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Categorical Combinatorics for Non Deterministic Strategies on Simple Games

  • Clément Jacq
  • Paul-André Melliès
Open Access
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10803)

Abstract

The purpose of this paper is to define in a clean and conceptual way a non-deterministic and sheaf-theoretic variant of the category of simple games and deterministic strategies. One thus starts by associating to every simple game a presheaf category of non-deterministic strategies. The bicategory of simple games and non-deterministic strategies is then obtained by a construction inspired by the recent work by Melliès and Zeilberger on type refinement systems. We show that the resulting bicategory is symmetric monoidal closed and cartesian. We also define a 2-comonad which adapts the Curien-Lamarche exponential modality of linear logic to the 2-dimensional and non deterministic framework. We conclude by discussing in what sense the bicategory of simple games defines a model of non deterministic intuitionistic linear logic.

1 Introduction

A new generation of 2-categorical and sheaf-theoretic game semantics is currently emerging in the field of programming language semantics. The games and strategies which determine them are more sophisticated mathematically, and also more difficult to define rigorously, than they were in the deterministic case. For that reason, it is timely to examine more closely the 2-categorical and sheaf-theoretic frameworks available to us in order to formulate these games and strategies in a suitably clean and conceptual way. In this investigation, one benefits from the efforts made in the past twenty-five years to give a clearer mathematical status to the previous generation of game semantics, which was (to a large extent) based on the notion of arena game. We recognize three main lines of work here:
  1. 1.

    the logical approach advocated by Girard, and formulated in ludics [3], polarized linear logic [7] or tensorial logic [12] with its connection to continuations and string diagrams,

     
  2. 2.

    the combinatorial approach advocated by Hyland, inspired by algebraic topology, and based on the combinatorial description of the structure of pointers in arena games [4],

     
  3. 3.

    the concurrent and asynchronous approach advocated by Melliès, based on the description of arena games as asynchronous games, and of strategies as causal concurrent structures playing on them, either in an alternated [9, 10, 11] or in a non-alternated way [18].

     

Interestingly, all the sheaf-theoretic frameworks designed for game semantics today are offsprings of the third approach based on asynchronous games: on the one hand, the notion of concurrent strategy in [19] is a sheaf-theoretic transcription of the notion of receptive ingenuous strategy formulated in [18]; on the other hand, the sheaf-theoretic notion of non-deterministic innocent strategy in [13, 17] relies on the diagrammatic and local definition of innocence in alternated asynchronous games [11]. For that reason, our purpose in this paper is to investigate the connection with the second approach, different in spirit and design, and to define a bicategory of simple games and non-deterministic strategies in the sheaf-theoretic style of Harmer et al. [4]. As we will see, our work also integrates a number of elements coming from the first approach, and more specifically, the discovery by Melliès that strategies are presented by generators and relations, and for that reason, are prone to factorisation theorems [14, 15]. Since we are interested in sheaf-theoretic models of computations, we should not forget to mention the pioneering work by Hirschowitz and Pous on models of process calculi [5], and its recent connection to game semantics [2].

In the present paper, we start from the category  Open image in new window of simple games and deterministic strategies between them, and we explain how to turn  Open image in new window into a bicategory  Open image in new window of simple games and non-deterministic strategies. As we will see, the construction of Open image in new window relies on the discovery of a number of elementary but fundamental fibrational properties of the original category  Open image in new window . Since our work is built on [4], let us recall that a simple game A is defined there as a contravariant presheaf \(A : \omega ^{op} \rightarrow \mathbf {Set}\) over the order category \(\omega = \begin{array}{ccccccc}0\rightarrow & {} 1\rightarrow & {} 2\rightarrow & {} \cdots \end{array}\) associated to the infinite countable ordinal \(\omega \). A simple game A is thus a family of sets \(A_{n}\) together with a function \(\pi _{n}:A_{n+1}\rightarrow A_{n}\) for all \(n\in \mathbb {N}\), depicted as:

One requires moreover that \(A_{0}\) is the singleton set. The intuition is that \(A\) is a rooted tree; that \(A_{n}\) contains its plays (or branches) of length n; and that \(\pi _{n}\) is the prefix function which transports every play of length \(n+1\) to its prefix of length n. In particular, every simple game \(A\) contains only one play of length 0, which should be thought as the empty play. Every simple game \(A\) should be moreover understood as alternating: here, the intuition is that every play of odd length \(2n+1\) ends with an Opponent move, and that every play of even length 2n ends with a Player move if \(n>0\).

