An SMT Theory of Fixed-Point Arithmetic

Baranowski, Marek S.; He, Shaobo; Lechner, Mathias; Nguyen, Thanh Son; Rakamarić, Zvonimir

doi:10.1007/978-3-030-51074-9_2

Cited by 13 publications

(11 citation statements)

References 36 publications

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“…Columns 3,4,and 5 (resp. 6,7,and 8) show the number of adversarial examples, the execution time, and the proportion of adversarial examples in the input region. Column 9 shows the error rate RealNum−EstimatedNum EstimatedNum , where RealNum is from our result, and EstimatedNum is from NPAQ.…”

Section: Robustness Analysismentioning

confidence: 99%

“…Existing techniques for quantized DNNs are mostly based on constraint solving, in particular, SAT/SMT solving [12,33,45,46]. Following this line, verification of BNNs with ternary weights [28,48] and quantized DNNs with multiple bits [7,22,24] were also studied. Recently, the SMT-based framework Marabou for real-numbered DNNs [31] has also been extended to support BNNs [1].…”

Section: Related Workmentioning

confidence: 99%

“…Some recent work also studies variants of BNNs [28,48], i.e., BNNs with ternary weights. For quantized DNNs with multiple bits (i.e., fixed-points), it is natural to encode them as quantifier-free SMT formulas, e.g., using bit-vector and fixed-point theories [7,22,24], so that off-the-shelf SMT solvers can be leveraged. In another direction, BDD-based approaches currently can tackle BNNs only [54].…”

Section: Introductionmentioning

confidence: 99%

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BDD4BNN: A BDD-Based Quantitative Analysis Framework for Binarized Neural Networks

Zhang

Zhao

Chen

et al. 2021

Lecture Notes in Computer Science

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Verifying and explaining the behavior of neural networks is becoming increasingly important, especially when they are deployed in safety-critical applications. In this paper, we study verification and interpretability problems for Binarized Neural Networks (BNNs), the 1-bit quantization of general real-numbered neural networks. Our approach is to encode BNNs into Binary Decision Diagrams (BDDs), which is done by exploiting the internal structure of the BNNs. In particular, we translate the input-output relation of blocks in BNNs to cardinality constraints which are in turn encoded by BDDs. Based on the encoding, we develop a quantitative framework for BNNs where precise and comprehensive analysis of BNNs can be performed. We demonstrate the application of our framework by providing quantitative robustness analysis and interpretability for BNNs. We implement a prototype tool and carry out extensive experiments, confirming the effectiveness and efficiency of our approach.

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Section: Robustness Analysismentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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BDD4BNN: A BDD-Based Quantitative Analysis Framework for Binarized Neural Networks

Zhang

Zhao

Chen

et al. 2021

Lecture Notes in Computer Science

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“…Moving from the idealized mathematical model of ANNs (infinite precision) to their quantized implementation (floating-or fixedpoint) makes the computational problem even harder [12]. Given the interest in deploying extremely quantized ANNs for low-power applications [11], a few recent studies have proposed new fixed-point SMT background theories [2,10].…”

Section: Introductionmentioning

confidence: 99%

QNNVerifier: A Tool for Verifying Neural Networks using SMT-Based Model Checking

Song¹,

Manino²,

Sena³

et al. 2021

Preprint

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QNNVerifier is the first open-source tool for verifying implementations of neural networks that takes into account the finite word-length (i.e. quantization) of their operands. The novel support for quantization is achieved by employing state-of-the-art software model checking (SMC) techniques. It translates the implementation of neural networks to a decidable fragment of first-order logic based on satisfiability modulo theories (SMT). The effects of fixed-and floatingpoint operations are represented through direct implementations given a hardware-determined precision. Furthermore, QNNVerifier allows to specify bespoke safety properties and verify the resulting model with different verification strategies (incremental and k-induction) and SMT solvers. Finally, QNNVerifier is the first tool that combines invariant inference via interval analysis and discretization of non-linear activation functions to speed up the verification of neural networks by orders of magnitude. A video presentation of QNNVerifier is available at https://youtu.be/7jMgOL41zTY. CCS Concepts:• Computing methodologies → Neural networks; • Software and its engineering → Formal software verification.

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“…As a result, specialized verification methods that take quantization into account need to be developed, due to more complex semantics of quantized neural networks. Groundwork on such methods demonstrated that special encodings of networks in terms of Satisfiability Modulo Theories (SMT) (Clark and Cesare 2018) with bitvector (Giacobbe, Henzinger, and Lechner 2020) or fixedpoint (Baranowski et al 2020) theories present a promising approach towards the verification of quantized networks. However, the size of networks that these tools can handle and runtimes of these approaches do not match the efficiency of advanced verification methods developed for standard networks like Reluplex (Katz et al 2017) and Neurify (Wang et al 2018a).…”

Section: Introductionmentioning

confidence: 99%

Scalable Verification of Quantized Neural Networks (Technical Report)

Henzinger¹,

Lechner²,

Žikelić³

2020

Preprint

Self Cite

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Formal verification of neural networks is an active topic of research, and recent advances have significantly increased the size of the networks that verification tools can handle. However, most methods are designed for verification of an idealized model of the actual network which works over real arithmetic and ignores rounding imprecisions. This idealization is in stark contrast to network quantization, which is a technique that trades numerical precision for computational efficiency and is, therefore, often applied in practice. Neglecting rounding errors of such low-bit quantized neural networks has been shown to lead to wrong conclusions about the network's correctness. Thus, the desired approach for verifying quantized neural networks would be one that takes these rounding errors into account. In this paper, we show that verifying the bitexact implementation of quantized neural networks with bitvector specifications is PSPACE-hard, even though verifying idealized real-valued networks and satisfiability of bit-vector specifications alone are each in NP. Furthermore, we explore several practical heuristics toward closing the complexity gap between idealized and bit-exact verification. In particular, we propose three techniques for making SMT-based verification of quantized neural networks more scalable. Our experiments demonstrate that our proposed methods allow a speedup of up to three orders of magnitude over existing approaches.

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An SMT Theory of Fixed-Point Arithmetic

Cited by 13 publications

References 36 publications

BDD4BNN: A BDD-Based Quantitative Analysis Framework for Binarized Neural Networks

BDD4BNN: A BDD-Based Quantitative Analysis Framework for Binarized Neural Networks

QNNVerifier: A Tool for Verifying Neural Networks using SMT-Based Model Checking

Scalable Verification of Quantized Neural Networks (Technical Report)

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