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We consider the probabilistic applicative bisimilarity (PAB), a coinductiverelation comparing the applicative behaviour of probabilistic untyped lambdaterms according to a specific operational semantics. This notion has beenstudied with respect to the two standard parameter passing policies,call-by-value (cbv) and call-by-name (cbn), using a lazy reduction strategy notreducing within the body of a function. In particular, PAB has been proven tobe fully abstract with respect to the contextual equivalence in cbv but not inlazy cbn. We overcome this issue of cbn by relaxing the laziness constraint: weprove that PAB is fully abstract with respect to the standard head reductioncontextual equivalence. Our proof is based on the Leventis Separation Theorem,using probabilistic Nakajima trees as a tree-like representation of thecontextual equivalence classes. Finally, we prove also that the inequality fullabstraction fails, showing that the probabilistic applicative similarity isstrictly contained in the contextual preorder.

The Benefit of Being Non-Lazy in Probabilistic {\lambda}-calculus