A Dialectica-Like Interpretation of a Linear MSO on Infinite Words A Dialectica-Like Interpretation of a Linear MSO on Inﬁnite Words (cid:63) Interpretation of a Linear MSO on

. We devise a variant of Dialectica interpretation of intuitionistic linear logic for LMSO , a linear logic-based version MSO over inﬁnite words. LMSO was known to be correct and complete w.r.t. Church’s synthesis, thanks to an automata-based realizability model. Invoking B¨uchi-Landweber Theorem and building on a complete axiomatization of MSO on inﬁnite words, our interpretation provides us with a syntactic approach, without any further construction of automata on inﬁnite words. Via Dialectica, as linear negation directly corresponds to switching players in games, we furthermore obtain a complete logic: either a closed formula or its linear negation is provable. This completely axiomatizes the theory the realizability model of LMSO . Besides, this shows that in principle, one can solve Church’s synthesis for a given ∀∃ -formula by only looking for proofs of either that formula or its linear negation.


Introduction
Monadic Second-Order Logic (MSO) over ω-words is a simple yet expressive language for reasoning on non-terminating systems which subsumes non-trivial logics used in verification such as LTL (see e.g. [27,2]). MSO on ω-words is decidable by Büchi's Theorem [6] (see e.g. [26,22]), and can be completely axiomatized as a subsystem of second-order Peano's arithmetic [25]. While MSO admits an effective translation to finite-state (Büchi) automata, it is a non-constructive logic, in the sense that it has true (i.e. provable) ∀∃-statements which can be witnessed by no continuous stream function.
On the other hand, Church's synthesis [8] can be seen as a decision problem for a strong form of constructivity in MSO. More precisely (see e.g. [29,12]), Church's synthesis takes as input a ∀∃-formula of MSO and asks whether it can be realized by a finite-state causal stream transducer. Church's synthesis is known to be decidable since Büchi-Landweber Theorem [7], which gives an effective solution to ω-regular games on finite graphs generated by ∀∃-formulae. In with input B ∈ 2 ω and output C ∈ 2 ω specifies functions F : 2 ω → 2 ω such that F (B) ∈ 2 ω P(N) is infinite whenever B ∈ 2 ω P(N) is infinite (resp. the complement of B is finite). One may also additionally require to respect the transitions of some automaton. For instance, following [28], in addition to either case of (1) one can ask C ⊆ B and C not to contain two consecutive positions: (∀n)(C(n) ⇒ B(n)) and (∀n)(C(n) ⇒ ¬C(n + 1)) (2) In any case, the realizers must be (finite-state) causal functions. A stream function F : Σ ω → Γ ω is causal (notation F : Σ → S Γ ) if it can produce a prefix of length n of its output from a prefix of length n of its input. Hence F is causal if it is induced by a map f : Σ + → Γ as follows: . Causal and f.s. causal functions form categories with finite products. Let S be the category whose objects are alphabets and whose maps from Σ to Γ are causal functions F : Σ ω → Γ ω . Let M be the wide subcategory of S whose maps are finite-state causal functions. (c) The conjunction of (2) with either side of (1) is realized by the causal function F : 2 → M 2 induced by the machine M : 2 → 2 displayed on Fig. 1 (left, where a transition a|b outputs b from input a), taken from [28]. The Logic MSO(M). Our specification language MSO(M) is an extension of MSO on ω-words with one function symbol for each f.s. causal function. More precisely, MSO(M) is a many-sorted first-order logic, with one sort for each simple type τ ∈ ST, and with one function symbol of arity (σ 1 , . . . , σ n ; τ ) for each map σ 1 ×· · ·× σ n → M τ . A term t of sort τ (notation t τ ) with free variables among x σ1 1 , . . . , x σn n (we say that t is of arity (σ 1 , . . . , σ n ; τ )) thus induces a map t : σ 1 × · · · × σ n → M τ . Given a valuation x i → B i ∈ σ i ω S[1, σ i ] for i ∈ {1, . . . , n}, we then obtain an ω-word t • B 1 , . . . , B n ∈ S [1, τ ] τ ω MSO(M) extends MSO with ∃x τ and ∀x τ ranging over S [1, τ ] τ ω and with sorted equalities t τ .
= u τ interpreted as equality over S [1, τ ] τ ω . Write |= ϕ when ϕ holds in this model, called the standard model. The full definition of MSO(M) is deferred to §4. 1.
An instance of Church's synthesis problem is given by a closed formula (∀x σ )(∃u τ )ϕ(u, x). A positive solution (or realizer) of this instance is a term t(x) of arity (σ; τ ) such that (∀x σ )ϕ(t(x), x) holds.
Eager Functions. A causal function Σ → S Γ is eager if it can produce a prefix of length n + 1 of its output from a prefix of length n of its input. More precisely, an eager F : Σ → S Γ is induced by a map f : Σ * → Γ as . . · B(n − 1)) (for all B ∈ Σ ω and all n ∈ N) Finite-state eager functions are those induced by eager (Moore) machines (see also [11]). An eager machine E : Σ → Γ is a Mealy machine Σ → Γ whose output function λ : Q E → Γ is does not depend on the current input letter. An eager E : Σ → Γ induces an eager function via the map (a ∈ Σ * ) → (λ E (∂ * E (a)) ∈ Γ ). We write F : Σ → E Γ when F : Σ → S Γ is eager and F : Σ → EM Γ when F is f.s. eager. All functions F : Σ → M 1, and more generally, constants functions F : Σ → S Γ are eager. Note also that if F : Σ → S Γ is eager, then F : Σ → EM Γ . On the other hand, if F : Σ → EM Γ is induced by an eager machine E then F is finite-state causal as beeing induced by the Mealy machine with same states and transitions as E, but with output function (q, a) → λ E (q).
Eager functions do not form a category since the identity of S is not eager. On the other hand, eager functions are closed under composition with causal functions.
Proposition 2. If F is eager and G, H are causal then H • F • G is eager.
Isolating eager functions allows a proper treatment of strategies in games and realizers w.r.t. the Dialectica interpretation. Since Σ + → Γ Σ * → Γ Σ , maps Σ → E Γ Σ are in bijection with maps Σ → S Γ . This easily extends to machines. Given a Mealy machine M : Σ → Γ , let Λ(M) : Σ → Γ Σ be the eager machine defined as M but with output map taking q ∈ Q M to (a → λ M (q, a)) ∈ Γ Σ . Eager f.s. functions will often be used with the following notations. First, let @ be the pointwise lift to M of the usual application function Γ Σ × Σ → Γ . We often write (F )G for @(F, G). Consider a Mealy machine M : Σ → Γ and the induced eager machine Λ(M) : Σ → Γ Σ . We have Given F : Γ → E Σ Γ , we write e(F ) for the causal @(F (−), −) : Γ → S Σ. Given F : Γ → S Σ, we write Λ(F ) for the eager Γ → E Σ Γ such that F = e(Λ(F )). We extend these notations to terms. Eager functions admit fixpoints similar to those of contractive maps in the topos of tree (see e.g. [4,Thm. 2.4]).

