Electronic Notes in Theoretical Computer Science 3 (1996)

URL: http://www.elsevier.nl/locate/entcs/volume3.html 10 pages

The Complexity of Local Proof Search in Linear Logic (Extended Abstract)

Patrick D. Lincoln^{1}

SRI International Computer Science Laboratory, Menlo Park CA 94025 USA

John C. Mitchell^{2}

Department of Computer Science, Stanford University, Stanford, CA 94305-9045 USA

Andre Scedrov^{3}

Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395 USA

Abstract

Proof search in linear logic is known to be difficult: the provability of propositional linear logic formulas is undecidable. Even without the modalities, multiplicativeadditive fragment of propositional linear logic, mall, is known to be pspace-complete, and the pure multiplicative fragment, mll, is known to be np-complete. However, this still leaves open the possibility that there might be proof search heuristics (perhaps involving randomization) that often lead to a proof if there is one, or always lead to something close to a proof. One approach to these problems is to study strategies for proof games. A class of linear logic proof games is developed, each with a numeric score that depends on the number of certain preferred axioms used in a complete or partial proof tree. Using recent techniques for proving lower bounds on optimization problems, the complexity of these games is analyzed for the fragment mll extended with additive constants and for the fragment mall. It is shown that no efficient heuristics exist unless there is an unexpected collapse in the complexity hierarchy.

1 lincoln@csl.sri.com Work supported under NSF Grant CCR-9224858 and ONR Grant N00014-95-C-0168.

 2 mitchell@cs.stanford.edu  http://theory.stanford.edu/people/jcm/home.html

Partially supported by NSF.

 3 scedrov@cis.upenn.edu http://www.cis.upenn.edu/~scedrov Partially supported

by NSF Grant CCR-94-00907, by ONR Grant N00014-92-J-1916, and by a Centennial Research Fellowship from the American Mathematical Society.

\begin{matrix}c \\ ◯ \end{matrix}1996 Elsevier Science B. V.

Lincoln, Mitchell, and Scedrov

1 Introduction

Linear logic, introduced in [10], is a refinement of classical logic often described as being resource sensitive because of its intrinsic ability to reflect computational states, events, and resources [11,26,27]. Several notions of game semantics for linear logic are investigated in [6,1,2,13,17,15,9].

Connections between linear logic proof search and probabilistic games considered in complexity theory are investigated in [20–22]. In particular, linear logic proof search may also be seen as a game. This game, the linear logic proof game, is played on linear logic formulas, and its moves are instances of inference rules of linear logic. There are two players, called proponent and opponent, and a separate verifier. Proponent’s goal is to play a sequence of moves that constitute a formal proof of an input formula, consisting of axioms and matching inference rules. Opponent tries to force the direction of proponent’s evidence in a way that makes it impossible for proponent to obtain a formal proof. Several versions of this game are discussed in [21,22], each with a numeric score that reflects the number of certain preferred axioms used in a complete or partial formal proof. The capabilities of the players may differ. While proponent is always omnipotent, in some versions of the game opponent’s decisions are based only on a fair coin toss.

Two fragments of propositional linear logic are considered here: the multiplicative additive fragment, mall, and the multiplicative fragment extended with additive constants, mll⊤. mall is pspace-complete [18]. It follows from the np-completeness of the pure multiplicative fragment, mll [16,19], that mll⊤ is np-complete. These are global hardness properties in that they provide lower bounds on proponent’s optimal strategy.

In chess and in many other intricate games, however, choosing the best next move often seems just as hard as developing a complete winning strategy. In other words, these games are locally hard. This property is studied here for the linear logic proof game. Let us say that an ϵ-heuristic, where ϵ ≥ 0, is a function from formulas to instances of inference rules (that is, proponent’s strategy) such that the optimum score arising from the use of this inference rule instance is within factor 1+ϵ of the optimal score. It is shown that unless p = np, there is no polynomial-time ϵ-heuristic for mll⊤. It is also shown that similar heuristics for mall cannot be computable in randomized polynomial time (bpp) unless bpp = pspace.

2 Linear logic proof games

Let p be a propositional atom, let A, B be mall formulas, let Γ, Δ, Θ, Ξ be finite multisets of mall formulas, and let Σ be a finite multiset of literals or constants 1, 0. We write Δ ⊎ Θ for the (disjoint) multiset union of Δ and Θ. As usual, we write Γ, A for the multiset obtained by adding an instance of A to Γ.

