Model Checking and Functional Program Transformations

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1 Model Checking nd Functionl Progrm Trnsformtions Axel Hddd LIAFA (Université Pris Diderot / CNRS) LIGM (Université Pris Est / CNRS) Astrct We study model for recursive functionl progrms clled higher order recursion schemes (HORS). We give new proofs of two verifiction relted prolems: reflection nd selection for HORS. The previous proofs re sed on the equivlence etween HORS nd collpsile pushdown utomt nd they lose the structure of the initil progrm. The constructions presented here re sed on shpe preserving trnsformtions, nd cn e pplied on ctul progrms without losing the structure of the progrm ACM Suject Clssifiction F.3.1 Specifying nd Verifying nd Resoning out Progrms Keywords nd phrses Higher-order recursion schemes, Model checking, Tree utomt Digitl Oject Identifier /LIPIcs.FSTTCS Introduction Recursion schemes were introduced in the erly 70s s model of computtion, descriing the syntcticl prt of functionl progrm [18, 6, 7, 8]. Originlly they only hndled order-0 (constnts) or order-1 (functions on constnts) expressions, ut not higher-order types. Higher-order versions of recursion schemes were lter introduced to del with functions tking functions s rguments [13, 9, 10]. Recently, the focus cme ck on higher-order recursion schemes when considering them s genertors of possily infinite trees [14, 19, 12]. Indeed, roughly speking recursion scheme is deterministic rewriting system on typed terms tht genertes d infinitum unique infinite tree. As the trees they generte re very generl nd s they cn cpture the computtion tree of (higher-order) functionl progrms, studying their logicl properties leds to very nturl nd chllenging prolems. The most populr one is (locl) model-checking: for given recursion scheme nd formul e.g. from mondic second order logic (MSO) or µ-clculus, decide whether the tree generted y the scheme stisfies the formul. Following prtil decidility results for the suclss of sfe recursion schemes [14, 5], Ong proved, using the notion of trversls, the decidility of MSO model-checking for the whole clss of trees generted y recursion schemes [19]. Since then, other proofs of this result hve een otined using different pproches: Hgue, Murwski, Ong nd Serre estlished the equivlence of schemes nd higher-order collpsile pushdown utomt (CPDA), nd then showed the MSO decidility y reduction to prity gmes on collpsile pushdown utomt [12]; following ides from [1], Koyshi nd Ong [17] developed the type system of [15] to otin new proof. Finlly, Slvti nd Wlukiewicz used Krivine mchines to estlish the MSO decidility of λ-y-clculus, which is typed lmd clculus with recursion, equivlent to higher order recursion schemes [21]. This work ws supported y the project AMIS (ANR 2010 JCJC AMIS). Axel Hddd; licensed under Cretive Commons License CC-BY 33rd Int l Conference on Foundtions of Softwre Technology nd Theoreticl Computer Science (FSTTCS 2013). Editors: Anil Seth nd Nisheeth K. Vishnoi; pp Leiniz Interntionl Proceedings in Informtics Schloss Dgstuhl Leiniz-Zentrum für Informtik, Dgstuhl Pulishing, Germny

2 116 Model Checking nd Functionl Progrm Trnsformtions Another importnt prolem is glol model-checking: for given recursion scheme nd formul, provide finite representtion of the set of nodes in the tree generted y the scheme where the formul holds. Brodent, Cryol, Ong nd Serre nswered this question using so-clled endogenous pproch to represent the set of nodes: they showed how to construct nother scheme generting the sme tree s the originl scheme except tht now those nodes where the formul holds re mrked [3]. They refer to this property s the logicl reflection. The technique they used relies on the equivlence etween schemes nd CPDA. Going further with the ide of mrking set of nodes, Cryol nd Serre considered in [4] the following prolem clled logicl selection nd generlising glol model-checking: for given recursion scheme nd n MSO formul ϕ[x] with one free set vrile, provide (if it exists) finite representtion of set S of nodes in the tree generted y the scheme such tht ϕ[s] holds. They show tht one cn construct nother scheme generting the sme tree s the originl scheme except tht now those nodes re mrked nd descrie set S with the previous property. Agin, they relied on the equivlence with CPDA. One interest for logicl reflection nd logicl selection is tht they cn e used to modify scheme in useful wy. Indeed, ssume some (syntx tree of ) progrm is descried y scheme nd tht it contins d ehviours: using logicl