Terminology: An element \(a\in A_{n}\) is called a position of degree n in the game \(A\). The position \(a\in A_{n}\) is called a P-position when its degree n is even, and a O-position when its degree n is odd. Given a position \(a\in A_{n+1}\), we write \(\pi (a)\) for the position \(\pi _{n}(a)\); similarly, given a position \(a\in A_{n+2}\), we write \(\pi ^2(a)\) for the position \(\pi _{n}\circ \pi _{n+1}(a)\). A simple game A is called O-branching when the function \(\pi :A_{2n+2}\rightarrow A_{2n+1}\) is injective, for all \(n\in \mathbb {N}\). This means that every Opponent position \(a\in A_{2n+1}\) can be extended in at most one way into a Player position \(b\in A_{2n+2}\), for all \(n\in \mathbb {N}\).

We start the paper by formulating a sheaf-theoretic notion of non-deterministic P-strategy on a simple game A. Recall that a deterministic P-strategy \(\sigma \) of a simple game A is defined in [4] as a family of subsets \(\sigma _{2n}\subseteq A_{2n}\) of P-positions, satisfying the following properties, for all \(n\in \mathbb {N}\):
(i)

Unique empty play — \(\sigma _{0}\) is equal to the singleton set \(A_{0}\),

(ii)

Closure under even prefixes — if \(a \in \sigma _{2n+2}\) then \(\pi ^2(a) \in \sigma _{2n}\),

(iii)

Determinacy — if \(a,b \in \sigma _{2n}\) with \(\pi (a) = \pi (b)\), then \(a=b\).

In order to generalize this definition to non-deterministic P-strategies, we find convenient to consider the full subcategory \(\omega _{P}\) of \(\omega \) consisting of the strictly positive even numbers, of the form 2n for \(n>0\); and the inclusion functor \(\iota _P:\omega _P\rightarrow \omega \). Define the presheaf \(A_P=A\circ \iota _P\) as the simple game A obtained as the restriction of the presheaf \(A:\omega ^{op}\rightarrow \mathbf {Set}\) to the subcategory \(\omega _P\):

The collection \(A_P\) thus consists of all the Player positions in A, except for the initial one \(*\in A(0)\). This leads us to the following definition of (non-deterministic) P-strategy on a simple game A:

Definition 1

A P-strategy \(\sigma \) on a simple game A is a presheaf \(S:\omega _P^{\mathrm {op}}\rightarrow \mathbf {Set}\) over the category \(\omega _P\) together with a morphism of presheaves \(\sigma : S\rightarrow A_P\). We write \(\sigma :A\) in that case. The presheaf \(S\) is called the support of the strategy \(\sigma \) and the elements of \(S_{2n}\) are called the runs of degree 2n of the strategy, for \(n\ge 0\).

In other words, a P-strategy \(\sigma \) on \(A\) is a family of sets \(S_{2n}\) indexed by strictly positive numbers \(n>0\), related between them by functions \((\pi _P)_{2n}:S_{2n+2}\rightarrow S_{n}\) pictured as:
together with a family of functions \(\sigma _{2n}:S_{2n}\rightarrow A_{2n}\) making the diagram below commute, for all \(n>0\):
To every simple game A, we associate the category Open image in new window of P-strategies over A, defined as the slice categorywhose objects are thus the strategies over A, and whose morphisms \(\theta :\sigma \rightarrow \tau \) between two strategies \(\sigma :S\rightarrow A\) and \(\tau :T\rightarrow A\) are the morphisms \(\theta :S\rightarrow T\) of presheaves satisfying the expected equation: \(\sigma =\tau \circ \theta \). We will call those simulations. One main contribution of the paper is the observation that the family of categories  Open image in new window can be organised into a pseudofunctorfrom the category Open image in new window of simple games and deterministic strategies. The pseudofunctor  Open image in new window is moreover monoidal, in the sense that there exists a family of functorsindexed by simple games AB. As a symmetric monoidal closed category, the category  Open image in new window is enriched over itself, with the simple game Open image in new window constructed from the simple games A and B. Here comes the nice point of the construction: the bicategory  Open image in new window is simply defined as the bicategory with simple games AB as objects, and withas category of morphisms between two simple games A and B. In other words, a morphism \(\sigma :A\rightarrow B\) in Open image in new window is a P-strategy \(\sigma :A\multimap B\), and a 2-cell \(\theta :\sigma \Rightarrow \tau : A\rightarrow B\) is a morphism \(\theta :\sigma \rightarrow \tau \) in the category Open image in new window . At this point, the fact that Open image in new window defines a bicategory is easily derived from the lax monoidal structure of the pseudofunctor Open image in new window . Recall that, as a symmetric monoidal closed category, the category  Open image in new window is enriched over itself. From a conceptual point of view, the construction of the bicategory  Open image in new window thus amounts to a change of enrichment category along the lax monoidal pseudofunctor  Open image in new window , transforming the Open image in new window -enriched category  Open image in new window into the (weak) \(\mathbf {Cat}\)-enriched category  Open image in new window .
Besides the construction of  Open image in new window , a great care will be devoted to the analysis of the Curien-Lamarche exponential comonad ! on the category  Open image in new window and to the recipe to turn it into an exponential 2-comonad on the bicategory  Open image in new window . The construction relies on the existence of a family of functorscalled “promotion” functors, and natural in the simple game A in the category  Open image in new window . In particular, the functorial part of the exponential 2-comonad  Open image in new window is defined as the composite:

where \(n_{A,B}:{!{(A\multimap B)}} \rightarrow {!{A}}\multimap {!{B}}\) is the canonical morphism in Open image in new window which provides the structure of a lax monoidal functor to the original comonad Open image in new window .

