Around finite second-order coherence spaces

TL;DR

This paper demonstrates finite and effective semantics for a polymorphic linear language using coherence spaces and hypercoherences, with implications for computational complexity and higher-order model checking.

Contribution

It introduces a finite, effective semantics for second-order linear logic models, extending to hypercoherence spaces and analyzing complexity bounds.

Findings

Denotations of formulas are finite in Girard's coherence space semantics.

The semantics are effectively computable, enabling practical applications.

Finiteness also holds for hypercoherence models, despite additional challenges.

Abstract

Many applications of denotational semantics, such as higher-order model checking or the complexity of normalization, rely on finite semantics for monomorphic type systems. We exhibit such a finite semantics for a polymorphic purely linear language: more precisely, we show that in Girard's semantics of second-order linear logic using coherence spaces and normal functors, the denotations of multiplicative-additive formulas are finite. This model is also effective, in the sense that the denotations of formulas and proofs are computable, as we show. We also establish analogous results for a second-order extension of Ehrhard's hypercoherences; while finiteness holds for the same reason as in coherence spaces, effectivity presents additional difficulties. Finally, we discuss the applications our our work to implicit computational complexity in linear (or affine) logic. In view of these applications, we study cardinality and complexity bounds in our finite semantics.

Figures

Equations

$$A,B:=X\mid X^{\perp }\mid 1\mid \perp \mid A\otimes B\mid A\text{parr}B\mid 0\mid \top \mid A\oplus B\mid A\text{with}B\mid \forall X.A\mid \exists X.B\mid \text{oc}A\mid \text{wn}A$$

$$\begin{array}[]{lcl!\qquad lcl !\qquad lcl}1^{\bot}&:=&\bot&\bot^{\bot}&:=&1&(\exists X.\,A)^{\bot}&:=&\forall X.\,A^{\bot}\\ (A\otimes B)^{\bot}&:=&A^{\bot}\parr B^{\bot}&(A\parr B)^{\bot}&:=&A^{\bot}\otimes B^{\bot}&(\forall X.\,A)^{\bot}&:=&\exists X.\,A^{\bot}\\ 0^{\bot}&:=&\top&\top^{\bot}&:=&0&(\oc A)^{\bot}&:=&\wn A^{\bot}\\ (A\oplus B)^{\bot}&:=&A^{\bot}\with B^{\bot}&(A\with B)^{\bot}&:=&A^{\bot}\oplus B^{\bot}&(\wn A)^{\bot}&:=&\oc A^{\bot}\end{array}$$

$$c_S=\{\ast \}\times S\subseteq [[1⊸X{]]}_{X\mapsto S}{c}_S^{\prime }=S\times \{\ast \}\subseteq [[X⊸1{]]}_{X\mapsto S}$$

$$c_X=\{F({\iota }_1,\dots ,{\iota }_n)(y)\mid ⟨\vec{Y}⊢y⟩\in c,{\iota }_i:Y_i\hookrightarrow X_i\text{for}i\in \{1\dots n\}\}$$

$$\mathrm{deg}(F)=\sup \{\mathrm{card}(|X_i|)\mid ⟨X_1,\dots ,X_n⊢x⟩\in \mathrm{NF}(F),i\in \{1,\dots ,n\}\}.$$

$$x::=a\in \mathrm{Var}\mid (x,x)\mid \mathrm{inl}(x)\mid \mathrm{inr}(x)\mid ⟨X⊢x⟩$$

$$[[A\text{rightarrowtriangle}B]]=(A\text{with}1)⊸B[[\forall X.A]]=\forall ([[A]])$$

$$\mathsf{HCohL}(n)(A,\forall (F))\cong \mathsf{HCohL}(n+1)(\mathrm{Const}(A),F)$$

$$|\mathrm{Tr}(F)|=\{x\in \mathrm{NF}(F)\mid \{x\}\in \Gamma \mathrm{NF}(F)\}\Gamma (\mathrm{Tr}(F))={\mathcal{P}}_{\mathrm{fin}}^{\ast }(\mathrm{Tr}(F))\cap \Gamma \mathrm{NF}(F)$$

$$F_f(X)=(|X|,\{S{\subseteq }_{\mathrm{fin}}^{\ast }|X|\mid f(\mathrm{Card}(S))=1\})$$

$$f:(⟨X_1,\dots ,X_n⊢x⟩\in P)\to |X_k|$$

$$f_{Y_1,\dots ,Y_n}^F:\{y\in |F(Y_1,\dots ,Y_n)|\mid \mathrm{NF}(y;F)\in P\}\to |Y_k|$$

$$f_{Y_1,\dots ,Y_n}^F:\{S{\subseteq }_{\mathrm{fin}}^{\ast }|F(Y_1,\dots ,Y_n)|\mid \mathrm{NF}(S;F)=P\}\to {\mathcal{P}}_{\mathrm{fin}}^{\ast }(|Y_k|)$$

$${\sigma }_F:(P\in {\mathcal{P}}_{\mathrm{fin}}^{\ast }(\mathrm{NF}(F)))\to \mathrm{PVal}(P)\to \{\odot ,\ominus ,\oplus \}$$

$$l_{F(Y_1,\dots ,Y_n)}(S)={\sigma }_F(\mathrm{NF}(S;F),(f\in {\mathrm{Proj}}_k(\mathrm{NF}(S;F))\mapsto l_{Y_k}(f_{Y_1,\dots ,Y_n}^F(S))))$$

$$\{w\in \{0,1{\}}^{\ast }\mid \pi (\overline{w}[A_i]\otimes \dots \otimes \overline{w}[A_n]){⟶}^{\ast }\mathtt{true}\}$$

$$\mathrm{deg}(F)=\inf \{d\in \mathbb{N}\mid \mathrm{Card}(|F(X)|)=O(\mathrm{Card}(|X|{)}^d)\}$$

$${\theta }_X\cong \{(\iota ,y)\mid ⟨Y⊢y⟩\in \theta ,\iota :Y\to X\text{ embedding}\}$$

$$\text{(functorial promotion)}\frac{⊢\Gamma ,A}{⊢\text{wn}\Gamma ,\text{oc}A}\text{(weakening)}\frac{⊢\Gamma }{⊢\Gamma ,\text{wn}A}\text{(contraction)}\frac{⊢\Gamma ,\text{wn}A,\text{wn}A}{⊢\Gamma ,\text{wn}A}$$

$$\text{(ax-rule)}\frac{}{⊢A,A^{\perp }}\text{(cut rule)}\frac{⊢\Gamma ,A⊢A^{\perp },\Delta }{⊢\Gamma ,\Delta }\text{(exchange rule)}\frac{⊢\Gamma ,A,B,\Delta }{⊢\Gamma ,B,A,\Delta }$$

$$\text{(⊗-rule)}\frac{⊢\Gamma ,A⊢B,\Delta }{⊢\Gamma ,A\otimes B,\Delta }\text{(parr-rule)}\frac{⊢\Gamma ,A,B}{⊢\Gamma ,A\text{parr}B}\text{(⊥-rule)}\frac{⊢\Gamma }{⊢\Gamma ,\perp }\text{(1-rule)}\frac{}{⊢1}$$

$$\text{(⊕-rule)}\frac{⊢\Gamma ,A_i}{⊢\Gamma ,A_1\oplus A_2}\text{for i∈{1,2}}\text{(with-rule)}\frac{⊢\Gamma ,A⊢\Gamma ,B}{⊢\Gamma ,A\text{with}B}\text{(⊤-rule)}\frac{}{⊢\Gamma ,\top }$$

$$\text{(∃-rule)}\frac{⊢\Gamma ,A[B/X]}{⊢\Gamma ,\exists X.A}\text{(∀-rule)}\frac{⊢\Gamma ,A}{⊢\Gamma ,\forall X.A}\text{for X not free in Γ}$$

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$$\{w\in \{0,1{\}}^{\ast }\mid \hat{\pi }(\overline{w}[A_i]\otimes \dots \otimes \overline{w}[A_n]){⟶}^{\ast }\mathtt{true}\}$$

Full text

Around finite second-order coherence spaces

Lê Thành Dũng Nguyễn

This model is also effective, in the sense that the denotations of formulas and proofs are computable, as we show. We also establish analogous results for a second-order extension of Ehrhard’s hypercoherences; while finiteness holds for the same reason as in coherence spaces, effectivity presents additional difficulties.

Finally, we discuss the applications our our work to implicit computational complexity in linear (or affine) logic. In view of these applications, we study cardinality and complexity bounds in our finite semantics.

1 Introduction

Polymorphism is a central topic in theoretical computer science since the sixties. A breakthrough in its logical understanding was its analysis by means of second order quantifiers, that is the introduction of System F (also known as the polymorphic $\lambda$-calculus) at the beginning of the seventies. This considerable success later led Jean-Yves Girard to develop a denotational semantics for System F [12], to get a deeper understanding of its computational features. Indeed, the general goal of denotational semantics is to give a “mathematical” counterpart to syntactic devices such as proofs and programs, thus bringing to the fore their essential properties. Sometimes this eventually results in improvements of the syntax: Linear Logic itself [13] arose precisely from the denotational model introduced in [12].

But denotational semantics is not just a matter of increasing our understanding of programming languages, it also has direct algorithmic applications. Let us mention:

• •

in the simply-typed lambda calculus (${\mathrm{ST}\lambda}$), the semantic evaluation technique for complexity bounds, see Terui’s recent paper [24] and references therein;

• •

in ${\mathrm{ST}\lambda}$ extended with a fixed-point combinator, the semantic approach to higher-order model checking (HOMC) advocated by Salvati and Walukiewicz [25, 26] (see also [1, 20]).

The following little-known theorem illustrates both kinds of applications. Indeed, it is a result in implicit computational complexity: it gives a machine-free characterization of a complexity class. At the same time, it is an instance of the correspondence between Church encodings and automata that HOMC generalizes to infinite trees.

Theorem 1.1** (Hillebrand and Kanellakis [19]).**

The languages decided by ${\mathrm{ST}\lambda}$ terms from Church-encoded binary strings to Church booleans, i.e. of type $\mathtt{Str}_{\mathrm{ST}\lambda}[A]\to\mathtt{Bool}_{\mathrm{ST}\lambda}$, are exactly the regular languages.