Let us describe several variations of the proof game discussed in [20], all involving the same moves. There are two players, called proponent and opponent, and a separate, polynomial-time verifier. Proponent’s goal is to play a number of moves demonstrating or giving evidence for a sequent. In order to do this, proponent plays proof rule instances. Opponent tries to force the direction of proponent’s evidence in a way that makes it impossible for proponent to win. Opponent plays special markers that may block one side of proponent’s & moves. If proponent plays a ⊗ move, then opponent does not block either of the premises. Note that opponent is absent in the case of mll⊤, that is, the game on mll⊤ sequents is a kind of solitaire game.

Polynomial-time verifier scores completed plays of the game. Various forms of the game differ in the way they are scored. The main objective of proponent is to never allow opponent to succeed in forcing an unprovable primitive sequent. However, in some forms of the game proponent will be more ambitious, that is, in addition to the main requirement, proponent will try to achieve the best score possible.

Let us first consider a simple version of the game against a randomized opponent, which can be described as an avg/max game played on mall sequents. The game may also be presented as a board game with tiles, where each tile is marked by a linear logic inference rule [20,21]. Proponent chooses the inference rule to be applied. In the case ⊗, proponent chooses a partition and requires both associated expressions to be evaluated. In the case ⊕, proponent chooses which of the two expressions will be evaluated. In the case &, opponent chooses by a fair coin toss which of the two expressions will be evaluated. In the case of a primitive sequent, verifier simply computes the value. Each sequent containing the constant ⊤, each identity axiom, and each primitive sequent containing only the constant 1 is scored 1 by verifier. All other primitive sequents are scored 0. Each completed play of the game is scored as the minimum of the scores of terminal sequents obtained in the play. Note that the number of moves is finite; indeed, it is polynomial in the size of a given mall sequent. Proponent wins when each encountered primitive sequent is an identity axiom or the constant 1.

Let us define the function μ, which represents the expected score when proponent plays optimally.

μ(Γ) = max{μ(Γ′; A) | Γ = Γ′, A},

μ(Γ; A ⊗ B) = max{min{μ(Δ, A), μ(Θ, B)} | Δ ⊎ Θ = Γ},

μ(Γ; A^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} B) = μ(Γ, A, B),

μ(Γ; A ⊕ B) = max{μ(Γ, A), μ(Γ, B)},

μ(Γ; A&B) = \frac{1}{2}[μ(Γ, A) + μ(Γ, B)],

μ(Γ; ⊥) = μ(Γ),

μ(Γ; ⊤) = 1,

μ(Σ) = \{ \begin{matrix}1 & if Σ is 1 or an axiom, \\ 0 & otherwise. \end{matrix}

Let us emphasize that, for any mall sequent ⊢ Ξ, the value μ(Ξ) is the maximum possible value satisfying these recursive conditions. Specifically, if any encountered sequent contains composite formulas, then several clauses regarding μ(Γ; A) might be applicable.

Proposition 2.1 A play of the simple linear logic proof game is won by proponent iff the score of the play is equal to 1. Furthermore, a mall sequent ⊢ Ξ is provable iff μ(Ξ) = 1. In addition, if ⊢ Ξ is unprovable and does not contain & , then μ(Ξ) = 0.

However, note that μ(Ξ) may be arbitrarily close to 1 if ⊢ Ξ is unprovable and contains & .

The more involved version of the linear logic proof game against a randomized opponent may also be presented as an avg/max game. The players’ moves and the winning condition are the same as in the simple game just described. However, in this version of the game, proponent also attempts to use as many certain preferred axioms as possible. Preferred axioms are, say, instances of a distinguished axiom ⊢ d, d^{⊥}, where the propositional atom d is fixed in advance. In this version, proponent gets one point for each instance of the distinguished axiom ⊢ d, d^{⊥} encountered in a play, but no points are awarded if a primitive sequent ⊢ Σ is any other identity axiom ⊢ p, p^{⊥}, or contains ⊤, or consists of a single constant ⊢ 1. If Σ is any other multiset of literals or constants, that is, a possibly empty multiset of literals or constants 1, 0 other than an identity axiom or the single constant 1, then proponent receives a penalty of −2^{n} points, where n is length of the original sequent. Each completed play of the game is now scored as the sum of the scores of primitive sequents obtained at the end of the play. Note that proponent wins iff this sum is ≥ 0.

Let us define the function ρ, which represents the expected score for proponent when proponent plays optimally.

