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Local Realizability Toposes and a Modal Logic For Electronic Notes in Theoretical Computer Science No URL httpwwwelsevier nll ocate entcs volume htm l pages Lo cal RealizabilityTop oses and a Mo dal Logic for Computability Extended Abstract Steven Awodey Department of Philosophy Carnegie Mel lon University Lars Birkedal School of Computer Science Carnegie Mel lon University Dana S Scott School of Computer Science Carnegie Mel lon University Abstract This work is a step toward developing a logic for typ es and computation that in cludes b oth the usual spaces of mathematics and constructions and spaces from logic and domain theory Using realizabilityweinvestigate a conguration of three top oses whichwe regard as describing a notion of relative computabilityAttention is fo cussed on a certain lo cal map of top oses whichwe study rst axiomatically and then by deriving a mo dal calculus as its internal logic The resulting framework is intended as a setting for the logical and categorical study of relative computability Intro duction We rep ort here on the current status of research on the Logic of Typ es and Computation at Carnegie Mellon University The general goal of this research program is to develop a logical framework for the theories of typ es and computability that includes the standard mathematical spaces alongside the many constructions and spaces known from typ e theory and domain theory One purp ose of this goal is to facilitate the study of computable op erations and maps on data that is not necessarily computable such as the space of all real numb ers Email awodeycmuedu birkedalcscmuedu danascottcscmuedu Lars Birkedal is supp orted in part by US National Science Foundation Grant CCR c Published by Elsevier Science B V Awodey et al Concretely in the research describ ed here we use the realizabilitytopos over the graph mo del P N of the untyp ed lamb da calculus together with the subgraph mo del given by the recursively enumerable subsets to repre sent the classical and computable worlds resp ectively There results a certain con guration of top oses that can b e regarded as describing a notion of rel ative computability We study this con guration axiomatically and derive a higherorder mo dal logic in which to reason ab out it The logic can then b e applied to the original mo del to formalize reasoning ab out computability in that setting Moreover the resulting logical framework provides a general categorical semantics and logical syntax for reasoning in a formal wayabout abstract computabilitywhich it is hop ed could also b e useful for formally similar concepts such as logical de nability In somewhat more detail Section b egins by recalling the standard re alizability top oses RT Aand RT A resulting from a partial combinatory algebra A and a subalgebra A We then identify a third category RT A A whichplays a key role very roughly sp eaking it represents the world of all continuous ob jects but with only computable maps b etween them This cat egory RT A A is a top os the relative realizability topos on A with resp ect to the subalgebra A The top oses RT AandRT A are not particularly wellrelated by them selves the purp ose of the relative realizabilitytopos RT A A is to remedy this defect The three top oses are related to each other as indicated in the following diagram in which the three functors on the left leg constitute a lo cal geometric morphism while the right leg is a logical morphism RT A A u H u H u u H u u u H u u u H u u u H u u u H u u zu u u RT A RT A The lo cal geometric morphism on the left is our chief concern and the fo cus of Section which also mentions some examples and prop erties of these fairly wellundersto o d maps of top oses When we rst encountered it wewere pleased to recognize our situation as an instance of one that FW Lawvere has already called attention to and dubb ed an adjoint cylinder or more colorfully a unity and identity of opposites In Section we present four axioms for lo cal maps of top oses and sketch the pro of that they are sound and complete Actually since the situation we are mainly interested inie realizabilityforces the lo cal map to b e lo calic we give the axioms in a form that implies this condition We simply mention here that a mo di cation of axiom ab out generators will accomo date all b ounded lo cal maps This axiomatization has b een found useful in working with the particular situation wehave in mind but its general utility for lo cal maps of top oses remains to b e seen Not to b e confused with the standard notion of computability relative to an oracle Awodey et al One application of sorts of the axioms for lo cal maps is the investigation of their logical prop erties These are given in Section in the form of a logical calculus involving two prop ositional op erations written and with left adjointto It turns out that satis es the S mo dal logic p ostulates for the boxop eration We here term the calculus a modal logic for computability since that is the interpretation wehave in mind but of course this mo dal logic can b e interpreted in any lo cal top os Weintend to use it to investigate the logical relations that hold in the relative realizability top os however this asp ect of our work is only just b eginning Note that any lo cal map also induces a closely related pair of adjoint op erations on logical types ob jects in addition to the ones on formulas sub ob jects studied here relating our work to The idea of a mo dal com putability op erator is due to the senior author January and was the original imp etus for this work parts of which are from the second authors do ctoral thesis The nal brief section of the pap er sp ells out the intended interpretation of the calculus in the relative realizability top os RT A A Realizability top oses for computability Let A K S b e a partial combinatory algebra PCA often we just denote it by its underlying set A The binary op erator is the partial application and combinators K and S are taken to b e part of the structure and not just required to exist Let A b e a subPCA of AthatisA is a subset of A containing K and S and closed under partial application Intuitivelywe are thinking of the realizers in A as continuous realizers and of theose in A as computable realizers This intuition comes from the main example where A is P N the graph mo del on the p owerset of the natural numb ers and A is RE the recursively enumerable subgraphmo del Note that the mo del P N has a con tinuum of countable subPCAs As another example one may consider van Oostens combinatory algebra B for sequential computation and its eective subalgebra B see e The PCAs A and A give rise to two realizability top oses RT A and RT A in the standard way One may think of RT Aasauniverse where all ob jects and all maps are realized bycontinuous realizers Likewise RT A may b e thought ofasauniverse where all ob jects and all maps are realized by computable realizers Unfortunately these two top oses are not very well related in particular it is not clear how to talk ab out computable maps op erating on continuous ob jects which is what one would liketodoforthe purp oses of eg computable analysis Thus one is led to intro duce an other realizability top os RT A A where intuitively equality on all ob jects is realized bycontinuous realizers and all maps are realized by computable Awodey et al realizers The top os RT A A is constructed by mo difying the underlying trip os for RT A in the following way The nonstandard predicates on a set I are still functions I P A into the p owerset of A and the Heyting prealgebra op erations are the same as in the trip os underlying RT A The mo di cation is in the de nition of the entailment relation wesay over I i there is a realizer a in A not just in Asuch that for all i in I allb i a b is de ned and a b i In the terminology of Pitts wehavechanged the designated truth values to b e those subsets of A whichhave a nonempty op intersection with A Denote this new trip os by P Set Cat Then RT A A isthetoposSetP represented by P Explicitly ob jects of RT A A are pairs X with X a set and X X X X P A a nonstandard equality predicate with computable realiz ers for symmetry and transitivity Morphisms from X toY are X Y equivalence classes of functional relations F X Y P A with computable realizers proving that F is a functional relation Two such functional rela tions F and G are equivalent i there are computable realizers showing them equivalent Wenow see that intuitivelyitmakes sense to think of ob jects as ob jects with continuous realizers for existence and equality of RT A A elements and of morphisms f F as computable maps since the realizers for the functionalityof F are required to b e computable Geometry of the realizabili ty top oses for computabil ity op Let Q Set Cat b e the standard realizability trip os on A ie the trip os underlying RT A Wenow de ne three Setindexed functors b etween Q and P Q P and P Q and r Q P These are de ned as follows Over I wehave I P A i i I I P AiA i I S r I P A i A i I PA where and are calculated as in P Theorem Under these denitions it fol lows that isageometric morphism of triposes from P to Q r is a geometric morphism of triposes from Q to P We rst learned ab out the top os RT A A from Thomas Streicher in February but the construction has actually b een known for a long time see Page item ii Awodey et al For al l I Set and r areboth ful l and faithful I I By Prop osition in these
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