(Here $\mathtt{Str}_{\mathrm{ST}\lambda}[A]=(A\to A)\to(A\to A)\to(A\to A)$ and $\mathtt{Bool}_{\mathrm{ST}\lambda}=o\to o\to o$; $o$ is a base type, while $A$ may be chosen depending on the language one wants to decide.)

To prove this, the main idea is to build a deterministic finite automaton (DFA) computing the denotation of the input string. Crucially, this relies on the existence of a finite semantics for ${\mathrm{ST}\lambda}$ – such as the category of finite sets – which will provide the states of the DFA. In general, this finiteness property, or finer cardinality bounds, are key to these applications.

This theorem also holds when replacing ${\mathrm{ST}\lambda}$ by propositional linear logic, which also admits finite semantics. In fact, Terui’s solution to the complexity of ${\mathrm{ST}\lambda}$ normalization at fixed order [24] relies on such a semantics. As for HOMC, Grellois and Melliès have developed an approach relying on models of linear logic [18, 17].

However, all of this concerns only monomorphic type systems, for a simple reason: as we shall soon see, System F does not admit any non-trivial finite semantics. A central message of this paper is that second-order quantification is not the only culprit here: one can also blame non-linearity, i.e. the possibility of duplicating data. What we show is that a semantics for a purely linear language with impredicative polymorphism can be finite:

Theorem 1.2**.**

Second-order Multiplicative-Additive Linear Logic ($\textsc{mall}_{2}$) admits a non-trivial (i.e. distinguishing the two inhabitants of $1\oplus 1$) finite semantics.

Recall that $\textsc{mall}_{2}$ is the fragment of second-order linear logic without the exponential modalities $\oc/\wn$ whose role is to allow111The possibility of duplication is expressed through the contraction rule $\oc A\multimap\oc A\otimes\oc A$. a controlled amount of non-linearity. We shall also prove the analogous property for the second-order affine $\lambda$-calculus.

1.1 Some immediate consequences of finite semantics

To illustrate the power of the above theorem, we find it instructive to explain first the impossibility of finite semantics for System F. It is a consequence of its ability to represent infinite data types, with definable destructors.

Proposition 1.3**.**

Let $\mathtt{Nat}_{F}=\forall X.\,(X\to X)\to(X\to X)$ be the type of System F natural numbers. Then any semantics distinguishing the two inhabitants of the type of booleans $\mathtt{Bool}_{F}=\forall X.\,X\to X\to X$ is injective on $\mathtt{Nat}$ – implying the latter has an infinite denotation.

Proof 1.4**.**

For any $n\in\mathbb{N}$, one can define the predicate $\mathtt{eq}_{n}:\mathtt{Nat}_{F}\to\mathtt{Bool}_{F}$ which tests if its argument is equal to222We identify natural numbers with their Church encoding. $n$. Thus, if $m$ and $n$ share the same denotation, then $\mathtt{eq}_{n}(n)=_{\beta}\mathtt{true}$ and $\mathtt{eq}_{n}(m)$ have the same denotation. Since we have assumed that $\mathtt{true}$ and $\mathtt{false}$ have different denotations, this means that $\mathtt{eq}_{n}(m)=_{\beta}\mathtt{true}$, i.e. $m=n$.

The argument above is robust enough to apply to a wide variety of situations. For instance, the existence of a finite semantics for ${\mathrm{ST}\lambda}$ immediately yields by contrapositive:

Theorem 1.5** (Statman).**

Equality is not definable on the Church integers in ${\mathrm{ST}\lambda}$.

Another such situation is the system $\mu$mall: propositional mall with fixed points [3]. Some functions analogous to $\mathtt{eq}_{n}$ can also be defined in $\mu$mall, using $\mu X.\,1\oplus X$ as the type of natural numbers. This leads to a first application of our Theorem 1.2:

Theorem 1.6**.**

There exists no faithful translation from $\mu$mall to $\textsc{mall}_{2}$.

Proof 1.7**.**

Else, $\mu$mall would admit a non-trivial finite semantics, by translating it into $\textsc{mall}_{2}$ and using the semantics of $\textsc{mall}_{2}$. By the argument above, this leads to a contradiction.

While $\mu$mall can be translated in full second-order linear logic, it was argued that both polymorphism and exponentials were required for such a translation [3, §2.3]. Our finite semantics provides a short rigourous proof of the necessity of exponentials, and pinpoints the main reason: $\textsc{mall}_{2}$ cannot represent infinite data types.

This divide between $\textsc{mall}_{2}$ and $\mu$mall is further explored in this paper in the context of implicit complexity; this is discussed in Section 1.3.

Remark 1.8**.**

Although ${\mathrm{ST}\lambda}$ can represent functions on infinite data types such as integers or strings, this generally involves a meta-level universal quantification, see e.g. Hillebrand and Kanellakis’s theorem. In $\textsc{mall}_{2}$, which already contains quantifiers, this way of sidestepping the issue does not work.

1.2 Concrete models: coherence spaces and hypercoherences

It turns out that a finite semantics of $\textsc{mall}_{2}$ has been lying around all along since the birth of linear logic, though we are not aware of anyone noticing this fact beforehand. It is none other than Girard’s coherence spaces [13], which he obtained as a simplification of his previous work on “qualitative domains” in System F [12], discovering linear logic along the way.

In this model, open types (i.e. formulas with free variables) are represented as normal functors333This categorical tool had been used previously to give the first quantitative semantics of the $\lambda$-calculus [14] – a work which is arguably one of the main inspirations for linear logic (linear $\lambda$-terms are interpreted in this model as monomials of degree 1, hence “linear”), and even differential linear logic [10].. To prove our finiteness theorem, we introduce a notion of normal functor of finite degree, which is preserved by $\textsc{mall}_{2}$ connectives and ensures finiteness. It is also equivalent to an asymptotic polynomial growth property.

Furthermore, this semantics is more concrete than its formulation using category-theoretic machinery could suggest. Thanks to a combinatorial presentation, we prove that it is an effective model: the denotations of types and terms are computable. Note that historically, effectivity was a major motivation for using coherence spaces instead of qualitative domains (see [12, Appendix C], especially Remark C.3).

We also study the interpretation of $\textsc{mall}_{2}$ in Ehrhard’s hypercoherences [9]. Although it was defined as a model of propositional linear logic, transposing the recipe of coherence spaces gives a model of second-order linear logic, with finite denotations for $\textsc{mall}_{2}$ formulas and proofs. In this semantics, effectivity stumbles upon the same issue as in qualitative domains: roughly speaking, the presence of “$n$-ary coherences” for arbitrary $n\geq 2$ (while in coherence spaces, $n=2$). We show that despite this, the hypercoherence model can be made effective for $\textsc{mall}_{2}$, using a notion of “specification by projections”.

1.3 Relevance to implicit computational complexity

The semantic developments we present here have already been applied to obtain some results on variants of Elementary Linear Logic (ELL) [8]. In this subsystem of linear logic, purely “geometric” restrictions inspired by the theory of proof nets enforce complexity constraints, following an approach pioneered by the characterization of polynomial time in Light Linear Logic (LLL) [16]. Thanks to our finite semantics, we can apply to second-order ELL some ideas from another tradition in implicit complexity, exemplified by Hillebrand and Kanellakis’s Theorem 1.1. (Although some previous works on “light logics” such as LLL and ELL make use of semantic arguments – for instance, Statman’s above-mentioned Theorem 1.5 has been applied to LLL in [7] – to our knowledge these applications have mostly consisted in proving inexpressivity results for monomorphic systems.)

• •

In [22], we characterize regular languages in the elementary affine $\lambda$-calculus [5]; as a side effect, this answers an open question on a pre-existing characterization of polynomial time (we refer to [22] for further discussion of the significance of this result). A crucial ingredient is the existence of a finite semantics of the second-order affine $\lambda$-calculus, which we shall prove in this paper using coherence spaces.

• •

In a joint work with P. Pradic [23], we define a class of queries over finite structures expressed in ELL, which lies between deterministic and non-deterministic logarithmic space. (We also obtain a somewhat contrived exact characterization of deterministic logarithmic space.) This relies on some cardinality and complexity bounds on the coherence semantics of $\textsc{mall}_{2}$, which we establish in the present paper.

As an illustration of the power of finite second-order coherence spaces, we prove a slight variation of the first item above (regular languages) in the setting of second-order ELL. By using some specific features of this model, we get a shorter proof than in [22], which is very close to the proof of Theorem 1.1 by Hillebrand and Kanellakis.

One should note that if we were to enrich ELL with type fixpoints, then, instead of regular languages, one would obtain a class containing at least444We believe that this class would be exactly P. But to adapt the P soundness theorem in [4] to ELL, one would need to work with some notion of proof net for second-order (elementary) linear logic with additives; since this is likely to involve technical complications, we have not attempted to do so. the languages decidable in polynomial time (as can be shown by adapting the proof of polynomial time completeness in [4]; see also the discussion in [22]). This gives a quantitative manifestation of the expressivity gap between $\mu$mall and $\textsc{mall}_{2}$ mentioned earlier in this introduction.

1.4 Plan of the paper

We first recall the second-order coherence space model and prove its finiteness and effectivity for $\textsc{mall}_{2}$ in Section 2. This is followed by a short discussion, in Section 3, on the adaptation of this finite semantics to the second-order affine $\lambda$-calculus. We define the second-order extension of the hypercoherence model in Section 4, and show how to make it effective.

2 Finite and effective second-order coherence spaces

Before we tackle the question of finiteness, we must first recall Girard’s model of second-order linear logic in coherence spaces. Since this model is not very well-known, the first two subsections will be expository with no new results. Finiteness is shown in Section 2.3, and effectivity in Section 2.4. The omitted proofs of this section are in Appendix A.

Syntax

The formulas of second-order linear logic are given by the grammar

[TABLE]

where $X$ belongs to a fixed countable set of variables. Propositional linear logic is the fragment made of formulas without quantifiers $\exists/\forall$, while $\textsc{mall}_{2}$ is the fragment without the the exponential modalities $\oc$/$\wn$.

The involutive linear negation $(-)^{\perp}$ is defined inductively on formulas by the rules in Figure 1. It is used to define linear implication as $A\multimap B:=A^{\perp}\parr B$.