Games.
Traditional solutions to Church's synthesis turn specifications to infinite two-player games with ω-regular winning conditions. Consider an MSO(M) formula ϕ(u τ , x σ ) with no free variable other than u, x. We see this formula as defining a two-player infinite game G(ϕ)(u τ , x σ ) between the Proponent P (∃loïse), playing moves in τ and the Opponent O (∀bélard), playing moves in σ . The Proponent begins, and then the two players alternate, producing an infinite play of the form The play χ is winning for P if ϕ((u k ) k , (x) k ) holds. Otherwise χ is winning for O. Strategies for P resp. O in this game are functions Hence finite-state strategies are represented by f.s. eager functions. In particular, a realizer of (∀x σ )(∃u τ )ϕ(u, x) in the sense of Church is a f.s. P-strategy in Most approaches to Church's synthesis reduce to Büchi-Landweber Theorem [7], stating that games with ω-regular winning conditions are effectivelly determined, and that the winner always has a finite-state winning strategy. We will use Büchi-Landweber Theorem in following form. Note that an O-strategy in the game G(ϕ)(u τ , x σ ) is a P-strategy in the game G ¬ϕ(u, (x)u) x (σ)τ , u τ .
Curry-Howard Approaches. Following the complete axiomatization of MSO on ω-words of [25] (see also [24]), one can axiomatize MSO(M) with a deduction system based on arithmetic (see §4.1). Consider an instance of Church's synthesis (∀x σ )(∃u τ )ϕ(u, x). Then we get from Theorem 1 the alternative for an eager term u(x) or a causal term x(u). By enumerating proofs and machines, one thus gets a (naïve) syntactic algorithm for Church's synthesis. But it seems however unlikely to obtain a complete classical system in which the provable ∀∃-statments do correspond to the realizable instances of Church's synthesis, because MSO(M) has true but unrealizable ∀∃-statments. Besides, note that while it is possible both for realizable and unrealizable instances to have In previous works [23,24], the authors devised intuitionsitic and linear variants of MSO on ω-words in which, thanks to automata-based polarity systems, proofs of suitably polarized existential statments correspond exactly to realizers for Church's synthesis. In particular, [24] proposed a system LMSO based on (intuitionistic) linear logic [13], such that via a translation (−) L : MSO → This paper goes further. We show that the automata-based realizability model of [24] can be obtained in a syntactic way, thanks to a (linear) Dialecticalike interpretation of a variant of LMSO, which turns a formula ϕ to a formula ϕ D of the form (∃u)(∀x)ϕ D (u, x), where ϕ D (u, x) essentially represents a deterministic automaton. While the correctness of the extraction procedure of [23,24] relied on automata-theoretic techniques, we show here that it can be performed syntactically. Second, by extending LMSO with realizable axioms, we obtain a system LMSO(C) in which, using an adaptation of the usual Characterization Theorem for Dialectica stating that ϕ˛ϕ D (see e.g. [15]), alternatives of the form (6) imply that for a closed ϕ, ⊥ is a linear negation. We thus get a complete linear system with extraction of suitably polarized ∀∃-statments. Such a system can of course not have a standard semantics, and indeed, LMSO(C) has a functional choice axiom which is realizable in the sense of both (−) D and [24], but whose translation to MSO(M) (which precludes (5)) is false in the standard model.

