Proof Generation in CDSAT

The main ideas in the CDSAT (Conflict-Driven Satisfiability) framework for SMT are summarized, leading to approaches to proof generation in CDSAT.Comment: In Proceedings PxTP 2021, arXiv:2107.0154

Automated reasoning for explainable artificial intelligence

Reasoning and learning have been considered fundamental features of intelligence ever since the dawn of the field of artificial intelligence, leading to the development of the research areas of automated reasoning and machine learning. This short paper is a non-technical position statement that aims at prompting a discussion of the relationship between automated reasoning and machine learning, and more generally between automated reasoning and artificial intelligence. We suggest that the emergence of the new paradigm of XAI, that stands for eXplainable Artificial Intelligence, is an opportunity for rethinking these relationships, and that XAI may offer a grand challenge for future research on automated reasoning

Set of support, demodulation, paramodulation: a historical perspective

This article is a tribute to the scientific legacy of automated reasoning pioneer and JAR founder Lawrence T. (Larry) Wos. Larry's main technical contributions were the set-of-support strategy for resolution theorem proving, and the demodulation and paramodulation inference rules for building equality into resolution. Starting from the original definitions of these concepts in Larry's papers, this survey traces their evolution, unearthing the often forgotten trails that connect Larry's original definitions to those that became standard in the field

Reasoning about quantifiers in SMT: the QSMA algorithm

This abstract summarizes the QSMA algorithm for satisfiability of arbitrary quantified formulas modulo a complete theory and an assignment to the free variables in the formula. It is the abstract corresponding to an invited talk

On conflict-driven reasoning

Automated formal methods and automated reasoning are interconnected, as formal methods generate reasoning problems and incorporate reasoning techniques. For example, formal methods tools employ reasoning engines to find solutions of sets of constraints, or proofs of conjectures. From a reasoning perspective, the expressivity of the logical language is often directly proportional to the difficulty of the problem. In propositional logic, Conflict-Driven Clause Learning (CDCL) is one of the key features of state-of-the-art satisfiability solvers. The idea is to restrict inferences to those needed to explain conflicts, and use conflicts to prune a backtracking search. A current research direction in automated reasoning is to generalize this notion of conflict-driven satisfiability to a paradigm of conflict-driven reasoning in first-order theories for satisfiability modulo theories and assignments, and even in full first-order logic for generic automated theorem proving. While this is a promising and exciting lead, it also poses formidable challenges

Conflict-driven reasoning in unions of theories

Many applications of automated reasoning require decision procedures for the satisfiability of a formula in a theory given by the union of a few theories. Reasoning in a union of theories can be approached in more than one way. The equality-sharing method, also known as Nelson-Oppen scheme, combines decision procedures for the component theories. Superposition-based theorem-proving strategies unite the presentations of the theories to reason about their union. CDSAT, which stands for Conflict-Driven SATisfiability, assumes that each theory is equipped with an inference system, called theory module, and coordinates the theory modules to reason in a conflict-driven manner in the union of the theories. A theory module is an abstraction of a decision procedure, made of inference rules that may correspond to axioms of the theory. Conflict-driven means that the system maintains a representation of a candidate partial model of the formula, and performs nontrivial inferences only to explain conflicts between the candidate model and the formula, so that the conflict can be solved by updating the partial model. CDSAT provides a framework where the theory modules cooperate to build the candidate model and to explain the conflicts. This talk presents CDSAT placing it in the big picture of multi-theory reasoning and conflict-driven reasoning

Six Decades of Automated Reasoning: Papers in Memory of Larry Wos (Preface)

Preface of the special issue of the Journal of Automated Reasoning in memory of Larry Wo

On SGGS and Horn clauses

SGGS (Semantically-Guided Goal-Sensitive reasoning) is a refutationally complete theorem-proving method that offers first-order conflict-driven reasoning and is model complete in the limit. This paper investigates the behavior of SGGS on Horn clauses, which are widely used in declarative programming, knowledge representation, and verification. We show that SGGS generates the least Herbrand model of a set of definite clauses, and that SGGS terminates on Horn clauses if and only if hyperresolution does, with the advantage that SGGS builds a model. We report on experiments applying the SGGS prototype prover Koala to Horn problems, with promising performances especially on satisfiable inputs

The Eos SMT/SMA-solver: a preliminary report

This is a preliminary report of work in progress on the development of the Eos SMT/SMA-solver. Eos is the first solver built from the start based on the CDSAT (Conflict-Driven SATisfiability) paradigm for solving satisfiability problems modulo theories and assignments. The latter means that assignments to first-order terms may appear in the input. CDSAT generalizes MCSAT (Model-Constructing SATisfiability), hence CDCL (Conflict-Driven Clause Learning), to theory combination. CDSAT reasons in a union of theories by combining in a conflict-driven manner theory inference systems, called theory modules. The current version of Eos has modules for propositional logic, equality with uninterpreted function symbols (UF), and linear real arithmetic. The module for propositional logic is a MiniSAT-inspired SAT solver. A key feature of MCSAT/CDSAT is theory conflict explanation by theory inferences: to this end, the Eos module for UF applies congruence closure inferences, and the Eos module for real arithmetic uses Fourier-Motzkin resolution; both rules may generate new (i.e., non-input) literals. The core solver in Eos implements the CDSAT transition system and several heuristics used in state-of-the-art CDCL-based SAT solvers. Some of these heuristics (e.g., random restarts) can be reused directly in the context of CDSAT, while others are adapted. Eos employs a generalization of the VSIDS heuristics to make decisions on both propositional and first-order terms, and the watched literals scheme for both BCP (Boolean Constraint Propagation) and deductions involving arithmetic terms and uninterpreted terms