Since we work in pre-existing semantics, we do not need to formally define the notion of denotational model of linear logic; we refer to [21] for an extensive survey of this topic. We will not be need to work with some precise proof system – e.g. sequent calculus – either, except in Section 5.

Coherence spaces (propositional case)

Recall that a coherence space is an undirected (reflexive) graph, i.e. a pair $X=(\mathopen{|}X\mathclose{|},\coh_{X})$ of a set $\mathopen{|}X\mathclose{|}$ – the web of $X$ – and a symmetric and reflexive relation $\coh_{X}\;\subseteq\mathopen{|}X\mathclose{|}\times\mathopen{|}X\mathclose{|}$ – its coherence relation. A subset $c\subseteq\mathopen{|}A\mathclose{|}$ is a clique of $A$ if its points are pairwise coherent for $\coh_{A}$; in this case we write $c\sqsubset A$.

The operations $(-)^{\perp},\otimes,\oplus,\oc$ are defined on coherence spaces as follows (cf. [15]):

• •

$\mathopen{|}X^{\perp}\mathclose{|}=\mathopen{|}X\mathclose{|}$ and $x\coh_{X^{\perp}}x^{\prime}\iff x\not\coh_{X}x^{\prime}\lor x=x^{\prime}$ (complement graph)

• •

$\mathopen{|}X\otimes Y\mathclose{|}=\mathopen{|}X\mathclose{|}\times\mathopen{|}Y\mathclose{|}$ and $(x,y)\coh_{X\otimes Y}(x^{\prime},y^{\prime})\iff x\coh_{X}x^{\prime}\land y\coh_{Y}y^{\prime}$

• •

$\mathopen{|}X\oplus Y\mathclose{|}=\mathopen{|}X\mathclose{|}\uplus\mathopen{|}Y\mathclose{|}$ and $z\coh_{A\oplus B}z^{\prime}\iff\exists Z\in\{X,Y\}:(z\in Z\land z^{\prime}\in Z\land z\coh_{Z}z^{\prime})$

• •

\mathopen{|}\oc X\mathclose{|}=\{\text{cliques of X}\} and $c\coh_{\oc X}c^{\prime}\iff c\cup c^{\prime}\sqsubset X$ (“set-based” exponential)

Furthermore, the multiplicative units $1,\bot$ are interpreted as the singleton space, and the additive units $\top,0$ as the empty space. This is enough to define inductively the denotation $\mathopen{\llbracket}A\mathclose{\rrbracket}$ of a formula $A$ in propositional linear logic, given an assignment of the free variables of $A$ to coherence spaces.

A proof $\pi:A$ is interpreted in the coherence space model as a clique $\mathopen{\llbracket}\pi\mathclose{\rrbracket}\sqsubset\mathopen{\llbracket}A\mathclose{\rrbracket}$. In terms of categorical semantics, the model is given as the category $\mathsf{CohL}$:

• •

whose objects are coherence spaces;

• •

whose morphisms between $X$ and $Y$ are cliques in $X\multimap Y$, composition being relational composition (this is meaningful since $\mathopen{|}X\multimap Y\mathclose{|}=\mathopen{|}X\mathclose{|}\times\mathopen{|}Y\mathclose{|}$).

2.1 Functors on embeddings and uniform families

In order to interpret second-order quantification, we want to give a first-class status to the map \{\text{assignments for variables in A}\}\to\{\text{possible values for \mathopen{\llbracket}A\mathclose{\rrbracket}}\} when $A$ is an open type (i.e. a formula with free variables). The first idea that comes to mind is to consider it as a functor on $\mathsf{CohL}$. But it stumbles on the fact that while the binary connectives are covariant bifunctors on $\mathsf{CohL}$, linear negation is a contravariant endofunctor. Instead, Girard’s idea is to work in a “category of embeddings” (see [6]) to make negation covariant.

Definition 2.1**.**

An embedding of a coherence space $X$ into a coherence space $Y$ is an injection $\iota:\mathopen{|}X\mathclose{|}\to\mathopen{|}Y\mathclose{|}$ such that $x\coh_{X}x^{\prime}\Leftrightarrow\iota(x)\coh_{Y}\iota(x^{\prime})$. We write $\iota:X\hookrightarrow Y$.

The category $\mathsf{CohI}$ has as objects coherence spaces, and as morphisms the embeddings.

Proposition 2.2**.**

$(-)^{\perp}$ is a covariant endofunctor of $\mathsf{CohI}$.

Proof 2.3**.**

If $X$ is an induced subgraph of $Y$, then $X^{\perp}$ is an induced subgraph of $Y^{\perp}$.

Remark 2.4**.**

In the same vein, the graph $\{(x,\iota(x))\mid x\in\mathopen{|}X\mathclose{|}\}$ of an embedding $\iota:X\hookrightarrow Y$ is a clique both in $X\multimap Y$ and in $X^{\perp}\multimap Y^{\perp}$.

Let us say, provisionally, that functors $F:\mathsf{CohI}^{n}\to\mathsf{CohI}$ are our semantical counterpart of open formulas with $n$ variables. A proof of such a formula should be a family of cliques $c_{X_{1},\ldots,X_{n}}\sqsubset F(X_{1},\ldots,X_{n})$, “uniform” in some way. The following notion of uniformity has been called the “mutilation property” by Girard [12]:

Definition 2.5**.**

A family $c_{X_{1},\ldots,X_{n}}\sqsubset F(X_{1},\ldots,X_{n})$ is uniform if for any embeddings $\iota_{i}:X_{i}\hookrightarrow Y_{i}$ ($i\in\{1,\ldots,n\}$), $c_{X_{1},\ldots,X_{n}}=F(\iota_{1},\ldots,\iota_{n})^{-1}(c_{Y_{1},\ldots,Y_{n}})$.

Remark 2.6**.**

At this point we have to point out a subtlety of the coherence space model: it is not a priori obvious that uniformity is closed under composition, in other words, that the pointwise composition of a uniform clique family for $F\multimap G$ with a uniform clique family for $G\multimap H$ is uniform for $F\multimap H$. Indeed, taking $n=1$ for simplicity, the uniformity condition for a family $f_{X}\sqsubset F(X)\multimap G(X)$ seen as a family of morphisms is expressed as a diagram

{F(Y)}$${G(Y)}$${F(X)}$${G(X)}$$\scriptstyle{f_{Y}}$$\scriptstyle{G(\iota)^{-}}$$\scriptstyle{F(\iota)^{+}}$$\scriptstyle{f_{X}}

in $\mathsf{CohL}$, where $j^{+}=\{(x,j(x))\mid x\in A\}\sqsubset A\multimap B$ and $j^{-}=\{(j(x),x)\mid x\in A\}\sqsubset B\multimap A$ for $j:A\hookrightarrow B$. Such diagrams cannot be “formally” pasted horizontally.

It turns out that uniform clique families do compose, but this issue is non-trivial and was overlooked in Girard’s papers. A proof is part of the folklore and often credited to Eugenio Moggi.

Remark 2.7**.**

In the relational semantics of linear logic, in which objects are sets and morphisms are relations, one could define an analogous notion of uniform subset family for a functor on the category of injections. But then, for the uniform families

[TABLE]

the composition $c^{\prime}_{S}\circ c_{S}$ is not uniform: it is equal to $\{(*,*)\}$ if $S\neq\varnothing$, and $\varnothing$ if $S=\varnothing$.

This issue with the second-order relational model is known and has been investigated by A. Bac-Bruasse (whose PhD thesis in French [2] is the main reference on the subject to our knowledge), T. Ehrhard and C. Tasson. What makes composition work in coherence spaces is the domain-theoretic stability555Stability is indeed a recurring pattern here: the uniformity condition is reminiscent of Berry’s stable order between functions on domains, and the preservation of pullbacks in normal functors (next section) is a categorification of stability. property of morphisms. This is our reason for working with coherence spaces instead of the simpler relational model.

2.2 Normal functors

The next goal is to interpret quantifiers. Let us look at the example of the formula $X\multimap X$, which admits a proof $\pi$ such that $\mathopen{\llbracket}\pi\mathclose{\rrbracket}_{X\mapsto S}=\{(s,s)\mid s\in S\}$ for any coherence space $S$. This uniform family should correspond to a clique in some space $\mathopen{\llbracket}\forall X.\,X\multimap X\mathclose{\rrbracket}$. The idea is to take some kind of “patterns with bound variables” as the points of this coherence space. Typically, $\{(s,s)\mid s\in S\}$ should correspond to the single pattern $\langle x\vdash(x,x)\rangle$ with a bound variable $x$ – and thus to the clique $\{\langle x\vdash(x,x)\rangle\}\sqsubset\mathopen{\llbracket}\forall X.\,X\multimap X\mathclose{\rrbracket}$. Observe:

• •

that the substitution of the variable $x$ by $s\in S$ in $(x,x)$ corresponds to the functoriality of $\mathopen{\llbracket}X\multimap X\mathclose{\rrbracket}:\mathsf{CohI}\to\mathsf{CohI}$ with respect to the embedding $\iota:\{x\}\hookrightarrow S$ such that $\iota(x)=s$;

• •

that $(x,x)\in F(\{x\})$, and $\{x\}$ is “minimal” or “initial” in the sense that any other $(s,s)\in F(S)$ is an image of $(x,x)$ via an embedding $\{x\}\hookrightarrow S$.

We thus arrive at the idea that “patterns with bound variables” should correspond to “minimal” spaces. To guarantee their existence, we need to put an additional condition on our functors $\mathsf{CohI}^{n}\to\mathsf{CohI}$. This is why Girard interprets open types by normal functors.

Definition 2.8**.**

A functor is normal if it preserves filtered colimits and finite pullbacks.

The name comes from Girard’s normal form theorem:

Theorem 2.9**.**

Let $F:\mathsf{CohI}^{n}\to\mathsf{CohI}$ be a functor, $\mathopen{|}F\mathclose{|}$ be the covariant presheaf obtained by taking the web, and $\mathsf{El}(\mathopen{|}F\mathclose{|})$ be its category of elements.

$F$ is normal if and only if, for any object $\vec{X}$ in $\mathsf{CohI}^{n}$ and point $x\in\mathopen{|}F(\vec{X})\mathclose{|}$, the slice category $\mathsf{El}(\mathopen{|}F\mathclose{|})/(\vec{X},x)$ admits a finite initial object $(\vec{X^{\prime}},x^{\prime})$.