A Monadic Linear Dialectica-like Interpretation
Gödel's "Dialectica" functional interpretation associates to ϕ(a) a formula ϕ D (a) of the form (∃u τ )(∀x σ )ϕ D (u, x, a). In usual versions formulated in higher-types arithmetic (see e.g. [1,15]), the formula ϕ D is quantifier-free, so that ϕ D is a prenex form of ϕ. This prenex form is constructive, and a constructive proof of ϕ can be turned to a proof of ϕ D with an explicit (closed) witness for ∃u. We call such witnesses realizers of ϕ. Even if Dialectica originally interprets intuitionistic arithmetic, it is structurally linear: in general, realizers of contraction only exist when the term language can decide ϕ D (u, x, a), which is possible in arithmetic but not in all settings. Besides, linear versions of Dialectica were formulated at the very beginning of linear logic [19,20,21].
In this paper, we use a variant of Dialectica as a syntactic formulation of the automata-based realizability model of [24]. The formula ϕ D essentially represents a deterministic automaton on ω-words and is in general not quantifier-free. Moreover, we extract f.s. causal functions, while the category M is not closed. As a result, a realizer of ϕ is an open (eager) term u(x) of arity (σ; τ ) satisfying ϕ D (u(x), x). Thanks to Büchi-Landweber Theorem, contractions on closed ϕ have realizers which are correct in the standard model. But this is generally not the case for open ϕ(a). We thus work in a linear system, in which we obtain witnesses for ∀∃(−) L -statements (and thus for realizable instances of Church's synthesis), but not for all ∀∃-statements.
Fix a set of atomic formulae At containing all (t τ . = u τ ).

The Multiplicative Fragment
Our linear system is based on full intuitionistic linear logic (see [14]). The formulae of the multiplicative fragment MF are given by the grammar: ϕ, ψ :: . Deduction is given by the rules of Fig. 2 and the axioms Each formula ϕ of MF can be mapped to an MSO(M)-formula ϕ (where I, , ⊗,`are replaced resp. by , →, ∧, ∨). Hence ϕ holds whenever ϕ.
The Dialectica interpretation of MF is the usual one rewritten with the connectives of MF, but for the disjunction`that we treat similarly as ⊗. Dialectica is such that ϕ D is equivalent to ϕ via possibly non-intuitionistic but constructive principles. The tricky connectives are implication and universal quantification. Similarly as in the intuitionistic case (see e.g. [15,1,30]), (ϕ ψ) D is prenex a form of ϕ D ψ D obtained using (LAC) together with linear variants of the Markov and Independence of premises principles. In our case, the equivalence ϕ˛ϕ D also requires additional axioms for ⊗ and`. We give details for the full system in §3. 3.
The soundness of (−) D goes as usual, excepted that we extract open eager terms: from a proof of ϕ(a κ ) we extract a realizer of (∀a)ϕ(a), that is an open eager term u(x, a) s.t. ϕ D (@(u(x, a), a), x, a). Composition of realizers (in part. required for the cut rule) is given by the fixpoints of Prop. 3. Note that a realizer of a closed ϕ is a finite-state winning P-strategy in G( ϕ D )(u, x).