reflection, one cn get new scheme where those d ehviours re mrked nd this ltter scheme cn then esily e modified to remove these ehviours. Using logicl selection, one cn e.g. select rnches in the syntx tree so tht given property is stisfied in the resulting sutree. A drwck of the pproch in [3, 4] tht goes ck nd forth etween schemes nd CPDA is tht the scheme tht is finlly otined is structurlly very fr from the originl one (the nmes of the non terminls s well s the shpe of the rewriting rules hve een lost): hence this is serious prolem if one is interested in doing utomted correction of progrms or even synthesis. Our min contriution in this pper is to provide proofs of oth logicl reflection nd selection without ppeling to CPDA s we only reson on schemes. Our constructions void the loss of the structure, i.e. the solution scheme is otined only y duplicting nd nnotting some prts of the initil scheme nd the trnsformtion is esily reversile. Our constructions re sed on the type system nd the gme used y Koyshi nd Ong in their proof of the model-checking decidility [17]. There is no known correspondence etween these proofs nd the former ones. In order to prove the logicl reflection, we introduce the notion of morphism inspired y denottionl semntics, nd we give morphism for MSO. In order to prove the logicl selection, we emed crefully winning strtegy of the model-checking gme into the scheme. In Section 2 we give the sic definitions nd in Section 3 we introduce the two prolems we re looking t in the pper. In Section 4 we introduce morphisms, explin how to emed morphism into scheme, nd give some possile pplictions. In Section 6, we show how to use morphisms to otin new proof of logicl reflection. In Section 7, we del with logicl selection. In Section 5, we present simple exmple tht descries how we cn use morphism descriing property to trnsform scheme. An extended version of this work is ville online: 2 Preliminries In this section we give the definition of higher order recursion scheme, which is deterministic rewriting system tht produces n infinite tree. It hndles terms of typed symols, i.e. ech symol is constnt or function tht cn hve functions s rguments. Terms my represent expressions of higher order progrmming lnguges, for exmple the term Apply Copy File

3 A. Hddd 117 mens pply the function Copy to the file. In this exmple n intuitive rewrite rule for the function Apply would e the term where the first rgument is pplied on the second rgument, written: Apply ϕ x ϕ x. Types. Types re defined y the grmmr τ ::= o τ τ nd o is clled the ground type. Considering tht is ssocitive to the right (i.e. τ 1 (τ 2 τ 3 ) cn e written τ 1 τ 2 τ 3 ), ny type τ cn e written uniquely s τ 1 τ k o. The integer k is clled the rity of τ. We define the order of type y order(o) = 0 nd order(τ 1 τ 2 ) = mx(order(τ 1 ) + 1, order(τ 2 )). For instnce o o o o is type of order 1 nd rity 3, (o o) (o o), tht cn lso e written (o o) o o is type of order 2.We let τ l τ e shorthnd for τ } {{ τ } τ. l times Terms. Let Γ e finite set of typed symols, nd Let Γ τ denote the set of symols of type τ. For ll type τ, we define the set of terms of type τ, T τ (Γ) s the smllest set contining the symols of type τ nd the ppliction of term of type τ τ to term of type τ for ll τ ; formlly: Γ τ T τ (Γ) nd τ {t s t T τ τ (Γ), s T τ (Γ)} T τ (Γ). We shll write T (Γ) for the set of terms of ny type, nd t : τ if t hs type τ. The rity of term t, rity(t), is the rity of its type. Remrk tht ny term t cn e uniquely written s t = α t 1... t k with α Γ nd 0 k rity(α). We sy tht α is the hed of the term t. For instnce, let Γ = {F : (o o) o o, G : o o o, H : (o o), : o}. Then F H nd G re terms of type o o; F (G ) (H (H )) is term of type o; F is not term since F is expecting first rgument of type o o while hs type o. A set of symols of order t most 1 (i.e. ech symol hs type o or o o o) is clled signture. In the following, we use the letters t, r, s to denote terms, nd given tuple of term t = (t 1,..., t k ) we my use the shorthnd s t to denote s t 1... t k. Contexts. Let t : τ, t : τ e two terms, x : τ e symol of type τ, then we write t [x t ] : τ for the term otined y sustituting ll occurrences of x y t in the term t. A τ-context is term C[ τ ] T (Γ { τ : τ}) contining exctly one occurrence of τ ; it cn e seen s n ppliction turning term into nother, such tht for ll t : τ, C[t] = C[ τ ] [ τ t]. In generl we will only tlk out ground type contexts where τ = o nd we will omit to specify the type when it is cler. For instnce, if C[ ] = F (H (H )) nd t = G then C[t ] = F (G ) (H (H )). Rewrite Rules. Given two disjoint sets of symols Γ, V where V is clled set of vriles, we define (fully pplied) rewrite rule on Γ nd V s pir of terms of T (Γ V) of type o written F x 1... x k e with F Γ, x 1,..., x k V such tht for ll i j x i x j, nd e T (Γ {x 1,..., x k }). Given set of rewrite rules R, we define the rewriting reltion T (Γ) 2 s t t iff there exists context C[ ], rewrite rule F x 1... x k e, nd term F t 1... t k : o such tht t = C[F t 1... t k ] nd t = C[e [x1 t 1]...