In this case, $(\vec{X^{\prime}},x^{\prime})$ is initial in its own slice category. We call an object of $\mathsf{El}(\mathopen{|}F\mathclose{|})$ enjoying this property a normal form.

Remark 2.10**.**

It is worth noting that this characterisation is one of many similar results. For instance, Joyal’s analytic functors have a weak finite normal form property (i.e. where the initial elements are only weakly initial), a variation corresponding to preservation of weak pullbacks and filtered colimits. Similarly, Kock’s characterisation of polynomial functors states that preservation of wide pullbacks is equivalent to the existence of normal forms (though not finite); in fact, Girard’s normal functors correspond to Kock’s finitary polynomial functors. See the discussion in [11, §1.18–1.21].

Definition 2.11**.**

Let $F$ be a normal functor. We define $\mathrm{NF}(F)$ to be its set of isomorphism classes of normal forms (for isomorphisms in $\mathsf{El}(\mathopen{|}F\mathclose{|})$).

Notation 2.12**.**

We use the notation $\langle\vec{X}\vdash x\rangle$ for normal forms $(\vec{X},x)\in\mathrm{NF}(F)$. Alternatively, if $\vec{X}=(X_{1},\ldots,X_{n})$, we may write $\langle X_{1},\ldots,X_{n}\vdash x\rangle$.

The set $\mathrm{NF}(F)$ summarizes in a way all the webs $|F(\vec{X}|)$ of the instantiations of $F$, as the proposition below shows. As for the uniform families of cliques of $F$, they are summarized by the trace of $F$, a coherence space built from $\mathrm{NF}(F)$.

Proposition 2.13**.**

Let $F:\mathsf{CohI}^{n}\to\mathsf{CohI}$ be a normal functor and $\vec{X}=(X_{1},\ldots,X_{n})$. Then $\mathopen{|}F(\vec{X})\mathclose{|}=\{F(\iota_{1},\ldots,\iota_{n})(y)\mid\langle\vec{Y}\vdash y\rangle\in\mathrm{NF}(F),\,\iota_{i}:Y_{i}\hookrightarrow X_{i}\;\text{for}\;i\in\{1\ldots n\}\}$.

Definition 2.14**.**

Let $F:\mathsf{CohI}^{n}\to\mathsf{CohI}$ be a normal functor. We endow $\mathrm{NF}(F)$ with a non-reflexive coherence relation: $\langle\vec{X}\vdash x\rangle\coh_{\mathrm{NF}(F)}\langle\vec{Y}\vdash y\rangle$ when for all $n$-tuples $\vec{Z}$ and all embeddings $\iota_{\vec{X},i}:X_{i}\hookrightarrow Z_{i}$ and $\iota_{\vec{Y},i}:Y_{i}\hookrightarrow Z_{i}$, $F(\iota_{\vec{X},1},\ldots,\iota_{\vec{X},n})(x)\coh_{F(\vec{Z})}F(\iota_{\vec{Y},1},\ldots,\iota_{\vec{Y},n})(y)$.

The trace $\mathrm{Tr}(F)$ is defined as the coherence space made of the self-coherent normal forms of $F$, equipped with the coherence relation above.

Proposition 2.15**.**

There is a bijection between the cliques $c\sqsubset\mathrm{Tr}(F)$ and the uniform families of cliques $c_{\vec{X}}\sqsubset F(\vec{X})$ for a normal functor $F:\mathsf{CohI}^{n}\to\mathsf{CohI}$, given by

[TABLE]

This leads to the interpretation of quantifiers. One interprets inductively any formula of second-order linear logic $A$ with $n$ free variables into a normal functor $\mathopen{\llbracket}A\mathclose{\rrbracket}:\mathsf{CohI}^{n}\to\mathsf{CohI}$: the connectives $\oplus,\otimes,(-)^{\perp}$ extend to “pointwise” operations on normal functors, and the case $A=\forall X.\,B$ is handled by the operation introduced below.

Proposition 2.16**.**

Let $F$ be a normal functor $F:\mathsf{CohI}^{n+1}\rightarrow\mathsf{CohI}$. The map on objects $\forall(F)(X_{1},\ldots,X_{n})=\mathrm{Tr}(F(X_{1},\dots,X_{n},-))$ extends to a normal functor $\forall(F):\mathsf{CohI}^{n}\to\mathsf{CohI}$.

2.3 Ensuring finiteness: normal functors of finite degree

We now come to our technical contributions, having just finished the exposition of Girard’s model. This section introduces a notion of degree of a normal functor, which will witness the finiteness of the interpretation of $\textsc{mall}_{2}$.

Definition 2.17**.**

Let $F:\mathsf{CohI}^{n}\rightarrow\mathsf{CohI}$ be a normal functor. We define the degree of $F$ as:

[TABLE]

We say $F$ is finite if it preserves finiteness of cardinality and is of finite degree.

Note that a normal functor may have finite but unbounded normal forms, so that its degree is in fact infinite. Typically, this is the case for the exponential modalities, which explains why the model is not finite for full second-order linear logic. We now give two characterizations of finite normal functors.

Proposition 2.18**.**

A normal functor $F$ is finite if and only if $\mathrm{NF}(F)$ is finite.

Theorem 2.19** (Finiteness = polynomial growth).**

Let $F:\mathsf{CohI}\rightarrow\mathsf{CohI}$ be a normal functor. There exists $d\in\mathbb{N}$ s.t. $\mathrm{card}(\mathopen{|}F(X)\mathclose{|})=\mathcal{O}(\mathrm{card}(\mathopen{|}X\mathclose{|}^{d}))$ if and only if $F$ is a finite normal functor. In that case, $\deg{F}$ is the smallest such $d$.

For applications such as the one in Section 5, the relevant notion of “finite semantics” is a model with finite sets of morphisms. Finite normal functors achieve this requirement.

Proposition 2.20**.**

A finite normal functor has finitely many uniform families of cliques.

Proof 2.21**.**

By Proposition 2.18 together with Proposition 2.15, since $\mathopen{|}\mathrm{Tr}(F)\mathclose{|}\subseteq\mathrm{NF}(F)$.

To obtain a finite semantics, our goal is therefore to show that inside the model of coherence spaces and normal functors, the finite ones constitute a submodel of $\textsc{mall}_{2}$.

Proposition 2.22**.**

If $F$ and $G$ normal functors in $\mathsf{CohI}^{n}\rightarrow\mathsf{CohI}$, then $\deg{F^{\bot}}=\deg{F}$, $\deg{F\otimes G}=\deg{F}+\deg{G}$, and $\deg{F\oplus G}=\max\{\deg{F},\deg{G}\}$.

Proposition 2.23**.**

For any normal functor $F:\mathsf{CohI}^{n+1}\rightarrow\mathsf{CohI}$, $\deg{\forall(F)}\leqslant\deg{F}$.

Theorem 2.24**.**

Finite normal functors are closed under $\textsc{mall}_{2}$ connectives.

Proof 2.25**.**

We still need to show that if $F,G:\mathsf{CohI}^{n}\to\mathsf{CohI}$ are finite normal functors, then $F\oplus G,F\otimes G$ and $\forall(F)$ preserve finiteness of cardinality. This is immediate for the first two, and the latter reduces to the case $n=1$: we must show that $\mathrm{Tr}(F)$ is finite. This follows from Proposition 2.18 since $\mathopen{|}\mathrm{Tr}(F)\mathclose{|}\subseteq\mathrm{NF}(F)$.

The above results, together with Proposition 2.20, entails the Theorem 1.2 claimed in the introduction. We can be a bit more precise:

Corollary 2.26**.**

Let $A$ be a formula of second-order linear logic. Suppose that in all subformulas of $A$ of the form $\oc B$ or $\wn B$, any type variable in $B$ is bound by a quantifier in $B$. Then $\mathopen{\llbracket}A\mathclose{\rrbracket}$ is a finite normal functor. In particular:

• •

this applies when $A$ is a $\textsc{mall}_{2}$ formula;

• •

when $A$ is closed, $\mathopen{\llbracket}A\mathclose{\rrbracket}$ is a finite coherence space.

2.4 Effectivity properties via a combinatorial description

We are now ready to revisit the example outlined at the start of Section 2.2, and discuss in more generality the “combinatorial” or “syntactic” presentation of the $\textsc{mall}_{2}$-definable coherence spaces. Our exposition here is inspired by the description of normal functors over the category of sets and injections in [2, §IV.5].

The idea is to see the webs $\mathopen{|}X_{i}\mathclose{|}$ in a normal form $\langle X_{1},\ldots,X_{n}\vdash x\rangle$ as sets of bound variables in $x$. Recall that these normal forms are considered up to isomorphism in a category of elements $\mathsf{El}(\mathopen{|}F\mathclose{|})$; these isomorphisms should be understood as $\alpha$-renamings. The initiality condition on normal forms means that all the variables in the $\mathopen{|}X_{i}\mathclose{|}$ appear free in $x$ – otherwise, one could take a smaller $X^{\prime}_{i}$. Note that the coherence spaces $X_{i}$ specify not only which variables are bound, but also the coherence relation between them.

In turn, this $x$ is a syntax tree with binders – indeed the interpretation of quantifiers uses (unary) normal forms. The $\textsc{mall}_{2}$ connectives induce a grammar of terms

[TABLE]

where $\mathopen{|}X\mathclose{|}\subset\mathrm{Var}$. The functorial action of a $\textsc{mall}_{2}$-definable functor $F$ on embeddings then corresponds to substitution – indeed an embedding $\iota_{i}:X_{i}\hookrightarrow Y_{i}$ is an assignment of variables.

The shape of the term is in fact heavily constrained by the $\textsc{mall}_{2}$ formula which $F$ interprets. With this point of view, one sees that $\deg(F)$ is the maximum number of leaves which a syntax tree in $\mathrm{NF}(F)$ can have.

With such a concrete description it becomes easier to see how one can compute operations on these variable types and cliques. For instance:

Proposition 2.27**.**

For any $\textsc{mall}_{2}$-definable functor $F$, the non-reflexive coherence relation on $\mathrm{NF}(F)$ (Definition 2.14) is decidable.

