  We give a new presentation of interactive realizability with a more explicit
syntax.
  Interactive realizability is a realizability semantics that extends the
Curry-Howard correspondence to (sub-)classical logic, more precisely to
first-order intuitionistic arithmetic (Heyting Arithmetic) extended by the law
of the excluded middle restricted to simply existential formulas, a system
motivated by its applications in proof mining.
  Monads can be used to structure functional programs by providing a clean and
modular way to include impure features in purely functional languages. We
express interactive realizers by means of an abstract framework that applies
the monadic approach used in functional programming to modified realizability,
in order to obtain more "relaxed" realizability notions that are suitable to
classical logic. In particular we use a combination of the state and exception
monads in order to capture the learning-from-mistakes nature of interactive
realizers at the syntactic level.