Polarized Exponentials
It is well-known that the structure of Dialectica is linear, as it makes problematic the interpretation of contraction: Example 3. Realizers of ϕ ϕ ⊗ ϕ for a closed ϕ are given by eager terms By the Büchi-Landweber Theorem 1, either there is an eager term U(x) such that ϕ D (U(x), x) holds, so that or there is an eager term X(u) such that ¬ ϕ D (u, e(X)(u)) holds, so that = 0 ω ) where t (B, C) = 0 n+1 1 ω for the first n ∈ N with C(n+1) = B(0) if such n exists, and such that t (B, C) = 0 ω otherwise. The game induced by ( In this game, P begins by playing a function 2 3 → 2, O replies in 2 3 , and then P and O keep on alternatively playing moves of the expected type. A finite-state winning strategy for O is easy to find. Let P begin with the function X.
Quantifier-free formulae are thus both positive and negative.

Example 5. Polarized contraction
gives realizers of all instances of itself. Indeed, with say ϕ D (a) = (∃u)ϕ D (u, −, a) and ψ D (a) = (∀y)ψ D (−, y, a), Λ(π 1 ) (for π 1 a M-projection on suitable types) gives eager terms U(u, a) and Y(y, a) such that We only have exponentials for polarized formulae. First, following the usual polarities of linear logic, we can let Hence !ϕ is positive for a positive ϕ and ?ψ is negative for a negative ψ. The following exponential contraction axioms are then interpreted by themselves: Second, we can have exponentials !(ψ − ) and ?(ϕ + ) with the automata-based reading of [24]. Positive formulae are seen as non-deterministic automata, and ? on positive formulae is determinization on ω-words (McNaughton's Theorem [17]). Negative formulae are seen as universal automata, and ! on negative formulae is to co-determinization (an instance of the Simulation Theorem [10,18]). Formulae which are both positive and negative (notation (−) ± ) correspond to deterministic automata, and are called deterministic. We let So !(ψ − ) and ?(ϕ + ) are always deterministic. The corresponding exponential contraction axioms are interpreted by themselves. This leads to the following polarized fragment PF (the deduction rules for exponentials are given on Fig. 4):

The Full System
The formulae of the full system FS are given by the following grammar: Deduction in FS is given by Fig. 2, Fig. 4 and (7). We extend − to FS with !ϕ := ?ϕ := ϕ . Hence ϕ holds when ϕ is derivable. The Dialectica interpretation of FS is given by Fig. 3 and (8) As usual, proving ϕ˛ϕ D requires extra axioms. Besides (LAC), we use the following (linear ) semi-intuitionistic principles (LSIP), with polarities as shown: as well as the following deterministic exponential axioms (DEXP): All these axioms but (LAC) are true in the standard model (via − ). Moreover:

Translations of Classical Logic
There are many translations from classical to linear logic. Two canonical possibilities are the (−) T and (−) Q -translation of [9] (see also [16]) targeting resp. negative and positive formulae. Both take classical sequents to linear sequents of the form !(−) ?(−), which are provable in FS in part. thanks to the PF rules For the completeness of LMSO(C) (Thm. 6, §4), we shall actually require a translation (−) L such that the linear equivalences (with polarities as displayed) are provable possibly with extra axioms that we require to realize themselves. In part., (10) implies (DEXP), and (−) L should give deterministic formulae. While (−) T and (−) Q can be adapted accordingly, (10) induces axioms which make the resulting translations equivalent to the deterministic (−) L -translation of [24]: where (PEXP) are the following polarized exponential axioms, with polarities as shown: If ϕ is provable in many-sorted classical logic with equality then FS + (DEXP) proves ϕ L . Note that ϕ L is deterministic and that ϕ L = ϕ.

