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The inreraciioii processes at work iii Hylund arid Oizg (HO) urid Abrniiisky, .laguciee.snii uiidMulmaria (A.JM) new game seninritics are two preexisting pavadiginatic iiiipleriieriiafioiis oj-linear head reduction: re~I7ecfively Kriviiie's RDsfrucr Machine and Girard4 Interactiori Abstract Mncli he.There is U simple oizd tiuirrral embedding ojA.lM-ganzes to MI-ganies, mappirig sirategies to srralegies and reducing A.lM dejiiiuhility (or JiiIl alntractioii) property lo H O 1s one.

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Abstract-Martin-Löf's type theory has strong existential elimination (dependent sum type) that allows to prove the full axiom of choice. However the theory is intuitionistic. We give a condition on strong existential elimination that makes it computationally compatible with classical logic. With this restriction, we lose the full axiom of choice but, thanks to a lazily-evaluated coinductive representation of quantification, we are still able to constructively prove the axiom of countable choice, the axiom of dependent choice, and a form of bar induction in ways that make each of them computationally compatible with classical logic.

Abstract. We give a decomposition of the equational theory of call-byvalue λ-calculus into a confluent rewrite system made of three independent subsystems that refines Moggi's computational calculus:-the purely operational system essentially contains Plotkin's βv rule and is necessary and sufficient for the evaluation of closed terms; -the structural system contains commutation rules that are necessary and sufficient for the reduction of all "computational" redexes of a term, in a sense that we define; -the observational system contains rules that have no proper computational content but are necessary to characterize the valid observational equations on finite normal forms. We extend this analysis to the case of λ-calculus with control and provide with the first presentation as a confluent rewrite system of SabryFelleisen and Hofmann's equational theory of λ-calculus with control. Incidentally, we give an alternative definition of standardization in callby-value λ-calculus that, unlike Plotkin's original definition, prolongs weak head reduction in an unambiguous way.

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We define a variant of Krivine realizability where realizers are pairs of a term and a substitution. This variant allows us to prove the normalization of a simply-typed call-by-need λ-calculus with control due to Ariola et al. Indeed, in such call-by-need calculus, substitutions have to be delayed until knowing if an argument is really needed. We then extend the proof to a call-by-need λ-calculus equipped with a type system equivalent to classical second-order predicate logic, representing one step towards proving the normalization of the call-by-need classical second-order arithmetic introduced by the second author to provide a proof-as-program interpretation of the axiom of dependent choice. The call-by-need evaluation strategyThe call-by-need evaluation strategy of the λ-calculus evaluates arguments of functions only when needed, and, when needed, shares their evaluations across all places where the argument is required. The call-by-need evaluation is at the heart of a functional programming language such as Haskell. It has in common with the call-by-value evaluation strategy that all places where a same argument is used share the same value. Nevertheless, it observationally behaves like the callby-name evaluation strategy (for the pure λ-calculus), in the sense that a given computation eventually evaluates to a value if and only if it evaluates to the same value (up to inner reduction) along the call-by-name evaluation. In particular, in a setting with non-terminating computations, it is not observationally equivalent to the call-by-value evaluation. Indeed, if the evaluation of a useless argument loops in the call-by-value evaluation, the whole computation loops, which is not the case of call-by-name and call-by-need evaluations.These three evaluation strategies can be turned into equational theories. For call-by-name and call-by-value, this was done by Plotkin through continuationpassing-style (CPS) semantics characterizing these theories [34]. For the call-byneed evaluation strategy, a specific equational theory reflecting the intensional behavior of the strategy into a semantics was proposed independently by Ariola and Felleisen [1], and by Maraist, Odersky and Wadler [26]. A continuationpassing-style semantics was proposed in the 90s by Okasaki, Lee and Tarditi [30]. However, this semantics does not ensure normalization of simply-typed call-byneed evaluation, as shown in [2], thus failing to ensure a property which holds in the simply-typed call-by-name and call-by-value cases.Continuation-passing-style semantics de facto gives a semantics to the extension of λ-calculus with control operators 6 . In particular, even though call-byname and call-by-need are observationally equivalent on pure λ-calculus, their different intentional behaviors induce different CPS semantics, leading to different observational behaviors when control operators are considered. On the other hand, the semantics of calculi with control can also be reconstructed from an analysis of the duality between programs and their evaluation c...