David Purser

Shaull Almagor, Ismaël Jecker, Filip Mazowiecki et al.

We consider weighted automata over the tropical semiring $\mathbb{Z}_\infty(min, +)$. Recently, it was shown that determinisation is decidable; in this paper we focus on the complexity when the alphabet is unary. In 2001, Lombardy showed this problem is decidable, a close inspection of his proof yields a coNP upper bound on the complexity. Earlier Gaubert showed that every weighted automaton in this setting can be effectively turned into an equivalent union of deterministic weighted automata. We prove Gaubert's result efficiently, presenting it as a generalisation of Chrobak's normal form for unary NFA. In particular, we prove that the equivalent union of deterministic weighted automata can be represented by a weighted automaton of quadratic size in the size of the original one, and this representation can be computed in polynomial time. Building on this, we show that determinisation, and even register minimisation (which generalises determinisation), is coNP-complete. We complete the paper with observations that the boundedness problem is also coNP-complete by reductions with determinisation. Lastly, we provide evidence that all of these problems are not FPT (by proving $coW_1$-hardness) when parametrised by the number of deterministic automata in the union.

Marco Gaboardi, Kobbi Nissim, David Purser

We study the problem of verifying differential privacy for loop-free programs with probabilistic choice. Programs in this class can be seen as randomized Boolean circuits, which we will use as a formal model to answer two different questions: first, deciding whether a program satisfies a prescribed level of privacy; second, approximating the privacy parameters a program realizes. We show that the problem of deciding whether a program satisfies $\varepsilon$-differential privacy is $coNP^{\#P}$-complete. In fact, this is the case when either the input domain or the output range of the program is large. Further, we show that deciding whether a program is $(\varepsilon,δ)$-differentially private is $coNP^{\#P}$-hard, and in $coNP^{\#P}$ for small output domains, but always in $coNP^{\#P^{\#P}}$. Finally, we show that the problem of approximating the level of differential privacy is both $NP$-hard and $coNP$-hard. These results complement previous results by Murtagh and Vadhan showing that deciding the optimal composition of differentially private components is $\#P$-complete, and that approximating the optimal composition of differentially private components is in $P$.