[x k t k ]]. We cll F t 1... t k : o redex. We define s the reflexive nd trnsitive closure of. Finlly we sy tht set of rewrite rules is deterministic 1 if for ll F Γ there exists t most one rule of the form F x 1... x k e. Trees. Let Σ e finite signture, m e the mximum rity in Σ nd : o e fresh symol. A (rnked) tree t over Σ is mpping t : dom t Σ, where dom t is prefix-closed suset of {1,..., m} such tht if u dom t nd t(u) = then {j u j dom t } = {1,..., rity()}. Given node u dom t, we define the sutree of t rooted t u s the tree t u such tht dom tu = {j u j dom t } nd for ll v dom tu, t u (v) = t(u v). Given : o k o nd 1 Although the rules re deterministics, there my e severl possile rules to pply in term. F S T T C S

4 118 Model Checking nd Functionl Progrm Trnsformtions some trees t 1,..., t k we use the nottion t 1... t k to denote the tree t whose domin is dom t = i i domti, t (ε) = nd t (i u) = t i (u). Note tht there is direct ijection etween ground terms of T o (Σ ) nd finite trees. Hence we will freely llow ourselves to tret ground terms over Σ s trees. We define the prtil order over trees s t t if dom t dom t nd for ll u dom t, t(u) = t (u) or t(u) =. Given (possily infinite) sequence of trees t 0, t 1, t 2,... such tht t i t i+1 for ll i, the set of ll t i hs supremum tht is clled the limit tree of the sequence. Higher Order Recursion Schemes. A higher order recursion scheme (HORS) G = V, Σ, N, R, S is tuple such tht: V is finite set of typed symols clled vriles; Σ is finite signture, clled the set of terminls; N is finite set of typed symols clled non-terminls; R is deterministic set of rewrite rules on Σ N nd V, such tht there is exctly one rule per non terminl nd no rule for terminls; S N is the initil non-terminl. We define inductively the -trnsformtion ( ) : T o (N Σ) T o (Σ { : o}) turning ground term into finite tree s: (F t 1... t k ) = for ll F N nd ( t 1... t k ) = t 1... t k for ll Σ. We define derivtion, s possily infinite sequence of terms linked y the rewrite reltion, nd we will minly look t derivtions where the first term of the sequence is equl to the initil non terminl. Let t 0 = S t 1 t 2 e derivtion, then one cn check tht (t 0 ) (t 1 ) (t 2 )..., hence it dmits limit. One cn prove 2 tht the set of ll such limit trees hs gretest element tht we denote G nd refer to s the vlue tree of G. Note tht G is the supremum of {t S t}. Given term t : o, we denote y G t the scheme otined y trnsforming G such tht it strts derivtions with the term t, formlly, G t = V, Σ, N {S }, R {S t}, S. One cn prove tht if t t then G t = G t. We cll G t the vlue tree of t in G. Prllel derivtion. Intuitively, the prllel rewriting from term t is otined y rewriting ll the redexes in t simultneously. Formlly, given term t = α t 1... t k with α Σ N, we define inductively the prllel rewriting t of t s if α Σ or k < rity(α), then t = α t 1... t k, if α N nd k = rity(α), let α x 1... x k e e the rewrite rule ssocited to α, we hve t = e [ i xi t i ]. Given t, t, we write t t for the reltion t = t. Notice tht if t t nd t t then t t (in prticulr t t ). Furthermore the derivtion S t 1 t 2... leds to G. c Exmple 1. Let G = Σ, V, N, S, R with Σ = {, : o o, c : o c o o}, V = {x : o, ϕ : o o}, N = {S : o, I, F : (o o) o, D, A : c (o o) o o} nd R = {S F, D ϕ x ϕ (ϕ x), A ϕ x ϕ ( x), I ϕ ϕ (I ϕ), F ϕ c (I (A ϕ)) (F (D ϕ))}.... Remrk the rewrite rule ssocited to D: it mens tht for ny function t, D t is simply the function otined y composing t with itself. The rule ssocited to I is lso interesting: for ny function t, I t leds to the infinite itertion of t. For exmple the term I cn e derived to otin ( ( ( (... )))). The tree G is depicted on the left, its rnches re lelled y c ω or c n ( 2(n 1) ) ω for ll n Prity tree utomton. A non-deterministic mx prity utomton (we will just refer to them s utomt in the following) is tuple A = Σ, Q, δ, q 0, Ω with Σ finite signture, Q finite set clled the set of sttes, δ {q (q 1,..., q rity() ) Σ, q, q 1,..., q rity() Q} 2 The existence of vlue tree is consequence the confluence property of HORS.