This may be used to establish the effectivity of our finite semantics of $\textsc{mall}_{2}$:

Theorem 2.28**.**

The function sending a $\textsc{mall}_{2}$ formula $A$ to $\mathrm{Tr}(\mathopen{\llbracket}A\mathclose{\rrbracket})$ is computable. Futhermore, the function taking a formula $A$ and a proof $\pi:A$ as input and returning the clique of $\mathrm{Tr}(\mathopen{\llbracket}A\mathclose{\rrbracket})$ corresponding to the uniform family $\mathopen{\llbracket}\pi\mathclose{\rrbracket}(\vec{X})\sqsubset\mathopen{\llbracket}A\mathclose{\rrbracket}(\vec{X})$ is computable.

Theorem 2.29**.**

Let $F:\mathsf{CohI}^{n}\to\mathsf{CohI}$ be a fixed $\textsc{mall}_{2}$-definable finite normal functor.

• •

The function $\vec{X}\mapsto F(\vec{X})$ is computable in logarithmic space.

• •

For any $c\sqsubset\mathrm{Tr}(F)$, the function $\vec{X}\mapsto c_{\vec{X}}\sqsubset F(\vec{X})$ is computable in logarithmic space.

Since the output of a logarithmic space algorithm has polynomial size, this is consistent with Theorem 2.19.

3 A finite semantics of the second-order affine $\lambda$-calculus

Let us discuss briefly how to adapt our finite coherence semantics from $\textsc{mall}_{2}$ to its intuitionistic affine variant. Indeed, the existence of a finite semantics for this variant is used in one of the applications to implicit complexity mentioned in the introduction [22].

The starting point is to give a semantics of the propositional affine $\lambda$-calculus. To do so, the natural idea is to use the well-known notion of affine function space between coherence spaces: $X\rightarrowtriangle Y=(X\multimap Y)\with Y$, so that every clique $c\sqsubset X\rightarrowtriangle Y$ can be written uniquely as the disjoint union $c=c_{1}\sqcup c_{2}$ of a linear function from $X$ to $Y$ ($c_{1}\sqsubset X\multimap Y$) and a constant part ($c_{2}\sqsubset Y$). Composition is defined as666Fore $r\subseteq S\times T$ and $s\subseteq S$, $r(s)$ denotes $\{y\mid\exists x\in s:(x,y)\in r\}$; this generalizes function composition and can be seen as a degenerate case of relational composition. $c^{\prime}\circ_{\mathrm{aff}}c=(c^{\prime}_{1}\circ c_{1})\sqcup(c^{\prime}_{2}\cup c^{\prime}_{1}(c_{2}))$ for $c\sqsubset X\rightarrowtriangle Y$ and $c^{\prime}\sqsubset Y\rightarrowtriangle Z$. Coherence spaces and affine functions form a category $\mathsf{CohA}$ which can also be seen as the Kleisli category for the comonad $(-\with 1)$, via the isomorphism $(X\multimap Y)\with Y\cong(X\with 1)\multimap Y$.

One issue is that this category does not quite fit into the definition of a “symmetric monoidal closed category with terminal unit”, which is the usual notion of denotational model for the affine $\lambda$-calculus. The reason is that it does not admit a monoidal product $*$ such that, for any coherence space $B$, $(-*B)$ is left adjoint to $(B\rightarrowtriangle-)$. Let us review the two main candidates:

• •

$*=\otimes$ fails: morally, an affine function from $A\otimes B$ can either use both $A$ and $B$ or none of them, but not one out of the two;

• •

$A*B=(A\with 1)\otimes(B\with 1)$ almost works, except for the fact that it is not associative777A similar problem afflicts the treatment of intuitionistic disjunction in domain-theoretic models..

To sidestep this issue, we do not attempt to interpret an affine tensor product – it does not exist anyway in the syntax of the affine $\lambda$-calculus (in practice, one uses the second-order encoding888Whose interpretation in our coherence space semantics coincides with $(A\with 1)\otimes(B\with 1)$. $A\odot B=\forall X.\,(A\rightarrowtriangle B\rightarrowtriangle X)\rightarrowtriangle X$). Instead, we give a semantics in a “closed multicategory”. In the absence of a well-established theory of multicategorical models of typed $\lambda$-calculi and their extension with second-order quantification, we merely give a concrete interpretation of second-order affine $\lambda$-terms.

An affine type in the grammar $A,B::=X\mid A\rightarrowtriangle B\mid\forall X.\,A$ with $n$ type variables is interpreted as a normal functor $\mathsf{CohI}^{n}\to\mathsf{CohI}$ by induction:

[TABLE]

An affine $\lambda$-term $t:B$ with $m$ free variables $x_{1}:A_{1},\ldots,x_{m}:A_{m}$ is mapped to a uniform family $\mathopen{\llbracket}t\mathclose{\rrbracket}(\vec{X})\sqsubset(\mathopen{\llbracket}A_{1}\mathclose{\rrbracket}(\vec{X})\with 1)\otimes\ldots\otimes(\mathopen{\llbracket}A_{m}\mathclose{\rrbracket}(\vec{X})\with 1)\multimap\mathopen{\llbracket}B\mathclose{\rrbracket}(\vec{X})$. The details are unsurpring and are given in Appendix B.

Remark 3.1**.**

One could try instead to use the Eilenberg–Moore category of coalgebras for the comonad $(-\with 1)$ instead of its Kleisli category. But in that case, even though the monoidal product can be defined, the function space cannot.

4 Hypercoherences

The remainder of the paper is dedicated to applying the ideas we developed in coherence spaces to Ehrhard’s hypercoherence [9] model of linear logic. To our knowledge, the treatment of quantifiers in hypercoherences has not appeared in the literature, but it is easily constructed by analogy with coherence spaces, as we do in Section 4.1. As mentioned in the introduction, obtaining an effective model will be harder than in the case of coherence spaces, and this is where most of our energy will be spent (Section 4.2). Some proofs are in Appendix C.

Notation 4.1**.**

Given a set $S$, we write $\mathcal{P}_{\mathrm{fin}}(S)$ (resp. $\mathcal{P}_{\mathrm{fin}}^{*}(S)$) for the set of finite (resp. finite non-empty) subsets of $S$. An alternative notation for $S^{\prime}\in\mathcal{P}_{\mathrm{fin}}(S)$ (resp. $S^{\prime}\in\mathcal{P}_{\mathrm{fin}}^{*}(S)$) is $S^{\prime}\subseteq_{\mathrm{fin}}S$ (resp. $S^{\prime}\subseteq_{\mathrm{fin}}^{*}S$).

Recall that a hypercoherence $X$ is a pair $(|X|,\Gamma(X))$ where $|X|$ is a set and $\Gamma(X)\subseteq\mathcal{P}_{\mathrm{fin}}^{*}(S)$ contains all singletons ($\forall x\in|X|,\,\{x\}\in\Gamma(X)$). Equivalently, one could specify a hypercoherence $X$ by the data of $|X|$ and $\Gamma^{**}(X)=\Gamma(X)\setminus\{\{x\}\mid x\in|X|\}$. A subset $c$ of $|X|$ is a clique of $X$ (as in coherence spaces, this is denoted $c\sqsubset X$) if $\mathcal{P}_{\mathrm{fin}}^{*}(c)\subseteq\Gamma(X)$.

Hypercoherences support the following operations:

• •

$X^{\perp}=(|X|,\mathcal{P}_{\mathrm{fin}}^{*}(X)\setminus\Gamma^{**}(X))$

• •

$X\otimes Y=(|X|\times|Y|,\{S\mid\pi_{1}(S)\in\Gamma(X)\land\pi_{2}(S)\in\Gamma(Y)\})$

• •

$X\oplus Y=(|X|+|Y|,\mathsf{inl}(\Gamma(X))\cup\mathsf{inr}(\Gamma(Y))$

As in coherence spaces, one builds a category $\mathsf{HCohL}$ whose objects are hypercoherences and whose morphisms between $X$ and $Y$ are the cliques of $X\multimap Y=(X\otimes Y^{\perp})^{\perp}$. These morphisms are composed by relational composition.

4.1 Variable and second-order hypercoherences

We follow the recipe of coherence spaces to extend this model to $\textsc{mall}_{2}$.

An embedding $X\hookrightarrow Y$ of hypercoherences is an injection $|X|\hookrightarrow|Y|$ which preserves both coherence and incoherence of subsets (equivalently, the graph of the injection is both a morphism $X\multimap Y$ and a morphism $X^{\perp}\multimap Y^{\perp}$). (Similarly, if $S\subseteq|X|$, the sub-hypercoherence of $X$ induced by $S$ is $(S,\mathcal{P}_{\mathrm{fin}}^{*}(S)\cap\Gamma(X))$.)

Definition 4.2**.**

The category $\mathsf{HCohI}$ has hypercoherences as objects, and embeddings as morphisms. A $n$-parameter variable hypercoherence is a normal functor $\mathsf{HCohI}^{n}\to\mathsf{HCohI}$.

A variable hypercoherence $F$ is (weakly) finite if $\mathrm{NF}(F)$ is finite – as in coherence spaces, $\mathrm{NF}(F)$ is defined from the presheaf $|F|$. Its degree $\deg(F)$ is defined as in Definition 2.17.

Definition 4.3**.**

Let $F:\mathsf{HCohI}^{n}\to\mathsf{HCohI}$ be a variable hypercoherence. A family $c_{X_{1},\ldots,X_{n}}\sqsubset F(X_{1},\ldots,X_{n})$ is uniform if for any embeddings $\iota_{i}:X_{i}\hookrightarrow Y_{i}$ ($i\in\{1,\ldots,n\}$), $c_{X_{1},\ldots,X_{n}}=F(\iota_{1},\ldots,\iota_{n})^{-1}(c_{Y_{1},\ldots,Y_{n}})$.

Proposition 4.4**.**

For all $n\in\mathbb{N}$, the $n$-parameter variable hypercoherences and uniform families of cliques form a category $\mathsf{HCohL}(n)$ which is a model of propositional mall.

The $n=0$ case is $\mathsf{HCohL}(0)=\mathsf{HCohL}$. As we saw in Remark 2.6, it is not quite trivial to show that $\mathsf{HCohL}(n)$ is a category, but since the morphisms in $\mathsf{HCohL}$ enjoy the domain-theoretic stability property, the proof used in the case of coherence spaces applies mutatis mutandis.

