Abstract
We present a technique to infer lower bounds on the worst-case runtime complexity of integer programs. To this end, we construct symbolic representations of program executions using a framework for iterative, under-approximating program simplification. The core of this simplification is a method for (under-approximating) program acceleration based on recurrence solving and a variation of ranking functions. Afterwards, we deduce asymptotic lower bounds from the resulting simplified programs. We implemented our technique in our tool LoAT and show that it infers non-trivial lower bounds for a large number of examples.
Supported by the DFG grant GI 274/6-1 and the Air Force Research Laboratory (AFRL).
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Notes
- 1.
Our implementation only supports addition, subtraction, multiplication, division, and exponentiation. Since we consider integer programs, we only allow programs where all variable values are integers (so in contrast to \(x=\frac{1}{2} x\), the assignment \(x=\frac{1}{2} x + \frac{1}{2} x^2\) is permitted). While our program simplification technique preserves this property, we do not allow division or exponentiation in the initial program to ensure its validity.
- 2.
Note that negations can be expressed by negating (in)equations directly, and disjunctions in programs can be expressed using multiple transitions.
- 3.
Programs with \(\mathsf {dh}_\mathcal {T}(\varvec{v}) = \omega \) result from non-termination or non-determinism. As an example, consider the program \(x =\texttt {random}(0,\omega )\); while \(x > 0\) do \(x = x -\,\! 1\) done.
- 4.
In the following, we often use arithmetic terms \(\mathcal {A}(\mathcal {V}_\ell )\) to denote functions \(\mathcal {V}_\ell \rightarrow \mathbb {R}\).
- 5.
For all \(x \in \mathcal {PV}\), \(\mathrm {ren}\circ \eta _1 \circ \eta _2(x) = \mathrm {ren}(\eta _1(\eta _2(x))) = \eta _2(x)[x_1/\eta _1(x_1), \ldots , x_n/\eta _1(x_n), tv _1/\mathrm {ren}( tv _1), \ldots , tv _m/\mathrm {ren}( tv _m)]\) if \(\mathcal {PV}= \{ x_1,\ldots ,x_n \}\) and \(\mathcal {TV}_{\ell _2} = \{ tv _1, \ldots , tv _m \}\).
- 6.
To ease the presentation, we restrict ourselves to binary operations f. For operations of arity \(m\), one would need limit vectors of the form \((\bullet _1, \ldots , \bullet _m)\).
- 7.
The other rules for \(\leadsto \) are implicitly labeled with the identical substitution \(\mathsf {id}\).
- 8.
\(\sigma = [ tv / \frac{1}{2}x^2 + \frac{1}{2} x - 1]\) satisfies the condition \(\varvec{v}(y\sigma ) \in \mathbb {Z}\) for all \(\varvec{v}\in \mathcal {V}\! al _{\ell _0}\) and \(y \in \mathcal {V}_{\ell _0}\).
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Acknowledgments
We thank S. Genaim and J. Böker for discussions and comments.
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Frohn, F., Naaf, M., Hensel, J., Brockschmidt, M., Giesl, J. (2016). Lower Runtime Bounds for Integer Programs. In: Olivetti, N., Tiwari, A. (eds) Automated Reasoning. IJCAR 2016. Lecture Notes in Computer Science(), vol 9706. Springer, Cham. https://doi.org/10.1007/978-3-319-40229-1_37
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