Completeness
In §3 we devised a Dialectica-like (−) D providing a syntactic extraction procedure for ∀∃(−) L -statements. In this Section, building on an axiomatic treatment of MSO(M), we show that LMSO, an arithmetic extension of FS + (LSIP) + (DEXP) + (PEXP) adapted from [24], is correct and complete w.r.t. Church's synthesis, in the sense that the provable ∀∃(−) L -statements are exactly the realizable ones. We then turn to the main result of this paper, namely the completeness of LMSO(C) := LMSO + (LAC). We fix the set of atomic formulae MSO(M) is many-sorted first-order logic with atomic formulae α ∈ At. Its sorts and terms are those given in §2, and standard interpretation extends that of §2 as follows:⊆ is set inclusion, E holds on B iff B is empty, N (resp. 0) holds on B iff B is a singleton {n} (resp. the singleton {0}), and S(B, C) (resp. B≤ C) holds iff B = {n} and C = {n + 1} for some n ∈ N (resp. B = {n} and C = {m} for some n ≤ m). We write x ι for variables x o relativized to N, so that (∃x ι )ϕ and (∀x ι )ϕ stand resp. for The logic MSO + [24] is MSO(M) restricted to the type o, hence with only terms for Mealy machines of sort (2, . . . , 2; 2). The MSO of [24] is the purely relational (term-free) restriction of MSO + . Recall from [24,Prop. 2.6], that for each Mealy machine M : The axioms of MSO(M) are the arithmetic rules of Fig. 5, the axioms (7) and the following, where M : 2 p → 2 and y, z, X are fresh.
From a proof of ϕ in LMSO(C) one can extract an eager term u(x) such that LMSO proves (∀x σ )ϕ D (u(x), x).

Completeness of LMSO(C)
The completeness of LMSO(C) follows from a couple of important facts. First, LMSO(C) proves the elimination of linear double negation, using (via Thm. 3) the same trick as in [24].
Combining Lemma 1 with (LAC) gives classical linear choice.
The key to the completeness of LMSO(C) is the following quantifier inversion.
Completeness of LMSO(C) then follows via (−) D , Proposition 5, completeness of MSO(M) and Büchi-Landweber Theorem 1. The idea is to lift a f.s. winning P- Theorem 6 (Completeness of LMSO(C)). For each closed formula ϕ, either

Conclusion
We provided a linear Dialectica-like interpretation of LMSO(C), a linear variant of MSO on ω-words based on [24]. Our interpretation is correct and complete w.r.t. Church's synthesis, in the sense that it proves exactly the realizable ∀∃(−) L -statements. We thus obtain a syntactic extraction procedure with correctness proof internalized in LMSO(C). The system LMSO(C) is moreover complete in the sense that for every closed formula ϕ, it proves either ϕ or its linear negation. While completeness for a linear logic necessarily collapse some linear structure, the corresponding axioms (DEXP) and (PEXP) do respect the structural constraints allowing for realizer extraction from proofs. The completeness of LMSO(C) contrasts with that of the classical system MSO(M), since the latter has provable unrealizable ∀∃-statements. In particular, proof search in LMSO(C) for ∀∃(−) L -formulae and their negation is correct and complete w.r.t.
Church's synthesis. The design of the Dialectica interpretation also clarified the linear structure of LMSO, as it allowed us to decompose it starting from a system based on usual full intuitionistic linear logic (see e.g. [3] for recent references on the subject). An outcome of witness extraction for LMSO(C) is the realization of a simple version of the fan rule (in the usual sense of e.g. [15]). We plan to investigate monotone variants of Dialectica for our setting. Thanks to the compactness of Σ ω , we expect this to allow extraction of uniform bounds, possibly with translations to stronger constructive logics than LMSO.

