Abstract
We study weakest precondition reasoning about the (co)variance of outcomes and the variance of run–times of probabilistic programs with conditioning. For outcomes, we show that approximating (co)variances is computationally more difficult than approximating expected values. In particular, we prove that computing both lower and upper bounds for (co)variances is \(\varSigma _2^0\)–complete. As a consequence, neither lower nor upper bounds are computably enumerable. We therefore present invariant–based techniques that do enable enumeration of both upper and lower bounds, once appropriate invariants are found. Finally, we extend this approach to reasoning about run–time variances.
This work was supported by the Excellence Initiative of the German federal and state government.
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Notes
- 1.
This contrasts MCMC–based analysis, as this is restricted to bounded programs.
- 2.
We restrict ourselves to a finite set of program variables for reasons of cleanness of the presentation. In principle, a countable set of program variables could be allowed.
- 3.
Notice that \(\mathbb {S}\) is countable and computably enumerable as \(\mathbb {V}\) is finite.
- 4.
We make use of the convention that \(\frac{0}{0} = 0\). Note that since our probabilistic choice is a discrete choice and our language does not support sampling from continuous distributions, the problematic case of “\(\frac{0}{0}\)” can only occur if executing C on input \(\sigma \) will result in a violation of an observation with probability 1.
- 5.
Note that, for obvious reasons, we restrict to computable expectations f, g only.
- 6.
Note that by “complement” we mean not exactly a set theoretic complement but rather all pairs \((C,\, \sigma )\) such that C does not terminate almost–surely on \(\sigma \).
- 7.
We write v for the expectation that in state \(\sigma \) returns \(\sigma (v)\).
- 8.
Here \(F_h^k(X)\) stands for k–fold application of \(F_h\) to X.
- 9.
Again, we stick to the convention that \(\frac{0}{0} = 0\).
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Kaminski, B.L., Katoen, JP., Matheja, C. (2016). Inferring Covariances for Probabilistic Programs. In: Agha, G., Van Houdt, B. (eds) Quantitative Evaluation of Systems. QEST 2016. Lecture Notes in Computer Science(), vol 9826. Springer, Cham. https://doi.org/10.1007/978-3-319-43425-4_14
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