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In this paper, we present a general realizability semantics for the simplytyped $\lambda\mu$-calculus. Then, based on this semantics, we derive both weakand strong normalization results for two versions of the $\lambda\mu$-calculusequipped with specific simplification rules. The novelty in our method, inaddition to its more systematic approach, lies in its applicability to abroader set of reduction rules. Furthermore, we have obtained a surprisingcharacterization of the typed $\lambda\mu$-terms with the help of oursaturation properties.

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