Reporting to a government agency has some constitutional protection in the law of defamation, which would require you to prove that the statements were made with reckless disregard of the truth (called "malice" in the law of defamation). On the other hand, what you may have a better chance with a claim called "abuse of process," which means that they used a legitimate process, and may have even provided information that they had reason to be believe was true, but used it for an illegitimate...
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That depends on whether the complaint the person made was reasonably believed by him to be true. If he accused you in reckless disregard of the truth, which may be hard to prove if the facts are at all muddy, then you may have a claim. Any claim against the police or the prosecutorial authorities is tougher. The standard for liability against them probably will be whether they probable cause to believe the assault occurred and probable cause to believe you committed it. Them even if they...