ABSTRACT. The standard approach to algorithm development is to focus on a specific problem and develop for it a specific algorithm. Codd’s introduction of the relational model in 1970 included two fundamental ideas: (1) relations provide a universal data representation formalism, and (2) relational databases can be queried using first-order logic. Realizing these ideas required the development of a meta- algorithm, which takes a declarative query and executes it with respect to a database. In this talk, I will describe this approach, which I call Logical Algorithmics, in detail, and trace a decades-long path from the comoutational complexity theory of relational queries to recent tools for Boolean reasoning.

ABSTRACT. Recent advances have enabled powerful distributed SAT solvers to emit proofs of unsatisfiability, which renders them as trustworthy as sequential solvers. However, this mode of operation is still lacking behind conventional distributed solving in terms of scalability. We argue that the core limiting factor of such approaches is the requirement of a single, persistent artifact at the end of solving that is then checked independently (and sequentially). As an alternative, we propose a bottleneck-free setup which exploits recent advancements in producing and processing LRAT information to immediately check all solvers’ reasoning on-the-fly during solving. In terms of clause sharing, the guarantee of a derived clause’s soundness is transferred from the sending to the receiving side via cryptographic signatures. Experiments with up to 2432 cores (32 nodes) indicate that our approach reduces the running time overhead incurred by proof checking by an order of magnitude, down to a median overhead of ≤42% over non trusted solving.

ABSTRACT. Unit propagation is known to be one of the most time-consuming procedures inside CDCL-based SAT solvers. Not surprisingly, it has been studied in depth and the two-watched-literal scheme, enhanced with implementation details that boost its performance, has emerged as the dominant method.

In pseudo-Boolean solvers, the importance of unit propagation is similar, but no dominant method exists: counter propagation and watched-based extensions are efficient for different types of constraints, which has opened the door to hybrid methods. However, probably due to the higher complexity of implementing pseudo-Boolean solvers, research efforts have not focused much on concrete implementation details for unit propagation but rather on higher-level aspects of other procedures, such as conflict analysis.

In this paper, we present (i) a novel methodology to precisely assess the performance of unit-propagation mechanisms, (ii) an evaluation of implementation variants of the propagation methods present in RoundingSat and (iii) a detailed analysis showing that hybrid methods outperform the ones based on a single technique. The final contribution of this paper is to show that a carefully implemented hybrid propagation method is considerably faster than the preferred propagation mechanism in RoundingSat, and that this improvement leads to a better overall performance of the solver.

ABSTRACT. Widespread use of artificial intelligence (AI) algorithms and machine learning (ML) models on the one hand and a number of crucial issues pertaining to them warrant the need for explainable artificial intelligence (XAI). A key explainability question is: given this decision was made, what are the input features which contributed to the decision? Although a range of XAI approaches exist to tackle this problem, most of them have significant limitations. Heuristic XAI approaches suffer from the lack of quality guarantees, and often try to approximate Shapley values, which is not the same as explaining which features contribute to a decision. A recent alternative is so-called formal feature attribution (FFA), which defines feature importance as the fraction of formal abductive explanations (AXp’s) containing the given feature. This measures feature importance from the view of formally reasoning about the model’s behavior. Namely, given a system of constraints logically representing the ML model of interest, computing an AXp requires finding a minimal unsatisfiable subset (MUS) of the system. It is challenging to compute FFA using its definition because that involves counting of AXp’s (equivalently, counting of MUSes), although one can approximate it. Based on these results, this paper makes several contributions. First, it gives compelling evidence that computing FFA is intractable, even if the set of contrastive formal explanations (CXp’s), which correspond to minimal correction subsets (MCSes) of the logical system, is provided, by proving that the problem is #P-hard. Second, by using the duality between MUSes and MCSes, it proposes an efficient heuristic to switch from MCS enumeration to MUS enumeration on-the-fly resulting in an adaptive explanation enumeration algorithm effectively approximating FFA in an anytime fashion. Finally, experimental results obtained on a range of widely used datasets demonstrate the effectiveness of the proposed FFA approximation approach in terms of the error of FFA approximation as well as the number of explanations computed and their diversity given a fixed time limit.