2 Non-deterministic P-strategies as P-cartesian Transductions

As explained in the introduction, a P-strategy  Open image in new window over a simple game A is defined as an object of the slice category (1) in the category \([\omega _P^{op},\mathbf {Set}]\) of contravariant presheaves over \(\omega _P\). We will use the fact that the slice category is equivalent to the category of contravariant presheavesover the Grothendieck category \(\mathbf {tree}({A_P})\) generated by the presheaf \(A_P\in [\omega _P^{op},\mathbf {Set}]\). The category \(\mathbf {tree}({A_P})\) has the P-positions of the simple game \(A\) as objects, and a morphism \(a\rightarrow a'\) between \(a\in A_{2p}\) and \(a'\in A_{2q}\) precisely when \(p\le q\) and \(\pi ^{2q-2p}(a')=a\). In other words, it is the order category associated to the tree of P-positions of the simple game \(A\).
We find convenient for later purposes to reformulate non-deterministic P-strategies in the following way. This paves the way to a comprehension theorem for the pseudofunctor  Open image in new window , which will be established in the next section. A transduction \(\theta :A\rightarrow B\) between two simple games \(A,B:\omega ^{op}\rightarrow \mathbf {Set}\) is defined as a natural transformation between the presheaves A and B, given by a family of functions \(\theta _n:A_n\rightarrow B_n\) making the square \(\Box _n\) diagram below commute, for all \(n\in \mathbb {N}\):
A transduction \(\theta :A\rightarrow B\) is called P-cartesian when \(\Box _{2n}\) is a pullback square for all \(n\in \mathbb {N}\); and O-cartesian when \(\Box _{2n+1}\) is a pullback square for all \(n\in \mathbb {N}\). We write Open image in new window for the category of simple games and transductions between them, and Open image in new window (resp. Open image in new window ) for the subcategory of P-cartesian (resp. O-cartesian) transductions. Note that the restriction functor
$$\begin{aligned} (-)_P \quad : \quad [\omega ^{op},\mathbf {Set}] \quad \longrightarrow \quad [\omega _P^{op},\mathbf {Set}] \end{aligned}$$
is a fibration, and that a transduction \(\theta :A\rightarrow B\) between simple games is P-cartesian precisely when it defines a cartesian morphism with respect to the fibration \((-)_P\). For that reason, a P-cartesian transduction \(\theta :A\rightarrow B\) is entirely characterized by the family of functions \(\theta _{2n}:A_{2n}\longrightarrow B_{2n}\) on the P-positions of the simple games A and B, for \(n\in \mathbb {N}\). From this follows easily that

Proposition 1

A P-strategy \(\sigma \) on a simple game A is the same thing as a simple game S together with a P-cartesian transduction \(S\rightarrow A\). The simple game S is uniquely determined by \(\sigma \) up to isomorphism. It is called the support (or run-tree) of \(\sigma \), and noted \(\{A\,|\,\sigma \}\), while the P-cartesian transduction is noted \(\mathsf {supp}_{\,\sigma } : \{A\,|\,\sigma \} \longrightarrow A\).

Note that the definition applies the general principle formulated in [18] that a strategy \(\sigma \) of a game A is a specific kind of map (here a P-cartesian transduction) \(S\rightarrow A\) from a given game \(S=\{A\,|\,\sigma \}\) to the game A of interest. One benefit of this principle is that it unifies the two concepts of game and of strategy, by regarding a strategy \(\sigma \) of a game A as a game S “embedded” in an appropriate way by \(S\rightarrow A\) inside the simple game A. This insight coming from [18] underlies for instance the construction in [19] of a category of non-deterministic strategies between asynchronous games.

Typically, consider the simple game \(A=\mathbb {B}_1\multimap \mathbb {B}_2\) where \(\mathbb {B}\) is the simple boolean game with a unique initial Opponent move q and two Player moves \(\mathsf {tt}\) for true and \(\mathsf {ff}\) for false; and where the indices 1, 2 are here to indicate the component of the boolean game \(\mathbb {B}\). The simple game A may be represented as the decision tree below:

where the sets of positions are defined as:
$$\begin{aligned} A_{1}=\{a\} \quad \quad A_{2}=\{b,a_1,a_2\} \quad \quad A_{3}=\{b_1,b_2\} \quad \quad A_{4}=\{b_{11},b_{12},b_{21},b_{22}\} \end{aligned}$$
and where the branches are induced by the prefix functions \(\pi _{n}:A_{n+1}\rightarrow A_{n}\) depicted on the picture above. For the reader’s convenience, we label every edge of A by the name of the move which would be used in the more familiar definition of simple games, where plays are defined as sequences of moves [1, 6]. Note that every position \(a\in A_{n}\) of degree n is determined by its occurrence, defined as the sequence of n moves from the root \(*\) to the position a in the tree \(A\). Typically, the P-position \(b\in A_{2}\) has occurrence \(q_2\cdot q_1\) and the P-position \(b_{21}\in A_{4}\) has occurrence \(q_2\cdot q_1 \cdot \mathsf {tt}_1 \cdot \mathsf {ff}_2\).
By way of illustration, we define the P-strategy Open image in new window as the presheaf below
$$\begin{aligned} \begin{array}{cccccccccc} &{}&{} *\mapsto \{*\} &{}\quad &{} a_1\mapsto \emptyset &{}\quad &{} a_2\mapsto \{x''\} &{}&{} \\ b\mapsto \{x'\} &{}\quad &{} b_{11}\mapsto \emptyset &{}\quad &{} b_{12}\mapsto \emptyset &{}\quad &{} b_{21}\mapsto \{z'\} &{}\quad &{} b_{22}\mapsto \{z'',z'''\} \end{array} \end{aligned}$$
on the Grothendieck category \(\mathbf {tree}({A_P})\) associated to the presheaf \(A_P\) of P-positions in \(A\). As explained in Proposition 1, the P-strategy \(\sigma \) may be equivalently defined as the simple game \(S=\{A\,|\,\sigma \}\) below
together with the P-cartesian transduction \(\mathsf {supp}_{\,\sigma }:\{A\,|\,\sigma \}\rightarrow A\) described as:
$$\begin{aligned} x\mapsto a \quad \quad x'\mapsto b \quad \quad x'' \mapsto a_2 \quad \quad y\mapsto b_1 \quad \quad z'\mapsto b_{21} \quad \quad z''\mapsto b_{22} \quad \quad z'''\mapsto b_{22} \end{aligned}$$
It is worth mentioning that the transduction \(\mathsf {supp}_{\,\sigma }\) may be recovered from the moves labelled on the run-tree \(S=\{A\,|\,\sigma \}\). This pictorial description provides a convenient way to describe how the non-deterministic P-strategy \(\sigma \) plays on A. Typically, when questioned by the initial move \(q_2\) of the game, the non-deterministic P-strategy \(\sigma \) answers \(\mathsf {tt}_2\) with the run \(x''\in S_{2}\) or asks the value of the input boolean by playing the move \(q_1\); when the Opponent answers with the move \(\mathsf {tt}_1\), the P-strategy reacts by playing the value \(\mathsf {ff}_2\) with the run \(z'\in S_{4}\) or by playing the value \(\mathsf {ff}_2\) with the runs \(z'', z'''\in S_{4}\). Note in particular that the P-strategy \(\sigma \) is allowed to play two different runs \(z'', z'''\in S_{4}\) of the same play \(b_{22}\in A_{4}\).

3 P-cartesian Transductions as Deterministic Strategies

In the previous section, we have seen how to regard every non-deterministic P-strategy Open image in new window as a P-cartesian transduction \(\mathsf {supp}_{\,\sigma }:\{B\,|\,\sigma \}\rightarrow B\) into the simple game B. Our purpose here is to show that every P-cartesian transduction \(\theta :A\rightarrow B\) can be seen as a particular kind of deterministic strategy of the simple game \(A\multimap B\).

Definition 2

(Total strategies). A deterministic strategy \(\sigma \) of a simple game A is total when for every O-position s such that the P-position \(\pi (s)\) is an element of \(\sigma \), there exists a P-position t in the strategy \(\sigma \) such that \(\pi (t)=s\).

Definition 3

(Back-and-forth strategies). Given two simple games A and B, a back-and-forth strategy f of the simple game \(A\multimap B\) is a deterministic and total strategy whose positions are all of the form (cab) where \(c:n\rightarrow n\) is a copycat schedule.

Back-and-forth strategies compose, and thus define a subcategory of  Open image in new window :

Definition 4

(The category Open image in new window ). The category  Open image in new window of back-and-forth strategies is the subcategory of Open image in new window whose objects are the simple games and whose morphisms \(f:A\rightarrow B\) are the back-and-forth strategies of \(A\multimap B\).

As a matter of fact, we will be particularly interested here in the subcategory Open image in new window of functional back-and-forth strategies in the category Open image in new window .

Definition 5

(Functional strategies). A functional strategy f of the simple game \(A\multimap B\) is a back-and-forth strategy such that for every position \(a\in A_{n}\) of degree n in the simple game \(A\), there exists a unique position \(b\in B_{n}\) of same degree in \(B\) such that \((c,a,b)\in f\), where \(c:n\rightarrow n\) is the copycat schedule.

