially more complex instances.

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.

Giuseppe Spallitta, Moshe Y. Vardi

In this work we investigate Weighted Model Enumeration (WME): given a Boolean formula and a weight function over its satisfying assignments, enumerate models while accounting for their weights. This setting supports weight-driven queries, such as producing the top-k models or all models above a threshold. While related to AllSAT, Weighted Model Counting, and MaxSAT, these paradigms do not treat selective enumeration under weights as a native solver task. We present CDCL-based algorithms for WME that integrate weight propagation, weight-based pruning, and weight-aware conflict analysis into both chronological and non-chronological backtracking frameworks. Chronological backtracking exploits implicit blocking and keeps the clause database compact, thereby reducing memory footprint and enabling efficient propagation. In contrast, non-chronological backtracking with clause learning supports explicit blocking and restarts. We show that both approaches are feasible and complementary, highlighting trade-offs in pruning effectiveness with weights and clarifying when each performs best. This work establishes WME as a solver-level reasoning task and provides a systematic exploration of its algorithmic foundations.

Gabriele Masina, Emanuale Civini, Massimo Michelutti et al.

In Knowledge Compilation (KC) a propositional knowledge base is compiled off-line into some target form, typically into deterministic decomposable negation normal form (d-DNNF) or one of its subcases, which is then used on-line to answer a large number of queries in polytime, such as clausal entailment, model counting, and others. The general idea is to push as much of the computational effort into the off-line compilation phase, which is amortized over all on-line polytime queries. In this paper, we present for the first time a novel and general technique to leverage d-DNNF compilation and querying to SMT level. Intuitively, before d-DNNF compilation, the input SMT formula is combined with a list of pre-computed ad-hoc theory lemmas, so that the queries at SMT level reduce to those at propositional level. This approach has several features: (i) it works for every theory, or theory combination thereof; (ii) it works for all forms of d-DNNF; (iii) it is easy to implement on top of any d-DNNF compiler and any theory-lemma enumerator, which are used as black boxes; (iv) most importantly, these compiled SMT d-DNNFs can be queried in polytime by means of a standard propositional d-DNNF reasoner. We have implemented a tool on top of state-of-the-art d-DNNF packages and of the MathSAT SMT solver. Some preliminary empirical evaluation supports the effectiveness of the approach.

Giuseppe Spallitta, Leonardo Duenas-Osorio, Moshe Y. Vardi

Many reasoning tasks require short partial satisfying assignments (implicants), sometimes focusing on a set of important variables. SAT-to-Ising-QUBO formulations are implicitly designed so that ground states correspond to total assignments, since the Ising/QUBO model assigns a value to every spin and has no native representation of unassigned variables. We introduce an Ising/QUBO framework that incorporates "don't-care" semantics into the quadratic model via a dual-polarity representation, enabling the retrieval of short implicants. The encoding supports implicant shrinking and projection through minor objective modifications. We provide parameter regimes under which ground states correspond to short partial satisfying assignments, achieving minimality and, when the quadratic penalty function permits, minimum-cardinality. We empirically evaluate the encoding with simulated annealing on random 3-SAT enumeration benchmarks and non-CNF formulas, showing that it leaves about one-third of variables unassigned on random 3-SAT formulas while preserving satisfiability, and that consecutive polarity-freezing rounds achieve minimality (and minimum-cardinality) with high probability.

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.