Integrating proxy theories and numeric model lifting for floating-point arithmetic

Ramachandran, Jaideep; Wahl, Thomas

doi:10.1109/fmcad.2016.7886674

Cited by 5 publications

(6 citation statements)

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“…The work presented in this paper builds on previous research on the use of approximations for solving FPA constraints [23,24]. UppSAT is also close in spirit to the framework presented by Ramachandran and Wahl [22] for efficiently solving FPA constraints based on the notion of 'proxy' theories, which correspond to our 'output theories.' This framework applies a relatively sophisticated method of reconstruction, by applying a fall-back FPA solver to a version of the input constraint in which all but one variables have been substituted by their value in a failing decoded model.…”

Section: Decision Procedures For Floating-point Arithmeticmentioning

confidence: 91%

“…More complex strategies might use a solver during the reconstruction to search for a model within some distance of a decoded (failing) model. A numeric model lifting strategy was proposed by Ramachandran and Wahl [22]. Their method identifies a subset of the model to be tweaked, and instantiates the formula as a univariate satisfiability check.…”

Section: Approximations In Uppsatmentioning

confidence: 99%

“…We present a comparatively simplistic implementation of this kind of approximation, due to the difficulty to refine approximations in real arithmetic in a meaningful way (real arithmetic already represents to infinite-precision arithmetic). Ramachandran and Wahl [22] describe a topological notion of refinement, that requires a back-end solver that handles the combined theory of real arithmetic and FPA. However, solving constraints over this combination of theories is challenging in itself, and efficient SMT solvers are not publicly available, to the best of our knowledge.…”

Section: Ra -The Real Arithmetic Approximation Of Fpamentioning

confidence: 99%

“…RA refinement is achieved by uniform refinement, and results in the full precision after a single iteration. In the case of the topological refinement proposed by Ramachandran and Wahl [22], the precision domain would be the same, but the precision itself would be compositional, i.e., a precision would be associated with each node of the formula. Essentially, the precision would represent a switch, deciding whether a node should be encoded in real arithmetic or floating-point arithmetic.…”

Section: Ra -The Real Arithmetic Approximation Of Fpamentioning

confidence: 99%

“…In this paper, we aim at a uniform description and exploration of the complete design space. We focus on the case of (quantifier-free) floating-point arithmetic (FPA) constraints, a particularly challenging domain that has been studied extensively in the SMT context over the past few years [3,12,23,22,13,24]. To enable uniform exploration of approximation, reconstruction, and refinement methods, as well as simple prototyping and comparative studies, we present UppSAT as a a general framework for building approximating solvers.…”

Section: Introductionmentioning

confidence: 99%

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Exploring Approximations for Floating-Point Arithmetic Using UppSAT

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Backeman

Wintersteiger

et al. 2018

Automated Reasoning

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We consider the problem of solving floating-point constraints obtained from software verification. We present UppSAT -an new implementation of a systematic approximation refinement framework [24] as an abstract SMT solver. Provided with an approximation and a decision procedure (implemented in an off-the-shelf SMT solver), UppSAT yields an approximating SMT solver. Additionally, UppSAT yieldsincludes a library of predefined approximation components which can be combined and extended to define new encodings, orderings and solving strategies. We propose that UppSAT can be used as a sandbox for easy and flexible exploration of new approximations. To substantiate this, we explore several approximations of floating-point arithmetic. Approximations can be viewed as a composition of an encoding into a target theory, a precision ordering, and a number of strategies for model reconstruction and precision (or approximation) refinement. We present encodings of floating-point arithmetic into reduced precision floating-point arithmetic, real-arithmetic, and fixed-point arithmetic (encoded into the theory of bit-vectors in practice). In an experimental evaluation we compare the advantages and disadvantages of approximating solvers obtained by combining various encodings and decision procedures (based on existing, state-of-the-art SMT solvers for floating-point, real, and bit-vector arithmetic).The first form of approximation uses floating-point operations of reduced precision, i.e., with a reduced number of bits for the significant and exponent. Approximations of this kind have previously been studied in [23,24], and found to be an effective way to boost the performance of bit-blasting-based SMT solvers, since the size of FPA circuits tends to grow quickly with the bit-width. The change of the actual formula lies in decreasing the number of bits used for each variable and operator.Example 2. We assume reduction to the floating-point (3, 3) format, i.e., the format in which 3 bits are used for the significant, and 3 bits for the exponent. The approximate formulaφ 3,3 is obtained by replacing the variables x and y with re-typed variants x 3,3 , y 3,3 , casting all floating-point literals to the new format, and replacing the addition operator + and comparison predicate ≤ with the operator + 3,3 and the predicate ≥ Even thoughφ 3,3 is satisfiable the models are not guaranteed models for the original formula, but only satisfies the reduced precision formula because of over/under-flows and rounding errors when working with only three precision and three significand bits. For example,m = {x → 2.0, y → 4.0}), satisfieŝ φ 3,3 because 2.0 3,3 + 3,3 1.75 3,3 = 4 3,3 .

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Section: Decision Procedures For Floating-point Arithmeticmentioning

confidence: 91%

Section: Approximations In Uppsatmentioning

confidence: 99%

Section: Ra -The Real Arithmetic Approximation Of Fpamentioning

confidence: 99%

Section: Ra -The Real Arithmetic Approximation Of Fpamentioning

confidence: 99%