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A CLASSICAL REALIZABILITY MODEL ARISING from a STABLE MODEL of UNTYPED LAMBDA CALCULUS Dedicated to Pierre-Louis Curien at the O

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A CLASSICAL REALIZABILITY MODEL ARISING from a STABLE MODEL of UNTYPED LAMBDA CALCULUS Dedicated to Pierre-Louis Curien at the O Logical Methods in Computer Science Vol. 13(4:24)2017, pp. 1–17 Submitted Jul. 6, 2014 www.lmcs-online.org Published Dec. 7, 2017 A CLASSICAL REALIZABILITY MODEL ARISING FROM A STABLE MODEL OF UNTYPED LAMBDA CALCULUS THOMAS STREICHER Fachbereich 4 Mathematik TU Darmstadt, Schloßgartenstr. 7, D-64289, Germany e-mail address: [email protected] Abstract. In [SR98] it has been shown that λ-calculus with control can be interpreted in any domain D which is isomorphic to the domain of functions from D! to the 2-element (Sierpi´nski)lattice Σ. By a theorem of A. Pitts there exists a unique subset P of D such that f 2 P iff f(d~) = ? for all d~ 2 P !. The domain D gives rise to a realizability structure in the sense of [Kri11] where the set of proof-like terms is given by P . When working in Scott domains the ensuing realizability model coincides with the ground model Set but when taking D within coherence spaces we obtain a classical realizability model of set theory different from any forcing model. We will show that this model validates countable and dependent choice since an appropriate form of bar recursion is available in stable domains. Dedicated to Pierre-Louis Curien at the occasion of his 60th Birthday Introduction In the first decade of this millenium J.-L. Krivine has developed his theory of classical realizability, see e.g. [Kri09, Kri11], for higher order logic and set theory. Whereas intu- itionistic realizability is based on the notion of a partial combinatory algebra (pca) classical realizability is based on a notion of realizability algebra as defined in [Kri11]. Both notions are incomparable since not every pca can be extended to a realizability algebra and there are realizability algebras which do not contain a pca as a substructure. Accordingly, not all classical realizability models appear as booleanizations of intuitionistic realizability models as studied in [vO08]. In the current paper, however, we concentrate on a particular classical realizability model which appears as a boolean subtopos of a relative realizability topos (see [vO08]). The starting point for this model is the observation from [SR98] that the recursive domain ! D =∼ ΣD gives rise to a model for λ-calculus with control. (Here Σ = f?; >g is the 2-element Sierpi´nskilattice and D! is the countable product of D.) Since D is a model of untyped λ-calculus it is in particular a pca. By a theorem of A. Pitts [Pit96] there exists a unique subset P of D such that t 2 P iff t(~s) = ? for all ~s 2 P !. Obviously, this subset P forms a sub-pca of D thus giving rise to the relative realizability topos E = RT(D; P ) as described in [vO08]. Notice that >D 2 D n P and thus U = f>Dg gives rise to a nontrivial truth value Key words and phrases: classical realizability, categorical logic, bar recursion. l LOGICAL METHODS c T. Streicher IN COMPUTER SCIENCE DOI:10.23638/LMCS-13(4:24)2017 CC Creative Commons 2 T. STREICHER in E different from both >E and ?E . This U (like any subterminal object of E) induces a closure operator (aka Lawvere-Tierney topology) jU (p) = (p ! U) ! U on E. As is well known the subtopos EU of jU -sheaves of E is boolean. We will show that EU is equivalent to the classical realizability topos K induced by the realizability structure whose set Λ of terms is D, whose set of stacks Π is D! and whose set PL of proof-like terms is P . We will show that K is equivalent to Set when D is the bifree ! solution of the domain equation D =∼ ΣD in Scott domains. However, when considering ! the solution of D =∼ ΣD in the category Coh of coherence spaces and Scott continuous and stable maps then the ensuing boolean topos K is not a Grothendieck topos and thus a fortiori not a forcing model1. We will show that K validates all true sentences of first order arithmetic and the principles of countable and dependent choice. ! 