Now we wish to define a notion of trace $\mathrm{Tr}(F)$ of a normal functor $F$ to interpret $\forall$. The fundamental thing to ensure is the adjunction (usual in categorical semantics)

[TABLE]

where $\mathrm{Const}$ is a “weakening” map sending a $n$-parameter hypercoherence to a $(n+1)$-parameter one. This reduces to the case999If one were to specialize this further to $\mathsf{HCohL}(0)(1,\mathrm{Tr}(F))\cong\mathsf{HCohL}(1)(1,F)$ – i.e. the cliques of $\mathrm{Tr}(F)$ are in bijection with the variable cliques of $F$ – then it would not determine $\mathrm{Tr}(F)$ uniquely, unlike the case of coherence spaces: in general, the structure of a hypercoherence cannot be fully recovered from its domain of cliques. That said, the adjunction defining $\mathrm{Tr}(-)$ can be derived from this bijection between cliques together with $\mathrm{Tr}(\mathrm{Const}(A)\multimap F)\cong A\multimap\mathrm{Tr}(F)$ (i.e. commutation between $\forall$ and $\parr$). $n=0$: $\mathsf{HCohL}(A,\mathrm{Tr}(F))\cong\mathsf{HCohL}(1)(\mathrm{Const}(A),F)$. So $\mathrm{Tr}$, being an adjoint functor, is unique up to natural isomorphism101010An isomorphism $X\cong Y$ in $\mathsf{HCohL}$ is just a bijection from $|X|$ to $|Y|$ sending $\Gamma(X)$ to $\Gamma(Y)$. (if it exists); we can just state the definition – mimicking coherence spaces – and check that the adjunction holds.

Definition 4.5**.**

For $x\in|F(X)|$, $\mathrm{NF}(x;F)$ denotes (the isomorphism class of) the normal form of $x$. For $S\subseteq|F(X)|$, $\mathrm{NF}(S;F)$ is the direct image of $S$ by the function $\mathrm{NF}(-;F)$.

Definition 4.6**.**

Let $F$ be a one-parameter variable hypercoherence. $N\subseteq_{\mathrm{fin}}^{*}\mathrm{NF}(F)$ is said to be coherent when for any $X$ and any $S\subseteq_{\mathrm{fin}}^{*}|F(X)|$, if $\mathrm{NF}(S;F)=N$ then $S\in\Gamma(F(X))$. The set of coherent sets of normal forms is denoted by $\Gamma\mathrm{NF}(F)$.

The trace of $F$ (notation: $\mathrm{Tr}(F)$) is the hypercoherence defined by

[TABLE]

and from this we define $\forall(G):\vec{X}\mapsto\mathrm{Tr}(G(\vec{X},-))$ for $G:\mathsf{HCohI}^{n+1}\to\mathsf{HCohI}$.

One can then routinely check that the adjunction holds. We therefore conclude:

Theorem 4.7**.**

Variable hypercoherences and uniform families of cliques form a semantics of second-order linear logic. Furthermore, by restricting to weakly finite variable hypercoherences, we obtain a finite semantics of $\textsc{mall}_{2}$.

4.2 Strong finiteness and effectivity

Unfortunately, the model of weakly finite variable hypercoherences is not effective. Let us give an exemple: let $f:\mathbb{N}\to\{0,1\}$ be any function, and

[TABLE]

This map on objects can be extended to a functor $F_{f}:\mathsf{HCohI}\to\mathsf{HCohI}$ which is in fact a weakly finite variable hypercoherence. But if $f$ is uncomputable, then $X\mapsto\Gamma(F_{f}(X))$ also is.

We are therefore seeking an additional condition on variable hypercoherences which would both guarantee effectivity and be preserved by $\textsc{mall}_{2}$ connectives. More precisely, our goal is to exhibit a class of variable hypercoherences $F$ such that $\Gamma(F)$ can be described canonically by some finite data – just as $\mathrm{NF}(F)$ fulfills this role for $|F|$ when $F$ is weakly finite. This is the purpose of the following defintions.

Definition 4.8**.**

Let $F$ be a $n$-parameter variable hypercoherence and $1\leq k\leq n$.

For any $P\subseteq\mathrm{NF}(F)$, a projection from $P$ on the $k$-th parameter is a dependent function

[TABLE]

i.e. it is a function $f$ defined on $P$ such that $f(\langle X_{1},\ldots,X_{n}\vdash x\rangle)\in|X_{k}|$. The set of projections from $P$ on the $k$-th parameter is written $\mathrm{Proj}_{k}(P)$.

Any $f\in\mathrm{Proj}_{k}(P)$ induces a family of functions indexed by hypercoherences $Y_{1},\ldots,Y_{n}$

[TABLE]

as follows: let $\mathrm{NF}(y;F)=\langle X_{1},\ldots,X_{n}\vdash x\rangle\in P$, then $F(\iota_{1},\ldots,\iota_{n})(x)=y$ for some (unique) embeddings $\iota_{i}:X_{i}\hookrightarrow Y_{i}$; one then takes $f^{F}_{Y_{1},\ldots,Y_{n}}(y)$ to be $\iota_{k}(f(x))$.

By direct image, this induces a family of functions

[TABLE]

Note that this could be extended to “$\mathrm{NF}(S;F)\subseteq P$” but most uses of this direct image will happen with $\mathrm{NF}(S;F)=P$.

We also write $\mathrm{Proj}(P)=\mathrm{Proj}_{1}(P)\cup\ldots\cup\mathrm{Proj}_{n}(P)$.

Notation 4.9**.**

Given a hypercoherence $X$ and $S\subseteq_{\mathrm{fin}}^{*}|X|$, we define $l_{X}(S)$ to be $\odot$ if $S$ is a singleton, $\ominus$ if $S\in\Gamma^{**}(X)$, $\oplus$ if $S\in\Gamma^{**}(X^{\perp})$.

(Mnemonic: $l_{A\oplus B}(S\cup T)=\oplus$ for all non-empty $S\subseteq|A|$ and $T\subseteq|B|$.)

Our eventual goal is to specify variable hypercoherences using projections as follows: given $S\subseteq_{\mathrm{fin}}^{*}|F(\vec{Y})|$, $l_{F(\vec{Y})}(S)$ should be determined by the $l_{Y_{k}}(f^{F}_{\vec{Y}}(S))$ for $f\in\mathrm{Proj}_{k}(\mathrm{NF}(S;F))$, $k\in\{1,\ldots,n\}$, following a sort of “truth table”. However, to make this specification canonical, one should ensure that all “rows” in the table are meaningful, i.e. serve to determine the (in)coherence of at least one $S\subseteq_{\mathrm{fin}}^{*}|F(\vec{Y})|$ for some $\vec{Y}$. This is the purpose of the following.

Definition 4.10**.**

Let $F$ be a $n$-parameter variable hypercoherence, and $P\subseteq\mathrm{NF}(F)$. We write $\mathrm{Val}(P)$ for the set of valuations on $P$, that is, of functions $\mathrm{Proj}(P)\to\{\odot,\ominus,\oplus\}$.

A valuation $v$ on $P$ is possible when there exist hypercoherences $Y_{1},\ldots,Y_{n}$ and a subset $S\subseteq|F(Y_{1},\ldots,Y_{n})|$ such that $\mathrm{NF}(S;F)=P$ and $v(f)=l_{Y_{k}}(f^{F}_{Y_{1},\ldots,Y_{n}}(S))$ for $f\in\mathrm{Proj}_{k}(P,F)$. The set of possible valuations on $P$ is denoted $\mathrm{PVal}(P)$.

Definition 4.11**.**

A specification by projections of $F$ is a dependent function

[TABLE]

such that, for all $Y_{1},\ldots,Y_{n}\in\mathsf{HCohL}$ and $S\subseteq_{\mathrm{fin}}^{*}|F(Y_{1},\ldots,Y_{n})|$,

[TABLE]

A variable hypercoherence is strongly finite if it is weakly finite and admits a specification by projections. (Note that if a specification by projections exists for $F$, it is unique.)

To justify the terminology, observe that for a weakly finite $F$, there are finitely many $P\subseteq_{\mathrm{fin}}^{*}\mathrm{NF}(F)$ and the sets $\mathrm{Proj}_{k}(P,F)$ are finite. Therefore, if $F$ admits a specification by projections, then this specification is a finite object, and so $\Gamma(F)$ is finitely described. This notion successfully excludes pathological examples such as our $F_{f}$ above:

Proposition 4.12**.**

If $F:\mathsf{HCohI}^{n}\to\mathsf{HCohI}$ is strongly finite, then $\vec{X}\mapsto F(\vec{X})$ is computable.

Proof 4.13**.**

All projections are computable, so it suffices to precompute a table encoding the specification, and to look up the relevant row.

Remark 4.14**.**

Our definition of specification by projections is very restrictive. For instance, if $\langle\varnothing,\ldots,\varnothing\vdash x\rangle\in P\subseteq_{\mathrm{fin}}^{*}\mathrm{NF}(F)$ and $\mathrm{Card}(P)\geq 2$, then the $S\subseteq_{\mathrm{fin}}^{*}|F(\vec{Y})|$ such that $\mathrm{NF}(S;F)=P$ are either all coherent or all incoherent, independently of $\vec{Y}$.

We still need to show that strongly finite variable hypercoherences are closed under all $\textsc{mall}_{2}$ connectives, and that the interpretations of $\textsc{mall}_{2}$ formulas and proofs can be effectively computed. The main lemma is:

Lemma 4.15**.**

There exists a criterion to determine, given $P\subseteq_{\mathrm{fin}}^{*}\mathrm{NF}(F)$ and $v\in\mathrm{Val}(P)$, whether the valuation $v$ is possible; this criterion is effective when $F$ is weakly finite. Note that $F$ is not part of the input; that means that $\mathrm{PVal}(P)$ depends only on $P$, not on any other information on $F$.

Proposition 4.16**.**

The normal functors $(X_{1},\ldots,X_{k})\mapsto A$ and $(X_{1},\ldots,X_{n})\mapsto X_{k}$ admit specifications by projections (and are therefore strongly finite).