5 A. Hddd 119 is clled the trnsition reltion, q 0 Q the initil stte, Ω : Q {1,..., m mx } for some m mx N the colouring function. Given n infinite tree t on Σ we define run r of A on t s tree on Σ Q = { q : o k o : o k o, q Q} such tht dom r = dom t, for ll u dom t, if r(u) = q then t(u) = nd there exists q q 1,..., q k δ such tht for ll i, r(u i) = t(u i) qi, nd r(ε) = t(ε) q0. We sy tht the utomton A ccepts the tree t, written t = A, if there exists run r on t such tht for every infinite rnch = ( 0, q 0 ) j 0 ( 1, q 1 ) j 1... in r, the gretest colour seen infinitely often in Ω(q 0 ), Ω(q 1 ), Ω(q 2 ),... is even. We define A q s the utomton otined y chnging in A the initil stte to q: A q = Σ, Q, δ, q, Ω, nd we sy tht A ccepts the tree t from stte q, if t = A q. Exmple 2. Let A = Σ, Q, q 0, δ, Ω e n utomton with Σ = {, : o o, c : o o o}, Q = {q 0, q 1, q 2 }, Ω = {q 0 0, q 1 1, q 2 2} c q 1, q 1 q 2, q 1 nd δ = {q 0 (q 0, q 0 ), q 0 q 1, q 2 q 2, q 2 q 1 }. The utomton A ccepts the trees whose rnches re lelled y c ω or y c ( ) ω. An ccepting run of A on the tree G of Exmple 1 is depicted on the right. q0 q1 q2 q1 q2 q1 q2 q1. c q0 q0 q1 q1 q2 q1 q1 q2. c q0 q0 q1 q1 q1 q1 q2. c q Logicl reflection nd logicl selection In this section we formlise the notion of nnotted tree nd we define the prolems of logicl reflection nd logicl selection we re interested in. Given signture Σ, set X nd tree t on the signture Σ, we define the signture Σ X = {(, x) : o k o : o k o Σ, x X} nd we sy tht the tree t on the signture Σ X is n X-nnottion of t, if dom t = dom t nd if for ll u dom t, t (u) = (t(u), x) for some x X. For exmple run of n utomton over tree t is Q-nnottion of t, with Q eing the set of sttes of the utomton. Furthermore, for ny tree t on the signture Σ X, we define Unl(t ) s the tree on Σ otined y turning ll nodes (, x) of t into i.e. the tree otined y removing the nnotted prt of the tree. Given set of nodes S in tree t, we define n S-mrking of t s {0, 1}-nnottion t of t such tht for ll nodes u, u S if nd only if t (u) = (t(u), 1). Given µ-clculus formul ϕ, we sy tht t is ϕ-reflection of t if it is mrking of the set of nodes u such tht the sutree of t rooted t u stisfies the formul ϕ. Given mondic second-order logic (MSO) formul ϕ[x] with first order free vrile, we sy tht t is ϕ[x]-reflection of t if it is mrking of the set of nodes u stisfying ϕ[u]. Given n MSO formul ϕ[x] with second order free vrile, we sy tht t is ϕ[x]-selection of t if it is mrking of set of nodes S stisfying ϕ[s]. Finlly, we define the µ-clculus reflection, MSO reflection, nd MSO selection s follow. Given clss R of genertors of trees (i.e. set of finitely descried elements such tht to ech g R we ssocite unique tree g ), we sy tht R is (effectively) reflective with respect to the µ-clculus (resp. MSO) if for ny µ-clculus formul ϕ (resp. ny MSO formul ϕ[x]) nd ny tree genertor g R, one cn construct nother genertor g such tht the g is ϕ-reflection (resp. ϕ[x]-reflection) of g. We sy tht R is (effectively) selective with respect to MSO if for ny MSO formul ϕ[x] nd ny genertor g R such tht there F S T T C S

6 120 Model Checking nd Functionl Progrm Trnsformtions exists suset S of nodes of g stisfying ϕ[s], one cn construct nother genertor g such tht g is ϕ[x]-selection. The min contriution of this pper is to give new proofs of the fct tht schemes hve these properties. In order to do so we use the equivlence results etween logic nd utomt, nd we rther prove utomt reflexivity nd utomt selectivity defined s follows. Given tree t on the signture Σ nd n utomton A = Σ, Q, q 0, δ, Ω, we define n A-reflection of A on t s 2 Q -nnottion t of t, such tht for ll nodes u, t (u) = (t(u), Q ) with Q Q eing the set of sttes q such tht A q ccepts the sutree of t rooted on u; we define n A-selection of A on t s n ccepting run of A on t. We sy tht clss R of genertors of trees is reflective with respect to utomt if for ny utomton A nd ny genertor g R, one cn construct nother genertor g such tht g is n A-reflection of g. We sy tht R is selective with respect to utomt if for ny utomton A nd ny genertor g R such tht g = A, one cn construct nother genertor g such tht g is n A-selection of g. From the equivlence etween logics nd utomt [20] we hve tht schemes re reflective with respect to the µ-clculus iff they re reflective with respect to utomt, nd they re selective with respect to MSO iff they re reflective with respect to utomt. Furthermore, it is shown in [3] tht µ-clculus reflection for schemes implies MSO reflection. 