1. Realizability structures induced by D =∼ ΣD ! Quite generally we might consider objects D =∼ ΣD in well pointed cartesian closed categories C with countable products and an object Σ having precisely two global elements (i.e. morphisms 1 ! Σ) > and ?. The set of global elements of D (which we also denote by D) can be endowed with the structure of a pca as follows: for t; s 2 D we define ts 2 D as (ts)(~r) = t(s:~r). For the set Λ of terms we take D and for the set Π of stacks we take D!. The push operation sends t 2 Λ and ~s 2 Π to t:~s, the stream with head t and tail ~s. For every ~s 2 Π let k~s 2 Λ be defined as k~s(t:~r) = t(~s). The control operator cc is given by cc(t:~s) = t(k~s:~s). A natural choice for the pole is fht; ~si j t(~s) = >g. But on this level of generality we do not know‚ how to choose a set PL of \proof-like ! terms". However, in case D is the bifree solution of D =∼ ΣD in some category of domains like 1) cpo's with bottom and Scott continuous functions 2) coherence spaces and stable (continuous) maps 3) observably sequential algorithms as in [CCF94] by a theorem of A. Pitts (see [Pit96]) there exists a unique subset P of D such that t 2 P iff t(~s) = ? for all ~s 2 P !. Such a P qualifies as a set PL of proof-like terms since P is closed under application, contains all elements definable in untyped λ-calculus and we also have cc 2 P . For later use we remark that the identity map on D is represented by i 2 P with i(t:~s) = t(~s). 2. Some triposes induced by (D; P ) Since P is a subpca of the pca D we may consider the relative realizability topos E = RT(D; P ) I I induced by the tripos P over Set where for a set I the fibre P is the preorder P(D) ; `I with φ `I iff 9t 2 P:8i 2 I:8s 2 φi: ts 2 i and for u : J ! I reindexing along u is given by precomposition with u. For the set ΣP of propositions of P we may take P(D) and for the truth predicate on ΣP we may take idP(D). Notice that ΣP contains an \intermediate" truth value U = f>Dg which is neither equivalent to ?ΣP = ; nor to >ΣP = D. Moreover, in RT(D; P ) the proposition U = f>Dg 1i.e. a category of sheaves over a complete boolean algebra or, equivalently, a Grothendieck topos where every epimorphism splits, see e.g. [Joh02] CLASSICAL REALIZABILITY MODELS ARISING FROM MODELS OF UNTYPED LAMBDA CALCULUS3 is equivalent to U _:U (since :U = ;) but not to D. Thus U _:U does not hold in RT(D; P ) for which reason the topos RT(D; P ) is not boolean. However, the truth value U gives rise to the (Lawvere-Tierney) topology jU on ΣP = P(D) which is defined as jU (A) = (A ! U) ! U for A 2 P(D). We may form the full subtripos PU of P I consisting of jU -closed predicates, i.e. φ 2 P(D) with jU ◦ φ `I φ. Since jU = :U ◦ :U with :U A = A ! U the fibres of PU are all boolean. We write EU = RT(D; P )U for the ensuing boolean subtopos of E = RT(D; P ). As described in the previous section P ⊆ D gives rise to a classical realizability structure with pole = fht; ~si j t(~s) = >g. We write E = RT(D; P ) or rather simply K for the ensuing classical‚ realizability topos which is induced‚ by the full‚ subtripos PK of P consisting I of those predicates φ 2 P(D) which factor through ΣK = fA 2 P(D) j A = Ag. We ‚‚ show now that Lemma 2.1. PK is equivalent to PU . Proof. First recall that on P(D) implication is given by A ! B = ft 2 D j 8s 2 A: ts 2 Bg = ft 2 D j 8s 2 A: λ~r:t(s:~r) 2 Bg from which it follows that ΣK is an exponential ideal in P(D), i.e. A ! B is in ΣK whenever B is in ΣK. Since U 2 ΣK the map jU sends P(D) to ΣK. Thus, postcomposition with jU gives rise to a tripos morphism from P to PK left adjoint to the inclusion of tripos PK into the tripos P (as induced by ΣK ⊆ P(D)). Since A ! jU (A) is uniformly realized by η = λx.λp.px 2 P and for A 2 ΣK the implication jU (A) ! A is uniformly realized by cc 2 P the adjunction above between P and PU 2 restricts to an equivalence between PU and PK. Thus K = RT(D; P ) and RT(D; P )U are equivalent boolean subtoposes of the relative realizability topos E = RT‚ (D; P ) which itself is not boolean. We write i : K ,!E for the corresponding injective geometric morphism. Its inverse image part i∗ : E!K (sheafification) is given by postcomposition with jU . Its (right adjoint) direct image part i∗ : K!E is nontrivial. As described in [vO08] it sends an object X in K to S(X), the object of singleton predicates on X in K considered as an object of E.
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