The following basic observation justifies our interest in the notion of functional strategy:

Proposition 2

For all simple games A, B, there is a one-to-one correspondence between the P-cartesian transductions \(A\rightarrow B\) and the functional strategies in \(A\multimap B\).

Proof

See Appendix E.

For that reason, we will identify P-cartesian transductions and functional strategies from now on. Put together with Proposition 1, this leads us to the following correspondence, which holds for every simple game A:

Proposition 3

The category Open image in new window is equivalent to the slice category Open image in new window .

The result may be understood as a preliminary form of comprehension: it states that every non-deterministic P-strategy  Open image in new window may be equivalently seen as a functional P-strategy
$$\begin{aligned} \mathsf {supp}_{\,\sigma } \quad : \quad \{A\,|\,\sigma \} \quad \longrightarrow \quad A \end{aligned}$$
(2)
in the category  Open image in new window of simple games and deterministic strategies, obtained by composing the equivalences stated in Propositions 1 and 3. Note that the simple game \(\{A\,|\,\sigma \}\) coincides with the run-tree S of the non-deterministic strategy \(\sigma \) formulated in Proposition 1 and that the functional strategy \(\mathsf {supp}_{\,\sigma }\) coincides with the P-cartesian transduction which “projects” the support S on the simple game A. The property (Proposition 3) is important from a methodological point of view, because it enables us to use the rich toolbox developed for simple games and deterministic strategies, in order to handle non-deterministic strategies inside the category  Open image in new window .

4 The Pseudofunctor Open image in new window

Suppose given a P-strategy  Open image in new window over the simple game A and a morphism \(f:A\rightarrow B\) in the category  Open image in new window .

Definition 6

The P-strategy Open image in new window over the simple game B is defined as the contravariant presheaf over \(\mathbf {tree}({B_P})\) which transports every P-position b of the simple game \(B\) to the disjoint union defined below:

The fact that (3) defines a presheaf over Open image in new window and that Open image in new window is a pseudofunctor (see Definition 24) is established in the Appendix F.

This construction equips the family of presheaf categories Open image in new window with the structure of a pseudofunctor Open image in new window Moreover, the pseudo-functor  Open image in new window has comprehension in the sense of Lawvere [8]. For every simple game B, the comprehension functor is defined as the compositewhich transports every non-deterministic P-strategy to the morphism (2) seen as an object of Open image in new window . One establishes that

Theorem 1

(Comprehension). For every simple game B, the comprehension functorhas a left adjoint functor
Given a deterministic strategy \(f:A\rightarrow B\), the contravariant presheaf \(\mathsf {image}(f)\) over the category \(\mathbf {tree}({B_P})\) transports every P-position b of the game \(B\) to the set below:
$$\begin{aligned} \mathsf {image}(f) \quad : \quad b \quad \mapsto \quad \Big \{\quad (e,a,b) \quad \Big | \quad (e,a,b)\in f\quad \Big \} \end{aligned}$$
Note that the presheaf \(\mathsf {image}(f)\) may be also described by the formulawhere \(*_A\) is the terminal object in the category Open image in new window of P-strategies over A. Note that the run-tree \(\{A\,|\,*_A\}\) of the P-strategy Open image in new window is the simple game A itself, with \(\mathsf {supp}_{\,*_A}\) the identity \(i_A:A\rightarrow A\). In other words, the P-strategy \(*_A\) has exactly one run over each position of the simple game A.

Also note that we will occasionally note positions of \(\mathsf {image}(f)\) \(b_{(e,a)}\) when there is need to emphasize the fact that \(\mathsf {image}(f)\) is a contravariant presheaf over \(\mathbf {tree}({B_P})\).

5 The Slender-Functional Factorisation Theorem

In order to establish the comprehension theorem, we prove a factorization theorem in the original category  Open image in new window , which involves slender and functional strategies.

Definition 7

A deterministic strategy f in a simple game \(A \multimap B\) is slender when for every P-position b in the simple game B, there exists exactly one P-position a of the simple game A and exactly one schedule e such that \((e,a,b) \in f\).

By extension, we say that a morphism \(f:A\rightarrow B\) in the category  Open image in new window is slender when the deterministic strategy f is slender in \(A\multimap B\). Note that every isomorphism \(f:A\rightarrow B\) in the category  Open image in new window is both slender and functional.

Proposition 4

Suppose that A and B are two simple games and that f is a deterministic strategy of \(A \multimap B\). Then, there exists a slender strategy \(g:A\rightarrow C\) and a functional strategy \(h:C\rightarrow B\) such that \(f=h \circ g\).

The simple game C is defined as \(\{B\,|\,\mathsf {image}(f)\}\) while the slender strategy \(g:A\rightarrow C\) is defined as
$$\begin{aligned} g \quad = \quad \Big \{\quad (e,a,(e,a,b)) \quad \Big | \quad (e,a,b)\in f\quad \Big \} \end{aligned}$$
and \(h:C\rightarrow B\) is the functional strategy \(\mathsf {supp}_{\,\mathsf {image}(f)}\) associated in Proposition 3 to the P-strategy Open image in new window .