Proposition 4.17**.**

If the $n$-parameter variable hypercoherences $F$ and $G$ can be specified by projections, then it is also the case for $F^{\perp}$, $F\otimes G$ and $F\oplus G$. Furthermore, if $F$ and $G$ are finite, then $\sigma_{F^{\perp}}$, $\sigma_{F\otimes G}$ and $\sigma_{F\oplus G}$ are computable from $\sigma_{F}$ and $\sigma_{G}$.

In the above proposition, finiteness may refer to either weak or strong finiteness: since we assume specifiability by projections, those two notions become equivalent by definition.

Proposition 4.18**.**

If $F\in\mathsf{HCohL}(n+1)$ admits a specification by projections, then so can $\forall(F)\in\mathsf{HCohL}(n)$. Furthermore, the function $(\mathrm{NF}(F),\sigma_{F})\mapsto(\mathrm{NF}(\forall(F)),\sigma_{\forall(F)})$ defined on finite $F$ is computable.

From these propositions, we see that $A\mapsto\mathopen{\llbracket}A\mathclose{\rrbracket}$ is computable in the second-order hypercoherence model. Since $(\pi:A)\mapsto\mathopen{\llbracket}\pi\mathclose{\rrbracket}\sqsubset\mathopen{\llbracket}A\mathclose{\rrbracket}$ is computable for essentially the same reasons as in coherence spaces, we may conclude:

Theorem 4.19**.**

Strongly finite variable hypercoherences form an effective model of $\textsc{mall}_{2}$.

5 Application: characterizing regular languages

Next, we illustrate the usefulness of our finite semantics of $\textsc{mall}_{2}$ on the following theorem.

Definition 5.1**.**

We consider the “stratified” Church encoding of strings: $\mathtt{Str}=\forall X.\mathtt{Str}[X]$, where $\mathtt{Str}[X]=\oc(X\multimap X)\multimap\oc(X\multimap X)\multimap\oc(X\multimap X)$.

Given a proof $\pi:\oc\mathtt{Str}\multimap\oc^{k}\mathtt{Bool}$ (with111111We also take $\mathtt{true}$ (resp. $\mathtt{false}$) to be the proof of $1\oplus 1$ proving the left (resp. right) occurrence of $1$. $\mathtt{Bool}=1\oplus 1$), the language decided by $\pi$ is defined as121212$\pi(\overline{w})$ denotes the proof obtained by applying a cut to $\pi$ and $\overline{w}$, and $\oc^{k}\mathtt{true}$ is the proof consisting of $k$ promotion rules with empty context applied to $\mathtt{true}$; cf. the sequent calculus recalled in Appendix D. $\mathcal{L}(\pi)=\{w\in\{0,1\}^{*}\mid\pi(\overline{w})\longrightarrow^{*}\oc^{k}\mathtt{true}\}$, where $\overline{w}$ is the Church encoding of $w$.

Theorem 5.2**.**

The type $\oc\mathtt{Str}\multimap\oc\oc\mathtt{Bool}$ in second-order Elementary Linear Logic ($\textsc{ell}_{2}$) captures the class of regular languages. In other words, the languages that can be expressed as $\mathcal{L}(\pi)$ for some proof $\pi:\oc\mathtt{Str}\multimap\oc\oc\mathtt{Bool}$ in $\textsc{ell}_{2}$ are exactly the regular languages.

Roughly speaking, $\textsc{ell}_{2}$ is a subsystem of full second-order linear logic where the rules governing the exponential modalities are restricted: promotion and dereliction are removed, and replaced with functorial promotion: from $\vdash A_{1},\ldots,A_{n},B$, infer $\vdash\wn A_{1},\ldots,\wn A_{n},\oc B$. This induces a sort of “stratification” on formulas and proofs, which is the reason why the number of $\oc$ modalities in the output type of an $\textsc{ell}_{2}$ function is significant.

The formal definition of $\textsc{ell}_{2}$ is given in Appendix D. Some parts of the proof of Theorem 5.2 are also relegated to this section of the appendix. To summarize:

• •

By encoding deterministic finite automata as proofs of $\oc\mathtt{Str}\multimap\oc\oc\mathtt{Bool}$, we show that every regular language can be decided by such a proof.

• •

Using the aforementioned stratification property of $\textsc{ell}_{2}$, we reduce the converse (only regular languages can be decided) to the lemma below. This reduction involves a “truncation at depth $k$” operation, similar to the one defined in [22] for the elementary affine $\lambda$-calculus, which might be of independent interest.

The lemma whose proof features coherence spaces actually applies to full second-order linear logic (with unrestricted exponentials). This is because the “geometric” properties specific to $\textsc{ell}_{2}$ have already been exploited in the previous step.

Lemma 5.3**.**

Let $\pi:\mathtt{Str}[A_{1}]\otimes\ldots\otimes\mathtt{Str}[A_{n}]\multimap\mathtt{Bool}$ be a proof in second-order linear logic where $A_{1},\ldots,A_{n}$ are closed $\textsc{mall}_{2}$ formulas. Then the following language is regular:

[TABLE]

where $\overline{w}[A_{i}]:\mathtt{Str}[A_{i}]$ is the instantiation of $\overline{w}$ on $A_{i}$.

Proof 5.4**.**

Let $B=\mathtt{Str}[A_{1}]\otimes\ldots\otimes\mathtt{Str}[A_{n}]$. By Corollary 2.26, we know that $\mathopen{\llbracket}B\mathclose{\rrbracket}$ is a finite coherence space. Indeed, if a subformula of $B$ is the form $\oc C$, then it cannot be a subformula of some $A_{i}$ since the $A_{i}$ are $\textsc{mall}_{2}$ formulas, so $C=A_{i}\multimap A_{i}$ for some $i\in\{1,\ldots,n\}$. Since $A_{i}$ is closed, the premise of Corollary 2.26 holds.

Let $x\in\{0,1\}$. The operation “add a $x$ at the end of the string” is definable by a proof $\mathtt{snoc}^{x}_{X}$ of $\mathtt{Str}[X]\multimap\mathtt{Str}[X]$. From this, we can derive $\mathtt{snoc}^{x}_{A_{1},\ldots,A_{n}}:B\multimap B$. This allows us to define a deterministic finite automaton (writing $\varepsilon$ for the empty string):

• •

whose states are the cliques of $\mathopen{\llbracket}B\mathclose{\rrbracket}$, with initial state $q_{I}=\mathopen{\llbracket}\overline{\varepsilon}[A_{1}]\otimes\ldots\otimes\overline{\varepsilon}[A_{n}]\mathclose{\rrbracket}$;

• •

whose transition function is $\delta(x,q)=\mathopen{\llbracket}\mathtt{snoc}^{x}_{A_{1},\ldots,A_{n}}\mathclose{\rrbracket}(q)$ for $x\in\{0,1\}$;

• •

whose accepting states are $\{q\sqsubset B\mid\mathopen{\llbracket}\hat{\pi}\mathclose{\rrbracket}(q)=\mathopen{\llbracket}\mathtt{true}\mathclose{\rrbracket}\}$.

Thanks to the compositionality of the coherence space model, when the DFA reads a word $w\in\{0,1\}$, it ends in the state $\mathopen{\llbracket}\overline{w}\mathclose{\rrbracket}$. This state is accepting if and only if $\mathopen{\llbracket}\hat{\pi}(\overline{w})\mathclose{\rrbracket}=\mathopen{\llbracket}\mathtt{true}\mathclose{\rrbracket}$; since the semantics is injective on $\mathtt{Bool}$, the DFA recognizes the language we want.

6 Conclusion

Motivated by applications to implicit complexity, we sought a finite semantics for $\textsc{mall}_{2}$, and obtained it by proving the finiteness of the pre-existing model of coherence spaces and normal functors. In retrospect, this is not so surprising: one advantage of coherence spaces (e.g. over Scott domains), that had already been stressed early in their history, is their tendency to give small and legible interpretations to formulas.

Another early observation by Girard was that the existential introduction in this model has a non-trivial computational contents, subsuming the cut rule – this was mentioned as being “key to a semantic approach to computation” [13, p. 57]. By going from $A[T]$ to $\exists X.\,A$, the information of the witness $T$ is compressed into some bounded data, and this is why the semantics can be finite. Let us reformulate this from the programming language point of view on existential types as abstract data types: the cliques of $\mathopen{\llbracket}\exists X.\,A\mathclose{\rrbracket}$ keep just enough information about the cliques of $\mathopen{\llbracket}A[T]\mathclose{\rrbracket}$ they originate from to determine their interaction with the generic (universally typed) programs which might use them.

Relatedly, observe that the syntactic model of propositional mall is finite, and the existential witnesses are the only reason why this is not the case in $\textsc{mall}_{2}$. One could also try to directly implement the above intuitions starting from the syntax; this will be the subject of an upcoming paper with P. Pistone, T. Seiller and L. Tortora de Falco.

Aside from finiteness, the present work also investigated in detail the question of effectivity. Almost no effort is needed in the case of coherence spaces, but to obtain an effective hypercoherence model of $\textsc{mall}_{2}$, we had to introduce the idea of specifications by projections.

Appendix A Omitted proofs of Section 2

Proof A.1** (Proof of Proposition 2.18).**

($\Longrightarrow$) There are finitely many coherence spaces of cardinality $\leq\deg(F)$, and their images by $F$ are all finite since $F$ preserves finiteness.

($\Longleftarrow$) $\deg(F)$ is the supremum of a finite subset of $\mathbb{N}$ and is therefore finite.

Proof A.2** (Proof of Theorem 2.19).**

Note that $\mathrm{Card}(|F(X)|)=O(\mathrm{Card}(|X|)^{d})$ implies that $F$ sends finite spaces to finite spaes. Thus, in the remainder of this proof, we assume that $F$ preserves finiteness of cardinality (otherwise, an equivalence between two false propositions is true). With this assumption, it suffices to prove that

[TABLE]

We decompose this equality into two inequalities.