ABSTRACT. Motivated by the application of quality assessment of logic locking we introduce Hierarchical Stochastic SAT (HSSAT) which generalizes Stochastic SAT (SSAT). We look into the complexity of HSSAT and for solving HSSAT formulas we provide a prototype solver which is exact both from an algorithmic and from a numerical point of view. Finally, we perform an intensive experimental evaluation of our HSSAT solver in the context of quality assessment of logic locking.

ABSTRACT. In applications, QBF solvers are often required to generate strategies. This typically involves a process known as strategy extraction, where a Boolean circuit encoding a strategy is computed from a proof. It has previously been observed that Craig interpolation in propositional logic can be seen as a special case of QBF strategy extraction. In this paper we explore this connection further and show that, conversely, any strategy for a false QBF corresponds to a sequence of interpolants in its complete (Herbrand) expansion. Inspired by this correspondence, we present a new strategy extraction algorithm for the expansion-based proof system Exp+Res. Its asymptotic running time matches the best known bound of O(mn) for a proof with m lines and n universally quantified variables. We report on experiments comparing this algorithm with a strategy extraction algorithm based on combining partial strategies, as well as with round-based strategy extraction.

ABSTRACT. In this paper, we explore the application of blocked clause elimination for projected model counting. This is the problem of determining the number of models ∥∃X.Σ∥ of a propositional formula Σ after eliminating a given set X of variables existentially. Although blocked clause elimination is a well-known technique for SAT solving, its direct application to model counting is challenging as in general it changes the number of models. However, we demonstrate, by focusing on projected variables during the blocked clause search, that blocked clause elimination can be leveraged while preserving the correct model count. To take advantage of blocked clause elimination in an efficient way during model counting, a novel data structure and associated algorithms are introduced. Our proposed approach is implemented in the model counter d4. Our experiments demonstrate the computational benefits of our new method of blocked clause elimination for projected model counting.

ABSTRACT. We introduce a SAT enabled algorithm for checking language emptiness of alternating finite automata (AFA), with transition relation encoded symbolically in a form of (compactly represented) logical formulae. A SAT solver is used to compute predecessors of AFA configurations, and at the same time, to evaluate the subsumption of newly found configurations in the antichain of the previously found ones. The algorithm could be naively implemented by an incremental SAT solver where the growing antichain is represented by adding new clauses. To make it efficient, we 1) force the SAT solver to prioritize largest/subsumption-strongest predecessors, and 2) store the antichain clauses in a special variant of trie that allows fast subsumption testing. The experimental results suggest that the resulting emptiness checker is very efficient compared to the state of the art, and that the trie data structure may even be beneficial in general SAT solving.

ABSTRACT. We propose a novel method for efficient handling of string constraints containing string-integer conversions, which often occur within verification of web programs. Our approach is based on a translation of the conversions together with regular constraints into a linear integer arithmetic formula. We combine the method with the recently introduced stabilization-based procedure for solving string (dis)equations with regular and length constraints and integrate the proposed method into the string solver Z3-Noodler. Our experimental evaluation shows that our technique is competitive and on some established benchmarks even several orders of magnitude faster than the state of the art.

ABSTRACT. In this paper, we suggest new SAT encodings of the partial-ordering based ILP model for the graph coloring problem (GCP) and the bandwidth coloring problem (BCP). The GCP asks for the minimum number of colors that can be assigned to the vertices of a given graph such that each two adjacent vertices get different colors. The BCP is a generalization, where each edge has a weight that enforces a minimal "distance" between the assigned colors, and the goal is to minimize the "largest" color used. For the widely studied GCP, we experimentally compare our new SAT encoding to the state-of-the-art approaches on the DIMACS benchmark set. Our evaluation confirms that this SAT encoding is effective for sparse graphs and even outperforms the state-of-the-art on some DIMACS instances. For the BCP, our theoretical analysis shows that the partial-ordering based SAT and ILP formulations have an asymptotically smaller size than that of the classical assignment-based model. Our practical evaluation confirms not only a dominance compared to the assignment-based encodings but also to the state-of-the-art approaches on a set of benchmark instances. Up to our knowledge, we have solved several open instances of the BCP from the literature for the first time.