Proposition 5

Suppose that \(s:U\rightarrow V\) and \(f:A\rightarrow B\) are two morphisms of the category Open image in new window . Suppose moreover that s is slender and that f is functional. Then, \(s:X\rightarrow Y\) is orthogonal to \(f:A\rightarrow B\) in the sense that for all morphisms \(u:X\rightarrow A\) and \(v:Y\rightarrow B\) making the diagram (a) commute, there exists a unique morphism \(h:Y\rightarrow B\) making the diagram (b) commute in the category  Open image in new window :
The deterministic strategy \(h:Y\rightarrow A\) is defined as
$$\begin{aligned} \begin{array}{rll} h \,\, = \,\,&{} \Big \{ (e,y,a) \Big | &{} \exists x\in X, b\in B, {e',e''\in \varUpsilon ,} \\ &{}&{} \begin{array}{lclclclcl} (e,y,b)\in v &{}\wedge &{} (c,a,b)\in f &{}\wedge &{} (e',x,y)\in s &{}\wedge &{} (e'',x,a) \in u \end{array} \Big \}\\ \uplus \,\,&{}\Big \{ (e,y,a) \Big | &{}\exists x\in X, b\in B, {e',e''\in \varUpsilon ,}\\ &{}&{}\begin{array}{lclclclcl} (e,y,b)\in v &{}\wedge &{} (c,a,b)\in f &{}\wedge &{} (e',x,\pi y)\in s &{}\wedge &{} (e'',x,\pi a) \in u\end{array} \Big \} \end{array} \end{aligned}$$
Note that the position b is uniquely determined by the position a because f is functional, and that the pair \((e',x)\) is uniquely determined by the position y because s is slender. Moreover, by determinism of \(u=h\circ s\), the schedule \(e''\) is entirely determined by the schedules e and \(e'\).

Theorem 2

(Factorization theorem). The classes Open image in new window of slender morphisms and Open image in new window of functional morphisms define a factorization system Open image in new window in the category Open image in new window .

It is a folklore result that, in that situation, the comprehension theorem (Theorem 1) follows from the factorization theorem. The reason is that the category Open image in new window is equivalent (by Proposition 3) to the full subcategory Open image in new window of functional strategies in the slice category  Open image in new window . Seen from that point of view, the comprehension functor \(\{B\,|\,-\}\) coincides with the embedding of Open image in new window into Open image in new window . It is worth noting that for every P-strategy  Open image in new window , one has an isomorphism
$$\begin{aligned} \sigma \quad \cong \quad \mathsf {image}(\mathsf {supp}_{\,\sigma }) \end{aligned}$$
in the category Open image in new window , and that one has an isomorphismin the category Open image in new window , for every morphism \(f:A\rightarrow B\) in the category Open image in new window . This provides an alternative way to define the pseudofunctor  Open image in new window .

6 The Bicategory Open image in new window of Simple Games and Non-deterministic Strategies

In this section, we explain how to construct a bicategory  Open image in new window of simple games and non-deterministic strategies, starting from the category  Open image in new window . The first step is to equip the pseudofunctor Open image in new window with a lax monoidal structure (See Definition 25), based on the definition of tensor product in the category  Open image in new window formulated in [4], see Appendix B for details. We start by observing that

Proposition 6

Suppose given two morphisms \(f:A\rightarrow B\) and \(g:C\rightarrow D\) in the category  Open image in new window of simple games and deterministic strategies. The morphism
$$\begin{aligned} f\otimes g: A\otimes C\longrightarrow B\otimes D \end{aligned}$$
is slender when f and g are slender, and functional when f and g are functional.

Proof

See Appendix G.

Note that the isomorphism \(\mathsf {image}(f \otimes g) \cong \mathsf {image}(f) \otimes \mathsf {image}(g)\) follows immediately from this statement and from the factorization theorem (Theorem 2), for every pair of morphisms \(f:A\rightarrow B\) and \(g:C\rightarrow D\) in the category  Open image in new window . The tensor product \(\sigma \otimes \tau \) of two P-strategies \(\sigma \) and \(\tau \) is defined in the same spirit, using comprehension:

Definition 8

Suppose that Open image in new window is a P-strategy of a simple game A and that Open image in new window is a P-strategy of a simple game B. The tensor product \(\sigma \otimes \tau \) is the P-strategy of the simple game \(A\otimes B\) defined as
$$\begin{aligned} \sigma \otimes \tau \quad = \quad \mathsf {image}(\,\mathsf {supp}_{\,\sigma } \otimes \mathsf {supp}_{\,\tau }\,). \end{aligned}$$
Here, the morphism \( \mathsf {supp}_{\,\sigma } \otimes \mathsf {supp}_{\,\tau } : \{A\,|\,\sigma \}\otimes \{B\,|\,\tau \} \rightarrow A\otimes B \) denotes the tensor product (computed in the original category  Open image in new window ) of the morphisms \(\mathsf {supp}_{\,\sigma }\) and \(\mathsf {supp}_{\,\tau }\). A direct description of Open image in new window is also possible, as the presheaf which transports every position (eab) of the simple game \(A\otimes B\) to the set-theoretic product below:
$$\begin{aligned} \sigma \otimes \tau \quad : \quad (e,a,b) \quad \mapsto \quad \sigma (a)\times \tau (b). \end{aligned}$$
As indicated in the introduction, the tensor product of P-strategies defines a family of functors Open image in new window which, together with the isomorphism of categories Open image in new window , equips the pseudofunctor  Open image in new window with a lax monoidal structure:

Theorem 3

The pseudofunctor  Open image in new window equipped with the family of functors \(m_{A,B}\) and \(m_1\) defines a lax monoidal pseudofunctor from Open image in new window to \((\mathbf {Cat},\times ,1)\).

Proof

See Appendix H.

The bicategory  Open image in new window of simple games and non-deterministic strategies is deduced from the lax monoidal pseudofunctor  Open image in new window in the following generic way, inspired by the idea of monoidal refinement system [16].

Definition 9

The bicategory Open image in new window has simple games A, B, C as objects, with the hom-category Open image in new window defined asthe composition functordefined as the composite
where \(comp_{A,B,C} : (B\multimap C) \otimes (A \multimap B) \longrightarrow (A \multimap C)\) is the morphism which internalizes composition in the symmetric monoidal closed category Open image in new window . In the same way, the identity in Open image in new window is defined as the composite

where the morphism \(id_A: 1\rightarrow (A\multimap A)\) internalizes the identity morphism in  Open image in new window .

Proposition 7

The bicategory Open image in new window is symmetric monoidal closed in the sense that there exists a family of isomorphisms
The isomorphism \(\varPhi _{A,B,C}\) is defined as the image by the pseudofunctor  Open image in new window of the isomorphism
$$\begin{aligned} \varphi _{A,B,C} \quad : \quad (A\otimes B)\multimap C\quad \cong \quad B\multimap (A\multimap C) \end{aligned}$$
in the category  Open image in new window between the underlying simple games. One benefit of our conceptual approach is that the monoidal closed structure of Open image in new window is neatly deduced from the monoidal closed structure of the original category  Open image in new window .

7 The Exponentional Modality on the Category  Open image in new window

Now that the monoidal bicategory  Open image in new window has been defined, we analyze how the exponential modality defined in [4] adapts to our sheaf-theoretic framework.

Definition 10

Let A be a simple game. !A is the simple game whose set \((!A)_n\) of positions of degree n consists of the pairs \((\phi , \overline{a})\) such that:
  • \(\phi \) is a O-heap over n and \(\overline{a}=(a_1,\dots ,a_n)\) is a sequence of positions of A,

  • for each \(k\in \{1,\dots ,n\}\), the sequence of positions in \(\overline{a}=(a_1,\dots ,a_n)\) corresponding to the branch of k in \(\phi \) defines a play
    $$\begin{aligned} \{a_k,a_{\phi (k)}, a_{\phi ^2(k)}, \dots \} \end{aligned}$$
    of the simple game A.

The predecessor function \(\pi _n:(!A)_{n+1}\rightarrow (!A)_{n}\) is defined as Open image in new window

Definition 11

Let f be a deterministic strategy of \(A \multimap B\). The deterministic strategy !f of \({!A} \multimap {!B}\) consists of the positions \((e,(\phi ,\overline{a}), (\psi , \overline{b}))\) such that \(\phi =e^*\psi \) and, for each branch of \((\phi ,e,\pi )\), the positions associated to that branch are played by f.

It is worth observing that the construction of \({!f}:{!A}\rightarrow {!B}\) can be decomposed in the following way. Consider the morphism
$$\begin{aligned} n_{A,B} \quad : \quad !\,(A \multimap B) \quad \longrightarrow \quad !A \, \multimap \,\, !B \end{aligned}$$
obtained by currying the composite morphism

in the symmetric monoidal closed category  Open image in new window , where we use the coercion morphism which provides the exponential modality Open image in new window with the structure of a lax monoidal functor.

Definition 12

(\(\# f\)). Given a deterministic strategy f of a simple game A, the deterministic strategy \(\#f\) of the simple game !A has positions the pairs \((\phi ,\overline{a})\) such that for each branch of \((\phi ,\overline{a})\), the positions associated to that branch are played by the deterministic strategy f.

Proposition 8

Given a morphism \(f:A\rightarrow B\) of the category  Open image in new window and its curried form \(\lambda a. f\, :\, 1\rightarrow A\multimap B\), the composite morphism

is the curried form \(\lambda x:{!A}.\,\,{!f}\) in the category  Open image in new window of the morphism \({!f}:{!A}\longrightarrow {!B}\).

