On the Complexity of Cockett-seely Polarized Games

In this paper the complexity of provability of polarized additive, multiplicative, and exponential formulas in the (initial) Cockett-Seely polarized game logic is discussed. The complexity is ultimately based on the complexity of finding a strategy in a formula which is, for polarized additive formulas, in the worst case linear in their size. Having a proof of a sequent is equivalent to having a strategy for the internal-hom object. In order to show that the internal-hom object can have size exponentially larger than the formulas of the original sequent we develop techniques for calculating the size of the multiplicative formulas. The structure of the internal hom object can be exploited and, using dynamic programming techniques, one can reduce the cost of finding a strategy in such a formula to the order of the product of the sizes of the original formulas. The use of dynamic techniques motivates the consideration of games as acyclic graphs and we show how to calculate the size of these graph games for the multiplicative and additive fragment and, thus, the cost of determining their provability using this dynamic programming approach. The final section of the paper points out that, despite the apparent complexity of the formulas, there is, for the initial polarized logic with all the connectives (additives, multiplicatives, and exponentials) a way of determining provability which is linear in the size of the formulas. Introduction It is a natural question in a logic to ask how hard is it to decide whether a sequent is provable. This paper examines this question for the Cockett-Seely polarized game logic [CS04] which we will henceforth refer to simply as polarized game logic. We shall show that this question can be answered in linear time. The decision procedure consists of translating the given sequent in the polarized game logic into a single polarized game using “negation” and the “mixed par” operation. As this is an internal-hom game, determining whether a strategy exists in this game corresponds to determining whether a proof exists for the sequent. To determine whether a strategy exists in a game requires, in the worst case, time proportional to the number of edges in the tree representing that game. Thus, it becomes important to estimate the size of this internal-hom game. While, in general, there is no simple expression for this size, it is often possible to estimate the “uniform size” of an internal-hom game. The uniform size is defined as the number of binary logical operations required, in the worst case, to determine whether a strategy is present. This makes it Research partially supported by NSERC, Canada. Diagrams were produced with the XY-pic package of K. Rose and R. Moore and inferences with M. Tatsuya’s proof.sty. 2000 Mathematics Subject Classification: 03F20, 03F20, 18A15. c © J. R. B. Cockett and C. A. Pastro, 2004. Permission to copy for private use granted. 1 2 possible to get a fairly precise estimate for the size of certain internal-hom games. These estimates allow us to demonstrate that there can be an exponential increase in the size of the internal-hom game over the original games. Fortunately it is possible to exploit the structure of the internal hom object and, using dynamic programming techniques, reduce the cost of finding a strategy in such an object to be bounded by the product of the sizes of the original game trees. This approach involves regarding games as if they are acyclic directed graphs, a direction already pioneered by Hyland and Schalk [HS02]. When this is done one can give precise expressions for the size of formulas built using the multiplicatives connectives and, thus, for the cost of the dynamic programming approach to finding a strategy. The categorical semantics of polarized games is outside the scope of this paper. In order to keep the paper largely self-contained we do, however, provide a description and brief discussion of the basic concepts. The reader who is interested in a more complete story should consult the paper of Cockett and Seely [CS04]. Such a course of action is recommended as, in the last section of this paper, we use this semantics, albeit in a very simple way, to show that for this model one can decide provability in linear time for the full logic with polarized additives, multiplicatives, and exponentials. Before starting this story we should emphasize that the multiplicative structure of this model is not the free structure. This model has “soft” multiplicative and exponential structures (see [CS04]). Provability in free polarized multiplicative structures has the same complexity as provability in the corresponding multiplicative fragment of linear logic (as any multiplicative proof net can be polarized). For multiplicative linear logic with atoms provability is known to be NP-complete [Kan92], and even without atoms this decision problem is just as hard: the multiplicative units can be used to simulate the effect of having atoms [LW94]. In the presence of atoms, additives, and exponentials (or even nonlogical axioms) these provability problems become undecidable for ordinary linear logic [LMSS92]. The results we obtain are somewhat curious as they apply to the finite fragment of precisely the same model whose depolarization was used by Abramsky and Jagadeesan [AJ94] to obtain full completeness for the (iso-mix) multiplicative fragment of multiplicative linear logic. This seems to highlight the important role units play in providing these logics with their underlying complexity. Note also that using the transformation to Laurent’s version of polarized logic [Lau02], discussed in [CS04], this model also provides, through the coKleisli construction for the !( ) comonad, a non-trivial model of intuitionistic logic which has a linear time provability. It is worth recalling that this is, for the free logic, a P-space complete problem [Stat79]. The outline of this paper is as follows. In Section 1 polarized games are introduced. In Section 2 polarized game logic is introduced. Section 3 describes the multiplicative and exponential structure on polarized games. Section 4 shows how provability can be turned into the question of finding a strategy in the internal-hom object. Section 5 discusses the size of the multiplicative formulas including internal-hom objects. In Section 6, we discuss the relationship between the dynamic programming solution to determining provability 3 and graph games. In particular, we describe simply formulas for the size of multiplicatives in graph games. Finally, in Section 7, we show that provability, despite the evident complexity of the underlying additive structures, can be evaluated for this model in linear time for the complete additive, multiplicative, and exponential logic. 1. Polarized games A polarized game is a finite 2-player input-output game of the sort studied by Abramsky and Jagadeesan [AJ94] and Hyland in [Hy97]. The two players are referred to as the “opponent” and the “player”. Games are often discussed using the terminology of processes; in these terms a game is a protocol for interaction while a map between games is a process which interacts through these protocols on the “input” (or domain) and “output” (or codomain) channels. On the output channel, the opponent messages come from the “environment” and the player messages are generated by the process or “system”. On the input channel, however, the roles are reversed: the opponent messages are generated by the “system” and the player messages come from the “environment.” There are two sorts of games: those in which the opponent has the first move and those in which the player has the first move. The games introduced in this paper are the games of the initial polarized game category [CS04]. Games are represented in a number of ways. An opponent game is denoted by O = (b1 :P1, . . . , bm :Pm) = l i∈I bi :Pi = ◦ b1 sss ss bm L L L L L P1 · · · Pm where I = {1, . . . , m} and each Pi is, inductively, a player game. A player game is denoted by P = {a1 :O1, . . . , an :On} = ⊔ j∈J aj :Oj = • a1 sss ss an L L L L L O1 · · · On where J = {1, . . . , n} and each Oj is, inductively, an opponent game. We allow (opponent and player) games with empty index sets; this gives two atomic games at which the inductive construction of games begins: