We consider the decision problem for quantifier-free formulas whose atoms are linear inequalities interpreted over the reals or rationals. This problem may be decided using satisfiability modulo theory (SMT), using a mixture of a SAT solver and a simplex-based decision procedure for conjunctions. State-of-the-art SMT solvers use simplex implementations over rational numbers, which perform well for typical problems arising from model-checking and program analysis (sparse inequalities, small coefficients) but are slow for other applications (denser problems, larger coefficients). We propose a simple preprocessing phase that can be adapted on existing SMT solvers and that may be optionally triggered. Despite using floating-point computations, our method is sound and complete - it merely affects efficiency. We implemented the method and provide benchmarks showing that this change brings a naive and slow decision procedure ("textbook simplex" with rational numbers) up to the efficiency of recent SMT solvers, over test cases arising from model-checking, and makes it definitely faster than state-of-the-art SMT solvers on dense examples.
Deep Dive into On using floating-point computations to help an exact linear arithmetic decision procedure.
We consider the decision problem for quantifier-free formulas whose atoms are linear inequalities interpreted over the reals or rationals. This problem may be decided using satisfiability modulo theory (SMT), using a mixture of a SAT solver and a simplex-based decision procedure for conjunctions. State-of-the-art SMT solvers use simplex implementations over rational numbers, which perform well for typical problems arising from model-checking and program analysis (sparse inequalities, small coefficients) but are slow for other applications (denser problems, larger coefficients). We propose a simple preprocessing phase that can be adapted on existing SMT solvers and that may be optionally triggered. Despite using floating-point computations, our method is sound and complete - it merely affects efficiency. We implemented the method and provide benchmarks showing that this change brings a naive and slow decision procedure (“textbook simplex” with rational numbers) up to the efficiency of r
Decision procedures for arithmetic theories are widely used for computer-aided verification. A decision procedure for a theory T takes as input a formula of T and outputs a Boolean: whether the formula is satisfiable. For many decidable and potentially useful theories, however, decision procedures are sometimes too slow to process problems beyond small examples. This is for instance the case of the theory of real closed fields (polynomial arithmetic over the real numbers). Excessive computation times arise from two sources: the Boolean structure of the formulas to be decided (propositional satisfiability is currently solved in exponential time in the worst case), and the intrinsic hardness of the theory. In recent years, SAT modulo theory (SMT) techniques have addressed the former source of inefficiency, by leveraging the power of efficient SAT (Boolean satisfiability) solvers to deal with the Boolean structure. SMT solvers combine a SAT solver with a decision procedure for conjunctions of atoms in T . If T is linear real arithmetic (LRA), then this decision procedure must decide whether a set of linear inequalities with rational or integer coefficients has rational solutions.
The problem of testing whether a set of linear inequalities has a solution and, if it has, to find a solution that maximizes some linear combination of the variables is known as linear programming and has been considerably studied in operational research. Very efficient implementations exist, whether commercial or not, and are able to solve very large problems. They are not directly applicable to our problems, however, if only because they operate over floating-point numbers and provide in general no assurance that their result is truthful, despite elaborate precautions taken against numerical instabilities. As a result, the decision procedures for LRA in SMT solvers are implemented with rational arithmetic, which is slower than floating-point, especially if coefficients become large, as often happens with dense linear problems: large coefficients force the use of costly extended precision arithmetic. It thus would seem desirable to leverage the speed and maturity of floating-point linear programming systems to enhance exact decision procedures.
This article describes a simple preprocessing phase that can be added, with minimal change, to existing rational simplex implementations used as decision procedures inside SMT solvers. The procedure was implemented on top of a naive and inefficient rational simplex implementation; the resulting procedure rivals recent SMT solvers.
A similar method has been proposed in the operational research field [4],1 but there are reasons why it may perform less well for the typical optimization tasks of operational research than for decision tasks. The novelty of this article is the application of this technique as a simple modification of existing SMT algorithms.
SMT solvers need a decision procedure capable of:
• being used incrementally: adding new constraints to the problem, and removing blocks of constraints, preferably without recomputing everything;
• telling whether the problem is satisfiable or not;
• if the problem is unsatisfiable, outputting a (preferably small or even minimal) unsatisfiable subset of the constraints;
• propagating theory lemmas, if possible at reasonable costs (from a conjunction
All current SMT solvers seem to decide general linear real arithmetic (as opposed to syntactic restrictions thereof such as difference logic) using the simplex algorithm. This algorithm is exponential in the worst case, but tends to perform well in practice; none of the current solvers seem to use a (polynomialtime) interior point method. Our method is a variant of the simplex algorithm; we shall thus first describe the “conventional” simplex.
We shall first give a brief summary on the dual simplex algorithm on which the LRA decision procedures in Yices2 [6, 5] and Z33 [3] are based. There otherwise exist many excellent textbooks on the simplex algorithm [2,15], though these seldom discuss the specifics of implementations in exact precision or incremental use.
Take a system of linear equations, e.g.
(1
The system is first made canonical. Inequalities are scaled so that each left hand side only has integer coefficients with no common factors. Then, each inequality is optionally negated so that the first coefficient appearing (using some arbitrary ordering of the variables) is positive. This ensures that two inequalities constraining the same direction in space (e.g. -y + 3z ≥ -1 and 2y -6z ≥ 3) appear with the exact same left-hand side. For each left-hand side that is not a variable, a new variable is introduced; the system is then converted into a number of linear equalities and bound constraints on the variables. For instance, the above system gets converted into:
The problem is thus formulated as deciding whether a product of intervals intersects a linear subspace given by a basis.
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