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We consider the probabilistic applicative bisimilarity (PAB)a coinductive relation comparing the applicative behaviour of probabilistic untyped λ-terms according to a specific operational semantics. This notion has been studied by Dal Lago et al. with respect to the two standard parameter passing policies, call-by-value (cbv) and call-by-name (cbn), using a lazy reduction strategy not reducing within the body of a function. In particular, PAB has been proven to be fully abstract with respect to the contextual equivalence in cbv [6] but not in lazy cbn [16].We overcome this issue of cbn by relaxing the laziness constraint: we prove that PAB is fully abstract with respect to the standard head reduction contextual equivalence. Our proof is based on Leventis' Separation Theorem [19], using probabilistic Nakajima trees as a tree-like representation of the contextual equivalence classes.Finally, we prove also that the inequality full abstraction fails, showing that the probabilistic applicative similarity is strictly contained in the contextual preorder.CCS Concepts: • Software and its engineering → Semantics; • Theory of computation → Program semantics.

We study the computational expressivity of proof systems with fixed point operators, within the 'proofs-as-programs' paradigm. We start with a calculus µLJ (due to Clairambault) that extends intuitionistic logic by least and greatest positive fixed points. Based in the sequent calculus, µLJ admits a standard extension to a 'circular' calculus CµLJ.Our main result is that, perhaps surprisingly, both µLJ and CµLJ represent the same first-order functions: those provably total in Π 1 2 -CA 0 , a subsystem of second-order arithmetic beyond the 'big five' of reverse mathematics and one of the strongest theories for which we have an ordinal analysis (due to Rathjen). This solves various questions in the literature on the computational strength of (circular) proof systems with fixed points.For the lower bound we give a realisability interpretation from an extension of Peano Arithmetic by fixed points that has been shown to be arithmetically equivalent to Π 1 2 -CA 0 (due to Möllerfeld). For the upper bound we construct a novel computability model in order to give a totality argument for circular proofs with fixed points. In fact we formalise this argument itself within Π 1 2 -CA 0 in order to obtain the tight bounds we are after. Along the way we develop some novel reverse mathematics for the Knaster-Tarski fixed point theorem.

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