($\leq$) Let $\langle X^{0}\vdash x\rangle\in\mathrm{NF}(F)$ and $d=\mathrm{Card}(|X^{0}|)$. Let $[n]=1\&\ldots\&1$ ($n$ times). For all $n$, there are $n^{d}$ embeddings $X^{0}\hookrightarrow X^{0}\otimes[n]$ which are the identity on the first component. If for two such embeddings $\iota$ and $\iota^{\prime}$, $F(\iota)(x)=F(\iota^{\prime})(x)$, then by uniqueness of the normal form $\iota$ and $\iota^{\prime}$ are isomorphic in the slice category $\mathsf{CohI}/(X^{0}\otimes[n])$; in the commuting triangle $\iota=\rho\circ\iota^{\prime}$, $\rho$ can only be the identity, since our considered family of embeddings differ only on the component $[n]$, and so $\iota=\iota^{\prime}$. Thus, $\iota\mapsto F(\iota)(x)$ is injective over the $n^{d}$ embeddings we consider, and therefore $\mathrm{Card}(|F(X^{0}\otimes[n])|)\geq n^{d}$ while $\mathrm{Card}(|X^{0}\otimes[n]|)=dn$.

($\geq$) We assume $d=\deg(F)<\infty$ (when $\deg(F)=\infty$, the inequality is true for trivial reasons). This entails that $\mathrm{NF}(F)$ is finite, by Proposition 2.18. Now let $X$ be a finite coherence space of cardinality $n$; each point $x\in|F(X)|$ has a normal form, so $\mathrm{Card}(|F(X)|)$ can be bounded by summing over $\langle X^{0}\vdash x\rangle\in\mathrm{NF}(F)$ the number of possible embeddings of $X^{0}$ in $X$. This number is at most $O(n^{\mathrm{Card}(X^{0})})$ and by definition $\mathrm{Card}(X^{0})\leq d$. In the end, using $\mathrm{Card}(\mathrm{NF}(F))=O(1)$, we get $\mathrm{Card}(|F(X)|)=O(n^{d})$.

Proof A.3** (Proof of Proposition 2.22).**

The case of negation is straightforward as $\mathopen{|}F(X_{1},\dots,X_{n})\mathclose{|}=\mathopen{|}F^{\bot}(X_{1},\dots,X_{n})\mathclose{|}$. For the $\oplus$ case, one easily checks that $\mathrm{NF}(F\oplus G)\cong\mathrm{NF}(F)\uplus\mathrm{NF}(G)$.

Only the $\otimes$ case needs to be carefully checked. Consider $\langle\vec{Z}\vdash(x,y)\rangle\in\mathrm{NF}(F\otimes G)$, and the corresponding normal forms $\langle\vec{X}\vdash x\rangle\in\mathrm{NF}(F)$ and $\langle\vec{Y}\vdash y\rangle\in\mathrm{NF}(G)$. Then one can show, from minimality of $\vec{Z}$, that131313For legibility purposes, we assume that $F$ preserves inclusions, and work with inclusions instead of embeddings; also, all operations are applied componentwise on the $n$-tuples $\vec{X}$, $\vec{Y}$, $\vec{Z}$. $\vec{X}\cup\vec{Y}\subseteq\vec{Z}$. Moreover, since $x\in F(\vec{Z})$ and $y\in G(\vec{Z})$, we have $\vec{X}\cup\vec{Y}\supseteq\vec{Z}$. Thus $\vec{X}\cup\vec{Y}=\vec{Z}$ and $\deg{(F\otimes G)}\leqslant\deg{F}+\deg{G}$. The converse inequality is obtained by noticing that if $\langle\vec{X}\vdash x\rangle\in\mathrm{NF}(F)$ and $\langle\vec{Y}\vdash y\rangle\in\mathrm{NF}(G)$, then $\langle\vec{X}\oplus\vec{Y}\vdash(x,y)\rangle\in\mathrm{NF}(F\otimes G)$.

Proof A.4** (Proof of Proposition 2.23).**

Suppose that $\langle Y_{1},\dots,Y_{n}\vdash\langle X_{1}\vdash x\rangle\rangle\in\mathrm{NF}(\forall(F))$. It suffices to check that $\langle Y_{1},\dots,Y_{n},X_{1}\vdash x\rangle\in\mathrm{NF}(F)$.

Proof A.5** (Proof of Theorem 2.29).**

For each point $x\in\theta_{X}$, there is a unique $(\iota,y)$ such that $F(\iota)(y)=x$; indeed, it is the initial object of the slice category $\mathsf{El}(\mathopen{|}F\mathclose{|})/(X,x)$. So we have a bijection

[TABLE]

This bijection is computable in logarithmic space: it is just a matter of performing substitutions on terms of size $O(1)$ (since $F$ is fixed), although the $\iota$’s are not of constant size (because of the representation of elements of $\mathopen{|}X\mathclose{|}$). Therefore the problem is reduced to enumerating the right-hand side without repetitions.

There are finitely many $\langle Y\vdash y\rangle$ in $\theta$. For each of them, each injection $Y\to X$ can be represented by $\mathrm{card}(\mathopen{|}Y\mathclose{|})\leq\deg(F)=O(1)$ elements of $\mathopen{|}X\mathclose{|}$: this takes $O(\log\mathrm{card}(\mathopen{|}X\mathclose{|}))$ space. Thus, all the injections can be enumerated in logarithmic space, and for each injection, whether it is an embedding can be determined in logarithmic space (using the coherence relation on $\mathrm{Tr}(F)$ which may be precomputed independently of the input.)

Appendix B Details for Section 3

Appendix C Omitted proofs of Section 4

Appendix D Details and proofs for Section 5

D.1 Proof of extensional completeness

Proposition D.1**.**

Any regular language can be expressed as $\mathcal{L}(\pi)$ for some $\textsc{ell}_{2}$ proof $\pi:\oc\mathtt{Str}\multimap\oc\oc\mathtt{Bool}$.

Proof D.2** (Proof sketch).**

To encode a deterministic finite automaton with set of states $Q$ and transition function, simply instantiate the input string at the type $1\oplus\ldots\oplus 1$ ($\mathrm{Card}(Q)$ times), and give it as arguments the linear functions representing $\delta(0,-)$ and $\delta(1,-)$.

D.2 Proof of soundness by reduction to Lemma 5.3

In this section, we fix $\pi:\oc\mathtt{Str}\multimap\oc\oc\mathtt{Bool}$ in $\textsc{ell}_{2}$, and prove the converse of the above. The first step is to understand the shape of $\pi$.

Lemma D.3**.**

Up to commutations, $\pi$ is of the form

$\hat{\pi}$

$\vdash\mathtt{Str}[A_{1}]^{\bot},\dots,\mathtt{Str}[A_{n}]^{\bot},\oc\mathtt{Bool}$

$\vdash\mathtt{Str}^{\bot},\dots,\mathtt{Str}^{\bot},\oc\mathtt{Bool}$

$\oc$

$\vdash\wn\mathtt{Str}^{\bot},\dots,\wn\mathtt{Str}^{\bot},\oc\oc\mathtt{Bool}$

$\vdash\wn\mathtt{Str}^{\bot},\oc\oc\mathtt{Bool}$

$\parr$

$\vdash\oc\mathtt{Str}\multimap\oc\oc\mathtt{Bool}$

Proof D.4**.**

A proof of $\vdash\wn\mathtt{Str}^{\bot},\dots,\wn\mathtt{Str}^{\bot},\oc\oc\mathtt{Bool}$ necessarily ends either with a structural rule or a promotion. From this and the invertibility of $\parr$, one obtains that the proof, up to commutation, ends with the following sequence of rules:

$\vdash\mathtt{Str}^{\bot},\dots,\mathtt{Str}^{\bot},\oc\mathtt{Bool}$

$\oc$

$\vdash\wn\mathtt{Str}^{\bot},\dots,\wn\mathtt{Str}^{\bot},\oc\oc\mathtt{Bool}$

$\vdash\wn\mathtt{Str}^{\bot},\oc\oc\mathtt{Bool}$

$\parr$

$\vdash\oc\mathtt{Str}\multimap\oc\oc\mathtt{Bool}$

Now, recall that $\mathtt{Str}^{\bot}=\exists X.\mathtt{Str}[X]^{\bot}$. Moreover, notice the introduction rule for $\exists$ commutes with all other rules except promotion and the introduction of $\forall$. We only need to show that the introduction rule for all $\exists$ connectives of the occurrences of $\mathtt{Str}^{\bot}$ are not followed by a promotion or a $\forall$ introduction. Ruling out promotion is easy, as all formulas in the conclusion sequent of a promotion rule have exponential as principal connectives. Moreover, it is possible to rule out $\forall$ introductions as folllws. If a $\forall$ introduction rule appears in the proof, the $\forall$ connective it introduces is part of an existential witness. But all variables existentially quantified appear under the scope of an exponential connective, and therefore can only precede a promotion rule.

Now, a crucial observation is that the $A_{i}$ in the previous lemma can be taken in $\textsc{mall}_{2}$ w.l.o.g. This is where the stratification property of $\textsc{ell}_{2}$ plays a key role.

Lemma D.5**.**

There is a proof $\pi^{\prime}$ whose witnesses $A_{1},\ldots,A_{n}$ are closed $\textsc{mall}_{2}$ formulas, and which decides the same language as $\pi$.

Proof D.6**.**

We define the truncation at depth 2 of a formula as follows: all subformulas of the form $!A$ (resp. $?A$) at depth 2, i.e. in the scope of two other nested $!/?$ modalities, are replaced by $1$ (resp. $\bot$). Note that the truncation at depth 2 of $!\mathtt{Str}$ and $!!\mathtt{Bool}$ are themselves.

This operation extends to proofs: any functorial promotion of conclusion $\vdash\wn B_{1},\ldots,\wn B_{m},\oc C$ is replaced by the only proof of $\vdash\bot,\ldots,\bot,1$, while contractions and weakenings are replaced by cuts with $1\vdash 1\otimes 1$ and $\vdash 1$. Note that this truncation is the identity on cut-free proofs of $!!\mathtt{Bool}$ and $!\mathtt{Str}$.

One may then check that truncation at depth 2 is compatible with cut-elimination, which means that one can replace $\pi$ by its truncation at depth 2 and still recognize the same language. Then the $A_{i}$ are replaced by their “truncation at depth 0” which are $\textsc{mall}_{2}$ formulas.

Proposition D.7**.**

$\mathcal{L}(\pi)$ is regular.

Proof D.8**.**

By Lemma 5.3, the language

[TABLE]

is regular. A examination of the cut-elimination process reveals that this language is none other than $\mathcal{L}(\pi)$.