More details about the original exponential modality in  Open image in new window will be found in Appendix C. By analogy with Proposition 6, we establish that

Proposition 9

Suppose that \(f:A\rightarrow B\) is a morphism in the category Open image in new window . Then, the morphism
$$\begin{aligned} !f \quad : \quad {!A} \quad \longrightarrow \quad {!B} \end{aligned}$$
is slender when f is slender, and functional when f is functional.

Proof

See Appendix I.

8 The Exponential Modality on the Bicategory  Open image in new window

In this section, we define the linear exponential modality Open image in new window on the symmetric monoidal closed bicategory  Open image in new window , in order to define a bicategorical model of intuitionistic linear logic. The construction is inspired by the observation made in the previous section (Proposition 8).

Definition 13

Given a P-strategy Open image in new window of a simple game A, the P-strategy \(\#\sigma \) of the simple game !A is defined as the image in Open image in new window of the morphism
$$\begin{aligned} !\,\mathsf {supp}_{\,\sigma } \quad : \quad !\,\{A\,|\,\sigma \} \quad \longrightarrow \quad {!A}. \end{aligned}$$
Note that the definition of \(\#\sigma \) induces a commutative diagram in the category  Open image in new window

where the top arrow is an isomorphism. Moreover, the definition of \(\#\sigma \) coincides with the previous definition (Definition 12) when the P-strategy \(\sigma =f\) happens to be deterministic.Consequently, for two games AB and a deterministic strategy \(f:A\multimap B\), we have Open image in new window and Open image in new window .

As mentioned in the introduction, this construction \(\sigma \mapsto \#\sigma \) defines a functorNow, remember that a morphism \(\sigma :A\rightarrow B\) of the bicategory  Open image in new window is defined as a P-strategyFor that reason, every such morphism \(\sigma :A\rightarrow B\) induces a P-strategyIn order to turn the P-strategy \(\#\sigma \) into a P-strategywe apply the functorto the P-strategy \(\#\sigma \), where
$$\begin{aligned} n_{A,B} \quad : \quad !\,(A \multimap B) \quad \longrightarrow \quad !A \, \multimap \,\, !B \end{aligned}$$
denotes the structural morphism of Open image in new window defined in the previous section. The construction may be summarized as follows:

Definition 14

The morphism \({!\sigma }:{!A}\rightarrow {!B}\) of the bicategory  Open image in new window associated to the morphism \(\sigma :A\rightarrow B\) is defined as the P-strategy

Theorem 4

With this definition, Open image in new window defines a pseudofunctor from the bicategory  Open image in new window to itself.

Proof

See Appendix J.

The family of morphisms
$$\begin{aligned} \delta _A : {!}A \rightarrow {!} {!}\,A \quad \quad \quad \quad \quad \quad \quad \varepsilon _A : {!}A \rightarrow A \end{aligned}$$
are defined with the same deterministic strategies in Open image in new window and Open image in new window as in the original category  Open image in new window . One checks that the families \(\delta \) and \(\varepsilon \) define natural transformations between pseudonatural functors on  Open image in new window (as defined in Definition 26), and that the 2-functor Open image in new window defines a 2-comonad in the appropriate bicategorical sense (see Definition 27). The family of morphisms
$$\begin{aligned} d_A : {!}A \rightarrow {!}\,A \otimes {!}\,A \quad \quad \quad \quad \quad \quad \quad e_A : {!}A \rightarrow 1 \end{aligned}$$
are defined with the same deterministic strategies in Open image in new window and Open image in new window as in the original category  Open image in new window , and one checks that they define natural transformations between pseudonatural functors on  Open image in new window . One obtains in this way that

Theorem 5

The bicategory p equipped with the exponential modality \(!:\) defines a bicategorical model of multiplicative intuitionistic linear logic.

The formal and rigorous verification of these facts would be extremely tedious if done directly on the bicategory Open image in new window of nondeterministic strategies. Our proof relies on the fact that the constructions of the model (Definitions 914) are performed by “push” functors Open image in new window above a structural morphism f living in the original category  Open image in new window . The interested reader will find part of the detailed proof in Appendix K.

9 Conclusion

We construct a bicategory  Open image in new window of simple games and non-deterministic strategies, which is symmetric monoidal closed in the extended 2-dimensional sense. We then equip the bicategory  Open image in new window with a linear exponential modality  Open image in new window which defines a bicategorical model of intuitionistic linear logic. This provides, as far as we know, the first sheaf-theoretic and non-deterministic game semantics of intuitionistic linear logic — including, in particular, a detailed description of the exponential modality.

Supplementary material

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Authors and Affiliations

  1. 1.Institut de Recherche en Informatique Fondamentale, Université Paris DiderotParisFrance

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