arXiv subjects
Publications and source records attributed to Andrew Reynolds.
We give an overview of recent techniques for implementing syntax-guided synthesis (SyGuS) algorithms in the core of Satisfiability Modulo Theories (SMT) solvers. We define several classes of synthesis conjectures and corresponding techniques that can be used when dealing with each class of conjecture.
SMT solvers have been used successfully as reasoning engines for automated verification and other applications based on automated reasoning. Current techniques for dealing with quantified formulas in SMT are generally incomplete, forcing SMT solvers to report "unknown" when they fail to prove the unsatisfiability of a formula with quantifiers. This inability to return counter-models limits their usefulness in applications that produce queries involving quantified formulas. In this paper, we reduce these limitations by integrating finite model finding techniques based on constraint solving into the architecture used by modern SMT solvers. This approach is made possible by a novel solver for cardinality constraints, as well as techniques for on-demand instantiation of quantified formulas. Experiments show that our approach is competitive with the state of the art in SMT, and orthogonal to approaches in automated theorem proving.
Separation Logic (SL) is a well-known assertion language used in Hoare-style modular proof systems for programs with dynamically allocated data structures. In this paper we investigate the fragment of first-order SL restricted to the Bernays-Schoenfinkel-Ramsey quantifier prefix $\exists^*\forall^*$, where the quantified variables range over the set of memory locations. When this set is uninterpreted (has no associated theory) the fragment is PSPACE-complete, which matches the complexity of the quantifier-free fragment. However, SL becomes undecidable when the quantifier prefix belongs to $\exists^*\forall^*\exists^*$ instead, or when the memory locations are interpreted as integers with linear arithmetic constraints, thus setting a sharp boundary for decidability within SL. We have implemented a decision procedure for the decidable fragment of $\exists^*\forall^*$SL as a specialized solver inside a DPLL($T$) architecture, within the CVC4 SMT solver. The evaluation of our implementation was carried out using two sets of verification conditions, produced by (i) unfolding inductive predicates, and (ii) a weakest precondition-based verification condition generator. Experimental data shows that automated quantifier instantiation has little overhead, compared to manual model-based instantiation.
This paper presents a complete decision procedure for the entire quantifier-free fragment of Separation Logic ($\seplog$) interpreted over heaplets with data elements ranging over a parametric multi-sorted (possibly infinite) domain. The algorithm uses a combination of theories and is used as a specialized solver inside a DPLL($T$) architecture. A prototype was implemented within the CVC4 SMT solver. Preliminary evaluation suggests the possibility of using this procedure as a building block of a more elaborate theorem prover for SL with inductive predicates, or as back-end of a bounded model checker for programs with low-level pointer and data manipulations.
This paper presents a framework to derive instantiation-based decision procedures for satisfiability of quantified formulas in first-order theories, including its correctness, implementation, and evaluation. Using this framework we derive decision procedures for linear real arithmetic (LRA) and linear integer arithmetic (LIA) formulas with one quantifier alternation. Our procedure can be integrated into the solving architecture used by typical SMT solvers. Experimental results on standardized benchmarks from model checking, static analysis, and synthesis show that our implementation of the procedure in the SMT solver CVC4 outperforms existing tools for quantified linear arithmetic.
This report describes several approaches for handling synthesis conjectures within an Satisfiability Modulo Theories (SMT) solver. We describe approaches that primarily focus on determining the unsatisfiability of the negated form of synthesis conjectures using new techniques for quantifier instantiation.
Satisfiability Modulo Theories (SMT) solvers incorporate decision procedures for theories of data types that commonly occur in software. This makes them important tools for automating verification problems. A limitation frequently encountered is that verification problems are often not fully expressible in the theories supported natively by the solvers. Many solvers allow the specification of application-specific theories as quantified axioms, but their handling is incomplete outside of narrow special cases. In this work, we show how SMT solvers can be used to obtain complete decision procedures for local theory extensions, an important class of theories that are decidable using finite instantiation of axioms. We present an algorithm that uses E-matching to generate instances incrementally during the search, significantly reducing the number of generated instances compared to eager instantiation strategies. We have used two SMT solvers to implement this algorithm and conducted an extensive experimental evaluation on benchmarks derived from verification conditions for heap-manipulating programs. We believe that our results are of interest to both the users of SMT solvers as well as their developers.
We introduce the first program synthesis engine implemented inside an SMT solver. We present an approach that extracts solution functions from unsatisfiability proofs of the negated form of synthesis conjectures. We also discuss novel counterexample-guided techniques for quantifier instantiation that we use to make finding such proofs practically feasible. A particularly important class of specifications are single-invocation properties, for which we present a dedicated algorithm. To support syntax restrictions on generated solutions, our approach can transform a solution found without restrictions into the desired syntactic form. As an alternative, we show how to use evaluation function axioms to embed syntactic restrictions into constraints over algebraic datatypes, and then use an algebraic datatype decision procedure to drive synthesis. Our experimental evaluation on syntax-guided synthesis benchmarks shows that our implementation in the CVC4 SMT solver is competitive with state-of-the-art tools for synthesis.
The affine oriented Brauer category is a monoidal category obtained from the oriented Brauer category (= the free symmetric monoidal category generated by a single object and its dual) by adjoining a polynomial generator subject to appropriate relations. In this article, we prove a basis theorem for the morphism spaces in this category, as well as for all of its cyclotomic quotients.