4 Morphisms 4.1 Definitions In the following we fix scheme G = Σ, N, V, S, R. We define typed domin D s set such tht ech element is typed, nd to ech element d : τ 1 τ 2 D is ssocited prtil mpping f d from D τ1 to D τ2, where D τ = {d D d : τ}. We write d d the element f d (d ). We define morphism : T (Σ N ) D from terms on Σ N to the domin D s mpping such tht (1) if t : τ then t : τ, (2) if t 0 : τ 1 τ 2 nd t 1 : τ 1, then t t is defined nd equl to t t. In the following, we refer to t s the D-vlue of the term t. Exmple 3. Let D = τ { τ : τ, τ : τ} such tht for ll τ 1, τ 2 : τ1 τ2 τ1 = τ2 ; τ1 τ2 τ1 = τ2 ; τ1 τ2 τ1 = τ2 ; If τ 2 = o then τ1 τ2 τ1 = τ2, otherwise τ1 τ2 τ1 = τ2. For t : τ, we define t s t = τ if t contins ground suterm nd t = τ otherwise. Then is morphism since t 1 t 2 contins ground suterm iff t 1 contins ground suterm, or t 2 contins ground suterm, or t 1 t 2 is ground. Note tht morphism is entirely defined y its vlue on Σ N, i.e. from those vlues one cn compute t for ny term t. Also remrk tht given context C[ ] nd two terms t nd t, if t = t then C[t] = C[t ]. We sy tht morphism is stle y rewriting if for t, t T such tht t t, t = t. Finlly, given set of terms T T (Σ N ), we sy tht the morphism recognises T if there exists suset D of D such tht t T if nd only if t D. In Exmple 3, the morphism recognises the set of terms contining ground term s suterm. 4.2 Emedding morphism into scheme We fix scheme G = V, Σ, N, R, S nd morphism : T (Σ N ) D on G, stle y rewriting, such tht for ll type τ, D τ is finite. We trnsform G into G = V, Σ, N, R, S which, while it is producing derivtion, evlutes t for ny suterm t of the current term nd nnottes the term with ll these D-vlues. The new symols of G re symols

7 A. Hddd 121 of G nnotted with elements of the domin D, nd we define trnsformtion ( ) + from terms of G to terms of G such tht the trnsformtion t + of t will e nnotted with the D-vlues of the suterms of t. The tree generted y G will e nnotted nd when one removes these nnottions, one retreives ck the tree generted y G. More precisely we show the following. Theorem 4 (Emedding morphism). Given two terms t, t : o T (Σ N ), if t G t, then t + G t +. In prticulr Unl( G t ) = G + t (where Unl is the function tht removes the nnottions). Remrk. This trnsformtion keeps the structure of the originl scheme i.e. the new symols re simple lelings of the originl ones, new rewrite rules re otined y duplicting some suterms nd leling the symols, very simple trnsformtion llows to get ck to the originl scheme, nd there is direct correspondence etween derivtions of the two schemes. 4.3 Applictions Emedding properties of suterms during derivtion, or properties of sutrees of the vlue tree, cn e useful for progrm nlysis: insted of sying There will e foridden ehviour in the progrm reflection llows to sy during the execution of progrm Here, the foridden ehviour will pper in this suexpression, ut the rest of the progrm is vlid. Furthermore, once property is emedded into scheme, one cn dd some new rewrite rules tht del explicitly with wether the property is vlid or not. For exmple one could replce ll foridden sutrees of the vlue tree y specil symol foridden, s illustrted in Section 5. Some morphisms generted y type systems re used in [15] to model check suclss of trivil cceptnce condition utomt (i.e. utomt where there is no colouring function, nd the cceptnce of tree simply sks if there exists run of the utomton on the tree). Then the construction llows one to reflect the ccepting sttes of the utomton (s defined in Section 3). This result hs een improved in [22] to del with the whole clss of trivil cceptnce condition utomt. In Section 6 we extend this result to show tht for ny prity tree utomton one cn crete morphism tht reflects the cceptnce of the utomton on the vlue tree. One cn crete model (for exmple in [2]) to decide whether or not ground term t would e productive or not (i.e. t ). Reflecting the productivity of terms into scheme G llows one to crete scheme G on the signture Σ { } such tht G = G, ut such tht no derivtion will crete some non productive terms. In [11] we developed this ide to compre evlution policies. 5 An exmple of scheme trnsformtion In this section, we present simple exmple tht descries how one cn use the emedding procedure to trnsform progrm. Let Mp({0, 1}) e the typed domin inductively defined y Mp({0, 1}) o = {0, 1}, Mp({0, 1}) τ τ is the set of totl functions from Mp({0, 1}) τ to Mp({0, 1}) τ. And given f : τ τ nd h : τ in Mp({0, 1}), f h = f(h). Let G = V, Σ, N, S, R e defined y V = {y : o o, x : o}, Σ = { : o 2 o, : o, c : o}, F S T T C S

