Knjiga apstrakata - Mathematical Institute of the Serbian Academy of ...

Formal Verification of Key Properties for Several ProbabilityLogics in the Proof Assistant CoqPetar MaksimovićMathematical Institute of the SerbianAcademy of Sciences and Arts,petarmax@mi.sanu.ac.rsHere, we present an encoding of four probability logics in the proof assistant Coq. These logics allow for reasoning onthe probability of events, each of them with its own specificities:• The logic LP P Q 1 , in which the probability of events can be any rational number from the unit interval, with iterationsof probability operators,• The logic LP P Q 2 , in which the probability of events can be any rational number from the unit interval, withoutiterations of probability operators,• The logic LP P F r(n)1 , in which the probability of events is restricted to the set {0, 1 n , . . . , n−1n, 1}, for a given positiveinteger n, with iterations of probability operators, and• The logic LP P F r(n)2 , in which the probability of events is restricted to the set {0, 1 n , . . . , n−1n, 1}, for a given positiveinteger n, without iterations of probability operators.For each of these logics, we encode their syntax, semantics, and axiom systems, and provide formal proofs of severalimportant meta-theorems, notably soundness, strong and simple completeness, and compactness or non-compactness, withthe strong completeness theorem being of particular importance, as it gives formal justification for the use of probabilisticSAT-checkers for problems such as:• determining whether probability estimates placed on certain events are consistent,• calculating, given probability estimates of certaing assumptions, the probability of the conclusion,which could arise in various expert systems applying any one of these logics to fields such as game theory, economy andmedicine.25