Categorical Combinatorics for Non Deterministic Strategies on Simple Games
Abstract
The purpose of this paper is to define in a clean and conceptual way a nondeterministic and sheaftheoretic variant of the category of simple games and deterministic strategies. One thus starts by associating to every simple game a presheaf category of nondeterministic strategies. The bicategory of simple games and nondeterministic 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 2comonad which adapts the CurienLamarche exponential modality of linear logic to the 2dimensional 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
 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.
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.
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 nonalternated way [18].
Interestingly, all the sheaftheoretic 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 sheaftheoretic transcription of the notion of receptive ingenuous strategy formulated in [18]; on the other hand, the sheaftheoretic notion of nondeterministic 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 nondeterministic strategies in the sheaftheoretic 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 sheaftheoretic 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].
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 Pposition when its degree n is even, and a Oposition 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 Obranching 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 \(n\in \mathbb {N}\).
 (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 \(a=b\).
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 (nondeterministic) Pstrategy on a simple game A:
Definition 1
A Pstrategy \(\sigma \) 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 \(n\ge 0\).
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 Nondeterministic Pstrategies as Pcartesian Transductions
Proposition 1
A Pstrategy \(\sigma \) on a simple game A is the same thing as a simple game S together with a Pcartesian transduction \(S\rightarrow A\). The simple game S is uniquely determined by \(\sigma \) up to isomorphism. It is called the support (or runtree) of \(\sigma \), and noted \(\{A\,\,\sigma \}\), while the Pcartesian 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 Pcartesian 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 nondeterministic 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:
3 Pcartesian Transductions as Deterministic Strategies
In the previous section, we have seen how to regard every nondeterministic Pstrategy Open image in new window as a Pcartesian transduction \(\mathsf {supp}_{\,\sigma }:\{B\,\,\sigma \}\rightarrow B\) into the simple game B. Our purpose here is to show that every Pcartesian 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 Oposition s such that the Pposition \(\pi (s)\) is an element of \(\sigma \), there exists a Pposition t in the strategy \(\sigma \) such that \(\pi (t)=s\).
Definition 3
(Backandforth strategies). Given two simple games A and B, a backandforth strategy f of the simple game \(A\multimap B\) is a deterministic and total strategy whose positions are all of the form (c, a, b) where \(c:n\rightarrow n\) is a copycat schedule.
Backandforth 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 backandforth 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 backandforth 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 backandforth 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 backandforth 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 onetoone correspondence between the Pcartesian transductions \(A\rightarrow B\) and the functional strategies in \(A\multimap B\).
Proof
See Appendix E.
For that reason, we will identify Pcartesian 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 .
4 The Pseudofunctor Open image in new window
Suppose given a Pstrategy 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 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.
Theorem 1
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 SlenderFunctional 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 Pposition b in the simple game B, there exists exactly one Pposition 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 \(f=h \circ g\).
Proposition 5
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 .
6 The Bicategory Open image in new window of Simple Games and Nondeterministic Strategies
In this section, we explain how to construct a bicategory Open image in new window of simple games and nondeterministic 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
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 Pstrategies \(\sigma \) and \(\tau \) is defined in the same spirit, using comprehension:
Definition 8
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 nondeterministic 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
where the morphism \(id_A: 1\rightarrow (A\multimap A)\) internalizes the identity morphism in Open image in new window .
Proposition 7
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 sheaftheoretic framework.
Definition 10

\(\phi \) is a Oheap 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 playof the simple game A.$$\begin{aligned} \{a_k,a_{\phi (k)}, a_{\phi ^2(k)}, \dots \} \end{aligned}$$
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.
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
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
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
where the top arrow is an isomorphism. Moreover, the definition of \(\#\sigma \) coincides with the previous definition (Definition 12) when the Pstrategy \(\sigma =f\) happens to be deterministic.Consequently, for two games A, B and a deterministic strategy \(f:A\multimap B\), we have Open image in new window and Open image in new window .
Definition 14
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.
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 9, 14) 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 nondeterministic strategies, which is symmetric monoidal closed in the extended 2dimensional 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 sheaftheoretic and nondeterministic game semantics of intuitionistic linear logic — including, in particular, a detailed description of the exponential modality.
Supplementary material
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