8 122 Model Checking nd Functionl Progrm Trnsformtions c c c Figure 1 The vlue tree of the scheme of Section 5. N = {S : o, H : o o, J : o o, F : (o o) o}, R contins the following rewrite rules: S (F H) (F J) H x x (H x) J x (J x) (J x) F y (y ) (y c). The vlue tree of G is (prtilly) depicted in Figure 1. We write [u, v] the mpping f : {0, 1} {0, 1} such tht f(0) = u nd f(1) = v for ll u, v. We define the morphism ϕ s follows: ϕ() = 0 ϕ(c) = 1 ϕ(s) = 1 ϕ() u v = u v ϕ(h) u = u ϕ(j) u = 0 ϕ(f ) [u, v] = u v. One cn check tht the morphism ϕ is stle y rewrite. Furthermore it recognises the property t hs type o, nd its vlue tree contins c, with the suset A = {1}. We construct scheme G = V, Σ, N, S, R tht consists of n emedding of the morphism ϕ inside the scheme G. V = {x : o, y 0, y 1 : o o}, Σ = { 0,0, 0,1, 1,0, 1,1 : o 2 o,, c : o}, N = {S : o, H 0, H 1, J 0, J 1 : o, F [0,0], F [0,1], F [1,0], F [1,1] : (o o) 2 o}, R contins the following rewrite rules: S 1,0 (F [0,1] H 0 H 1 ) (F [0,0] J 0 J 1 ) H 0 x 0,0 x (H 0 x) H 1 x 1,1 x (H 0 x) J 0 x 0,0 (J 0 x) (J 0 x) J 1 x 0,0 (J 1 x) (J 1 x) F [0,0] y 0 y 1 0,0 (y 0 ) (y 1 c) F [0,1] y 0 y 1 0,1 (y 0 ) (y 1 c) F [1,0] y 0 y 1 1,0 (y 0 ) (y 1 c) F [1,1] y 0 y 1 1,1 (y 0 ) (y 1 c). Let us explin how the rewrite rule relted to F [0,1] hs een produced. Recll the originl rule: F y (y ) (y c). The first occurrence of y is pplied to, therefore it should e nnotted with ϕ() = 0. Similrly, the second occurrence of y should e nnotted with ϕ(c) = 1. This justifies the occurrence of y 0 nd y 1 on the left hnd prt of the rule. The nnottion [0, 1] mens tht this rule will e pplied to n rgument whose evlution is the mpping [0, 1], thus the evlution of y is equl to [0, 1] ϕ() = [0, 1] 0 = 0 nd y similr resoning the evlution of y c is equl to 1. Therefore the occurrence of should e nnotted with (0, 1). We wnt to trnsform the scheme in order to forid the derivtion of suterm when its ssocited vlue tree will not include ny c. For instnce, the occurrence of c would correspond to completed service, nd thus such sitution witnesses useless derivtion. In the emedded scheme, this cn e detected y pplying the evlution of the hed over

9 A. Hddd 123 its nnottion. For instnce F [0,0] t 1 t 2 is the nnottion of term F t whose vlue is ϕ(f ) [0, 0] = 0. Therefore we might turn the rule ssocited to F [0,0] into F [0,0] y 1 y 2 foridden, where foridden : o is new terminl dded to the scheme. Here is the whole set of rewrite rules trnsformed this wy. S 1,0 (F [0,1] H 0 H 1 ) (F [0,0] J 0 J 1 ) H 0 x foridden H 1 x 1,1 x (H 0 x) J 0 x foridden J 1 x foridden F [0,0] y 0 y 1 foridden F [0,1] y 0 y 1 0,1 (y 0 ) (y 1 c) F [1,0] y 0 y 1 1,0 (y 0 ) (y 1 c) F [1,1] y 0 y 1 1,1 (y 0 ) (y 1 c). 6 Logicl reflection In the following we present morphism sed on [17] tht recognises the cceptnce of prity tree utomton. Using the construction introduced in Section 4.2, one cn construct scheme tht reflects the ccepting sttes of the utomton, which is equivlent to reflect the sutrees ccepted y formul of the µ-clculus. In [3], µ-clculus reflection (nd MSO reflection) on schemes is lredy proven, ut this construction uses the equivlence etween schemes nd collpsile pushdown utomt, nd the successive trnsformtions (scheme pushdown utomton reflective pushdown utomton reflective scheme) lose the structure of the scheme. In our construction, the structure of the scheme is the preserved, in the sense of Remrk 4.2. Since our proof of the logicl reflection, s well s the one of logicl selection in Section 7, is uild on top of the MSO model checking proof of Koyshi nd Ong [17], we first recll their construction nd then we explin how to use this result to otin the logicl reflection. 6.1 Koyshi-Ong result We fix non deterministic prity tree utomton A = Σ, Q, q I, δ, Ω nd scheme G = Σ, N, V, S, R. We let rity mx, order mx, nd m mx e the mximum rity in Σ N, order in Σ N, colour in Ω(Q). The ide of the result is to define type system, nd to use this type system in the construction of two plyer prity gme, such tht Eve wins the gme if nd only if the utomton ccepts the vlue tree of the scheme. The type system. Koyshi nd Ong introduced set of judgement rules tht llow to type term y n element of the typed set Mp, clled the set of mppings. Mppings of type o re the sttes Q, nd mppings of type τ τ re of the form (θ 1, m 1 )... (θ k, m k ) θ with for ll i θ i is mpping of type τ, m i is color, nd θ is mpping of type τ. The judgements re of the form Γ t θ mening tht under the environment Γ, one cn judge t with the mpping θ. The environment Γ ssocites some mpping nd colors to non terminl nd vriles, nd gives some restriction on the judgements one cn mke. Terminls re judged ccording to the trnsition of the utomton, i.e. : o k o Σ my e judged s (q 1, m 1 )... (q k, m k ) q with for ll i, m i = mx(ω(q i ), Ω(q)) nd q q 1,..., q k δ. This type system keeps trck of the colours in order to know exctly wht colour hs een seen long the term. It is given formlly in the extended version. F S T T C S

