Reasoning About Vectors: Satisfiability Modulo a Theory of Sequences

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Abstract

Dynamic arrays, also referred to as vectors, are fundamental data structures used in many programs. Modeling their semantics efficiently is crucial when reasoning about such programs. The theory of arrays is widely supported but is not ideal, because the number of elements is fixed (determined by its index sort) and cannot be adjusted, which is a problem, given that the length of vectors often plays an important role when reasoning about vector programs. In this paper, we propose reasoning about vectors using a theory of sequences. We introduce the theory, propose a basic calculus adapted from one for the theory of strings, and extend it to efficiently handle common vector operations. We prove that our calculus is sound and show how to construct a model when it terminates with a saturated configuration. Finally, we describe an implementation of the calculus in cvc5 and demonstrate its efficacy by evaluating it on verification conditions for smart contracts and benchmarks derived from existing array benchmarks.

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Notes

1. We number elements in a sequence starting at 0.

2. In Bjørner et al. [9], the second argument j denotes the end index, while here it denotes the length of the sub-sequence, in order to be consistent with the theory of strings in the SMT-LIB standard.

3. It is possible to obtain a similar encoding using the theory of datatypes (see, e.g., [5]); however, here we use uninterpreted functions which are simpler and more widely supported by SMT solvers.

4. In practice, this is implemented by associating each introduced variable with a witness term as described in Reynolds et al. [26].

5. Note that this goes beyond Nelson–Oppen combination because the theories $$T_\textsf{LIA}$$ and $$T_\textsf{Seq}$$ are not disjoint. As a consequence, the exchanged (dis)equalities are not limited to shared variables.

6. Recall that the meaning of $$\textsf{nth}$$ is fixed by the theory only for certain inputs.

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Acknowledgements

This work was funded in part by the Stanford Center for Blockchain Research, NSF-BSF Grant Numbers 2110397 (NSF) and 2020704 (BSF), ISF Grant Number 619/21, and Meta Novi. Part of the work was done when the first author was an intern at Meta Novi.

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Correspondence to Yoni Zohar.

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Sheng, Y., Nötzli, A., Reynolds, A. et al. Reasoning About Vectors: Satisfiability Modulo a Theory of Sequences. J Autom Reasoning 67, 32 (2023). https://doi.org/10.1007/s10817-023-09682-2