Normalization properties of $\lambda\mu$-calculus using realizability semantics

In this paper, we present a general realizability semantics for the simply typed $\lambda\mu$-calculus. Then, based on this semantics, we derive both weak and strong normalization results for two versions of the $\lambda\mu$-calculus equipped with specific simplification rules. The novelty in our method, in addition to its more systematic approach, lies in its applicability to a broader set of reduction rules without relying on the usual postponement technique. Our approach is original in that it introduces a parameter into the definition of the model, thus establishing a general result which we can then apply to systems with different sets of reduction rules by adjusting the parameter accordingly. Our saturation conditions also lead to a neat characterization of typable $\lambda\mu$-terms.