10 124 Model Checking nd Functionl Progrm Trnsformtions The gme. Now we define gme G A in which Eve s sttes will e triples mde of non terminl, mpping nd colour, nd Adm s sttes will e environments. Eve chooses n environment tht cn judge the rewrite rule of the current nonterminl with the current tomic mpping, while Adm picks one inding in the current environment. Intuitively, Eve tries to show well-typing of the terms with respect to the rewrite rules, tht would induce well-coloured run of the utomton, nd Adm tries to show tht she is wrong. From the stte (F, θ, m), Eve hs to find n environment Γ such tht she cn prove Γ r F : θ, then Adm picks F nd θ in Γ nd sks Eve to prove tht θ is chosen correctly ccording to the rewrite rule of F. If t some point of ply, Eve cnnot find correct environment, she loses the ply; if she cn choose the empty environment, Adm would hve nothing to choose then she wins the ply; if the ply is infinite, Eve wins if nd only if the gretest colour seen infinitely often is even. The gme is lso given formlly in the extended version. Theorem 5 (Koyshi, Ong 09). The tree G is ccepted y A from the stte q if nd only if Eve hs winning strtegy from the vertex (S, q, Ω(q)) in the gme G A. 6.2 A morphism for utomt reflection From the Eve s winning strtegy, we define morphism : T (Σ N ) D: the domin D contins the sets of mppings of the sme type: for ll τ, D τ = 2 Mpτ. Given nonterminl F, F = {θ m Eve wins from (F, θ, m)}, given Σ, = {θ : θ}, nd given d : τ 1 τ 2 D nd d : τ 1 D d d = {θ (θ 1, m 1 )... (θ k, m k ) θ d i θ i d }. Theorem 6. The morphism recognises the sttes of the utomton, i.e. for ech stte q Q of the utomton, it recognises the set T q = {t : o G t = A q } which is the set of ground terms whose ssocited vlue tree is recognised y the utomton from stte q. Furthermore, it is stle y rewriting. Using the construction of Section 4.2, we hve the following result. Corollry 7 (Automt Reflection). Higher order recursion schemes re reflective with respect to utomt (hence they re reflective with respect to µ-clculus, nd to MSO). 7 Selection Given signture Σ, we recll the selection prolem: given n MSO formul ϕ[x] hving one free mondic second order vrile X, nd scheme G such tht G stisfies the formul X ϕ[x], produce nother scheme G on the signture Σ {0, 1} such tht their exists set S stisfying ϕ[s] nd G is S-mrking of G. Note tht MSO selection implies MSO reflection. Indeed, eing le to mrk the nodes u stisfying the formul ϕ[x] is equivlent to eing le to mrk (unique) set stisfying ψ[x] = x x X ϕ[x]. As mentioned in Section 3, MSO selection is equivlent to utomt selection; we show construction of scheme nnotting itself with n ccepting run of the utomton. Since we cnnot emed this prolem into morphism, we define nother construction, similr to the one of Section 4.2. The construction is lso sed on the Koyshi-Ong result presented in Section 5. The min difference etween reflexivity nd selectivity is tht to construct reflection of n utomton we emedded the winning region of the gme into the scheme, while here we will emed the winning strtegy of the gme to prove the selection. Theorem 8 (Automt Selection). Higher order recursion schemes re selective with respect to utomt (hence to MSO).

11 A. Hddd 125 Proof sketch. In the following, we give n informl glimpse on the proof, the full (technicl) proof cn e found in the extended version. Tke scheme G nd n utomton A such tht G = A. We wnt to construct scheme G such tht G is n ccepting run of A on G. The first oservtion is tht there re some gret similrities etween the structure of the proof trees in the type system system nd the definition of term. Indeed, one cn type proofs nd one cn pply proofs to one nother to get new proofs. For exmple tke two terms t 0 : o o nd t 1 : o nd ssume tht we hve proof P 0 of t 0 (θ 1, m 1 ) θ nd proof P 1 of t 1 θ 1 under some environments. Then one cn put together P 0 nd P 1 to otin proof of t 0 t 1 θ. We cn see this proof s P 0 P 1 : the ppliction of P 1 to P 0. In the ctul construction, the trnsformed scheme will not del with such proofs, ut we use this similrity etween proofs nd terms to crete nnottions of terms. Given term t nd proof P of judgement Γ t θ, we will define the nnotted term t P where ech symol is nnotted y n tomic mpping nd color, verifying tht if non terminl F is nnotted y (θ, m) then Γ ssicite F to (θ, m). The term t P is somehow trce of the proof P. Now given rewrite rule F x 1...x k e in the originl scheme, we wnt to define for ll nnotted versions F (θ,m) n ssocited rewrite rule in the trnsformed scheme. If (F, θ, m) is winning vertex of the gme, then Eve cn choose n environment Γ such tht Γ e θ. We tke proof P of this judgement nd we define F (θ,m) x 1... x k e P. Since Eve hs chosen the environment Γ with respect to her winning strtegy, then for ll nnotted non terminls H (θ,m ) ppering in e P, (H, θ, m ) is winning in the gme. In prticulr, if the initil non terminl of the trnsformed scheme is S (θ,m) with (S, θ, m) winning in the gme (s it will