A.1 Proof of Proposition 3 (Fixpoints for Eager Functions)
We split Proposition 3 into two statments.
Consider an eager function We are going to define a causal function and let us look at how can be induced induced by a functionh : Γ + → Σ defined fromf . We should haveh That is, for n > 0,h(b 1 . . . b n ) = a n , where the a k 's are given by the recurrence: In terms of as follows: This easily gives an eager machine for fix(F ) given an eager machine for F . where Proposition 10 will mostly be used in the following context. Consider a Mealy term u(z, y) of sort (ς, σ; τ ) and an eager term t(x, y) of sort (τ, σ; ς). Intuitivelly, the term u(t(x, y), y) is "eager in x", in the sense that it can be seen as an eager Moore function Moreover H is finite-state whenever F and G are finite-state.
Proof. Assume that F and G are induced respectivelly by Then for all B ∈ Σ ω , all C ∈ Γ ω and all n ∈ N we have @ G F (B, C), B, C , F (B, C) , C (n) = @ G F (B, C), B, C (n) , F (B, C)(n) , C(n) = @ g F (B, C) n, B n, C n , f (B n, C n) , C(n) In particular, given F as in Lemma 4 and given G : Corollary 4. If G : Θ × Γ → S ∆ is causal and F : Σ × Γ → E Θ is eager, then there is an eager H : Σ × Γ → E ∆ Γ such that for all B ∈ Σ ω and all C ∈ Γ ω , we have @ (H(B, C), C) = G (F (B, C), C) Moreover, H is finite-state whenever so are F and G.
Returning to u(z, y) and t(x, y), consider H defined from G := u and F := t as in Corollary 4. Note that H is finite state since so are u and t . By applying Proposition 10 to if there are eager terms v(u, y, a) of sort (τ, ς, υ; (κ)υτ ) and x(u, y, a) of sort (τ, ς, υ; (σ)υςτ ) such that ϕ D u , @(x(u, y, a), u, y, a) , a FS+Ax ψ D @(v(u, y, a), u, a) , y , a In particular, ϕ(a) − → ψ(a) stands for − → w.r.t. the system FS without further axioms.
We will prove Theorem 2 with the following inductive invariant. Let

Proof. We have
and we have to provide eager terms u(u, x, a) of sort (τ, σ, υ; (τ )υτ ) and x(u, y, a) of sort (τ, σ, υ; (σ)υστ ) such that ϕ D u , @(x(u, x, a), u, x, a) , a FS+Ax ϕ D @(u(u, x, a), u, a) , x , a For x(u, x, a), consider the M-projection We obtain x(u, x, a) by composising the eager term for For u(u, x, a), we take the eager term obtained by composing the eager function Then we are done since FS proves @(u(u, x, a), u, a) . = σ u and @(x(u, x, a), u, x, a) .
The proof of Proposition 11 relies on the fixpoints of (finite-state) eager functions given by Proposition 3 (see also §A.1). For legibility reasons, we may refrain from writing free variables of terms explicitely and manipulate explicit substitutions.
In such a case, given a term t(. . . , x, . . . ) with free variable x and a term u of the appropriate sort, we write t[u/x] for the substitution of x by u in t. Let By assumption, there are eager terms From this data, our goal is to produce eager terms v(u 0 , x 2 , a) of sort (τ 0 , σ 2 , υ; (τ 2 )υτ 0 ) and y(u 0 , x 2 , a) of sort (τ 0 , σ 2 , υ; (σ 0 )υσ 2 τ 0 ) such that We would like v and y to satisfy the following equations in FS: But the variables x σ1 1 and u τ1 1 , which are free in u 1 and x 1 , should not occur in the terms y, v. We are thus lead to solve the following equations in FS with terms y 1 (u 0 , x 2 , a) of sort (τ 0 , σ 2 , υ; σ 1 ) v 1 (u 0 , x 2 , a) of sort (τ 0 , σ 2 , υ; τ 1 ) Assuming (11) satisfied, we are done since We now turn to the resolution of (11). We first discuss the construction of y 1 and v 1 and then turn to y and v.
then a by taking v 1 := (u 1 [y 1 /x 1 ])u 0 a one obtains a term satisfying the corresponding equation in (11). The Mealy term induces a finite-state causal function while the eager term u 1 (u 0 , x 1 , a) induces (via Prop. 9) a f.s. eager function
The cases of !(ϕ − ) and ?(ϕ + ) are easy. The cases of !(ϕ + ) and ?(ϕ − ) amount to the corresponding rules in PF and follow by taking terms similar to those of Lemma 5. We only detail some cases. We first consider cases of !(ϕ − ) and ?(ϕ + ).
(1) We have and the result follows from (3) We have and the result follows from the that FS proves !ψ − !ψ ⊗ !ψ for all ψ.
(1) We have and we conclude as in Lemma 5, using that FS proves !ψ ψ for all ψ.
We thus have (∃z.ϕ(a, z)) D = (∃u)(∃z)(∀x)ϕ D (u, x, a, z) and (∀z.ϕ(a, z)) D = (∃u (τ )κ )(∀x)(∀z)ϕ D ((u)z, x, a, z) In both cases we assume a υ , z κ and t(a) to be of sort (υ; κ) (1) We have to find eager terms u(u, x, a), z(u, x, a) and x(u, x, a) such that ϕ D (u, @(x(u, x, a), u, x, a), a, t(a)) − ϕ D (@(u(u, x, a), u, a), x, a, @(z(u, x, a), a)) We let u and x be given as in Lemma 5. As for z(u, x, a), we take the eager term obtained from the composite is a suitable Mealy projection. (2) Note that the variable u now has type (τ )κ. We have to find eager terms u(u, x, a), z(u, x, a) and x(u, x, a) such that z(u, x, a), z)), @(x(u, x, a), u, x, a), a, @(z(u, x, a), a) − ϕ D (@(u(u, x, a), u, x, a), x, a, t(a)) First, for z(u, x, a), we take as above the eager term obtained from the composite It thus remains to find u and x such that ϕ D @(u, t(a)), @(x(u, x, a), u, x, a), a, t(a) − ϕ D (@(u(u, x, a), u, x, a), x, a, t(a)) We then take for x the same term as in Lemma 5. It remains to deal with u. Consider the Mealy termũ for Then we take for u the eager term The Equality Axioms (7). Realization of the equality axioms (7), follows from the fact that atomic formulae are interpreted by themselves, so these axioms are interpreted by instances of themselves. This concludes the proof of Theorem 8.