e), ny non terminl in ny term of ny derivtion will e nnotted in order to represent winning vertex in the gme. Therefore we do not need to cre out which rewrite rule is chosen for F (θ,m) when (F, θ, m) is not winning. As we sid, the initil non terminl is (S, q 0, Ω(q 0 )) which is winning in the gme since A ccepts G from stte q 0. Any terminl will e nnotted y some (q 1 q k q, m). Then to otin elements of Σ Q, we just turn ny (q1 qk q,m) into non terminl nd we dd rewrite rule trnsforming it into q. The intuition of why this construction works is the following, sed on the proof of the soundness of Koyshi nd Ong construction. To derivtion in the trnsformed scheme we ssocite tree of plys in the gme. The terms will e leled y some F (θ,m) tht Eve hs chosen in the environments she picked, nd ech time Adm choose one such F (θ,m), it is rewritten ccording to Eve s strtegy. Due to the colour constrints in the type systems, we cn show tht from the point where F (θ,m) is creted to the point where it is on the hed of redex, the mximum colour tht hs een seen is m. Furthermore, we hve tht long n infinite rnch, there is n infinite sequence of nonterminls tht re rewritten such tht ech non terminl is creted when the previous one is rewritten. This mens tht we cn mp n infinite rnch to n infinite ply in the gme. Furthermore the gretest colour seen infinitely often long this rnch is equl to the gretest colour seen infinitely often in the sequence of mximum colours ppering etween the non terminls of the sequence. And this is equl to the gretest colour seen infinitely often in the ply. Since Eve wins in the gme, this colour is even, then for ny rnch of the vlue tree of G the gretest colour seen infinitely often is even, hence it is n ccepting run. 8 Conclusion We hve given new shpe preserving constructions for logicl reflection nd logicl selection using scheme-only pproch, which cn e useful for correction or synthesis of progrms. F S T T C S

12 126 Model Checking nd Functionl Progrm Trnsformtions The complexity is the sme s in the solutions proposed so fr, i.e. the prolem is n- EXPTIME complete, nd the size of the new scheme is n-exp the size of the originl one n eing the order of the scheme. As possile continution of these work, we my e interesting to see if these results scle for ctul progrm verifiction, nd if they cn e included in tools like T-RecS [16], model-checker for HORS. References 1 Klus Aehlig. A finite semntics of simply-typed lmd terms for infinite runs of utomt. In CSL 06, volume 4207 of LNCS, pges , R. M. Amdio nd P.-L. Curien. Domins nd Lmd-Clculi. CTTCS, Christopher H. Brodent, Arnud Cryol, C.-H. Luke Ong, nd Olivier Serre. Recursion schemes nd logicl reflection. In LICS 10, pges , Arnud Cryol nd Olivier Serre. Collpsile pushdown utomt nd leled recursion schemes. In LICS 12, pges , Didier Cucl. On infinite terms hving decidle mondic theory. In MFCS 02, volume 2420 of LNCS, pges , Bruno Courcelle. A representtion of trees y lnguges I. TCS, 6: , Bruno Courcelle. A representtion of trees y lnguges II. TCS, 7:25 55, Bruno Courcelle nd Murice Nivt. The lgeric semntics of recursive progrm schemes. In MFCS 78, volume 64 of LNCS, pges 16 30, Werner Dmm. Higher type progrm schemes nd their tree lnguges. In Theoreticl Computer Science, 3rd GI-Conference, volume 48 of LNCS, pges 51 72, Werner Dmm. Lnguges defined y higher type progrm schemes. In ICALP 77, volume 52 of LNCS, pges , Axel Hddd. IO vs oi in higher-order recursion schemes. In FICS 12, pges 23 30, Mtthew Hgue, Andrzej S. Murwski, C.-H. Luke Ong, nd Olivier Serre. Collpsile pushdown utomt nd recursion schemes. In LICS 08, pges , Klus Indermrk. Schemes with recursion on higher types. In MFCS 76, volume 45 of LNCS, pges , Teodor Knpik, Dmin Niwiński, nd Pwel Urzyczyn. Higher-order pushdown trees re esy. In FOSSACS 02, volume 2303 of LNCS, pges , Noki Koyshi. Types nd higher-order recursion schemes for verifiction of higher-order progrms. In POPL 09, pges , Noki Koyshi. A prcticl liner time lgorithm for trivil utomt model checking of higher-order recursion schemes. In FOSSACS 11, pges , Noki Koyshi nd C.-H. Luke Ong. A type system equivlent to the modl mu-clculus model checking of higher-order recursion schemes. In LICS 09, pges , M. Nivt. On the interprettion of recursive progrm schemes. In Symp. Mt., C.-H. Luke Ong. On model-checking trees generted y higher-order recursion schemes. In LICS 06, pges 81 90, M.O. Rin. Decidility of second-order theories nd utomt on infinite trees. In Trns. Amer. Mth. Soc., Sylvin Slvti nd Igor Wlukiewicz. Krivine mchines nd higher-order schemes. In ICALP 11, pges , Sylvin Slvti nd Igor Wlukiewicz. Using models to model-check recursive schemes. In Typed Lmd Clculi nd Applictions, pges Springer, 2013.

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