B.2 Realization of Additional Axioms (Proposition 4)
We decompose Proposition 4 as follows.
Lemma 12. The axioms (LSIP) are realized in FS: In each case we are done by taking terms similar to those of Lemma 5: (1) We have Lemma 13. The axiom (LAC) is realized in FS: amounts to the following.
The second one is to notice that characterization for polarized formulae is provable within the polarized fragment PF augmented with the following polarized weakening of (LSIP) (with polarities as displayed): We only detail Lemma 15, as it corresponds to the statment of Theorem 3.
Proof of Lemma 15. Consider first the case of ϕ(a) ψ(a). By assumption, for θ either ϕ or ψ, we have The case of ϕ ± (a) ψ ± (a) is trivial. The other cases are given by the following derviations (where we did not display the free variable a).
-Case of (ϕ − ψ + ) D . We first show For the converse implication, we use the axioms where D is obtained from the axiom and D is obtained from the axiom The converse direction is given by Consider now the case of (∀a)ϕ − (a). Let
The result follows by cutting   (7). But they follow from the fact that α L = α for each atomic formula α ∈ At.
Proof of Proposition 5. We now turn to Prop. 5, namely the equivalence of (DEXP) + (PEXP) with So we can as well prove Proposition 5 with (−) T ± instead of (−) L . We split this into two statments.
-The arithmetic rules of Fig. 5 follow from the fact that α L = α for each atomic formula α ∈ At. -The induction scheme of LMSO requires one hypothesis to be under an exponential modality !(−) to accomodate arbitrary negative formulae; the situation is resolved by cutting with the LMSO axiom enabling to remove exponentials over deterministic formulae. By the induction hypothesis (and since (−) L commutes over substitution), we have proofs of Clearly, it is equivalent to the following scheme where we make the universal quantification implicit by using formulae with free variables which translate to the following, which is then clearly derivable from the corresonding scheme in LMSO by instantiating the universal quantifiers by the free variables We then easily derive LMSO ϕ L (u(x), x) whence the result.
-For the arithmetic rules of Fig. 5, this follow from the fact that atomic formulae are interpreted by themselves, so these axioms are interpreted by instances of themselves. -For comprehension, this follows from the fact that the axiom is a deterministic formula, so its realizers are trivial, and the fact that each instance is interpreted by an instance of the axiom. Using an instance of choice available thanks to the fact that O may be wellordered, this is in turn equivalent to It follows that we only need to prove ∀F ∈ P O P .∀õ ∈ O P .∃p ∈ P.∃f ∈ O P .
[F (f ) = p and f (p) =õ(p)] But this is now easy: given F ∈ P O P andõ ∈ O P , we can take f :=õ and p := F (õ) to conclude.
In particular, given alphabets Σ and Γ , there are functions