ρ(Γ) = ρ_{n}(Γ), where n = |Γ|,

ρ_{n}(Γ) = max{ρ_{n}(Γ′; A) | Γ = Γ′, A},

ρ_{n}(Γ; A ⊗ B) = max{ρ_{n}(Δ, A) + ρ_{n}(Θ, B) | Δ ⊎ Θ = Γ}, ρ_{n}(Γ; A^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} B) = ρ_{n}(Γ, A, B), ρ_{n}(Γ; A ⊕ B) = max{ρ_{n}(Γ, A), ρ_{n}(Γ, B)},

ρ_{n}(Γ; A&B) = \frac{1}{2}[ρ_{n}(Γ, A) + ρ_{n}(Γ, B)],

ρ_{n}(Γ; ⊥) = ρ_{n}(Γ), ρ_{n}(Γ; ⊤) = 0,

ρ_{n}(Σ) = \{ \begin{matrix}1 & if Σ is the distinguished axiom d, d^{⊥}, \\ 0 & if Σ is another axiom or 1, \\ −2^{n} & otherwise. \end{matrix}

Observe again that, for any mall sequent ⊢ Ξ, the value ρ(Ξ) is the maximum possible value satisfying these recursive conditions, now with n fixed as the number of symbols in Ξ.

Proposition 2.2 A play of the advanced linear logic proof game is won by proponent iff the score of the play is ≥ 0. Furthermore, a mall sequent ⊢ Ξ is provable iff ρ(Ξ) ≥ 0. In addition, if ⊢ Ξ is unprovable and does not contain the distinguished propositional atom d, then ρ(Ξ) ≤ −1.

Game score functions considered here are intrinsic to the proof system mall. In particular, they are invariant with respect to certain permutability properties, important “structural” properties of mall. That is, our game score functions are invariant with respect to invertible inference rules of mall. The following theorem is proved by induction on the length of Γ.

Theorem 2.3 The following equalities hold, with n the length of the longest sequent in each equality:

• ρ_{n}(Γ, A ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} B) = ρ_{n}(Γ, A, B),

• ρ_{n}(Γ, A&B) = \frac{1}{2}(ρ_{n}(Γ, A) + ρ_{n}(Γ, B)),

• If Γ does not contain &, then ρ_{n}(Γ, A ⊕ B) = max{ρ_{n}(Γ, A), ρ_{n}(Γ, B)},

and similarly for μ.

Let us also observe that many isomorphisms of linear logic are respected by the score functions we consider, e.g., A ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} (B ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} C) ≅ (A ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} B) ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} C, A ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} (B&C) ≅ (A ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} B)&(A ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} C), A ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ⊥ ≅ A, A ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} B ≅ B ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} A, A ⊗ B ≅ B ⊗ A, A⊗1 ≅ A, A⊗(B ⊗C) ≅ (A⊗B)⊗C, etc. Notable exceptions are A&⊤ ≅ A and A&(B&C) ≅ (A&B)&C.

3 Lower bounds for local proof search

The optimal score functions μ, ρ and the corresponding optimal strategies for proponent are pspace-hard to compute on mall-formulas and np-hard to compute on mll⊤ formulas. It is shown in [21,22] that μ, ρ are just as hard to approximate. Some of these properties may be obtained directly from the np-hardness of mll⊤ and the pspace-hardness of mall, while others involve recent complexity-theoretic techniques for proving lower bounds on optimization problems [4,3,29,8,7,14].

But how hard is it to make one good move? In this paper we investigate lower bounds on such local proof search heuristics. For instance, we show that on mll⊤ formulas, it is np-hard to compute a local proof search heuristic up to any constant factor. That is, unless p=np, there is no polynomial-time strategy for proponent satisfying the condition that, given a mll⊤ formula A, chooses a proof rule instance so that even if one continued playing the game from then on optimally, the achieved score would be within a factor of the true optimal score ρ(A). Such a heuristic could be used to solve any problem in np in polynomial time. Therefore such heuristics are not likely to exist. Surprisingly, it suffices to require the above condition only for provable formulas A.

The formal definition follows.

Definition 3.1 Let ϵ ≥ 0. An ϵ-heuristic is a function from mall sequents to proof rule instances such that, for every formula A, the score h(A) achieved by playing optimally from then on is within multiplicative factor 1 + ϵ of the optimal achievable score OP T (A) ≥ 0, that is,

\frac{OP T (A)}{1 + ϵ} ≤ h(A) ≤ (1 + ϵ)OP T (A).

where in the simple version of the linear logic proof game, OP T (A) = μ(A), and in the advanced version on provable formulas, OP T (A) = ρ(A).

The following theorem is one of our main results.

Theorem 3.2 Let ϵ ≥ 0. If p≠np, then there are no polynomial-time computable ϵ-heuristics for the advanced linear logic proof game on provable mll⊤ formulas.

Note that the analog for the simple version of the game is false, because μ(A) = 1 for any provable formula A. On the other hand, the analogs for either version of the game on all mll⊤ formulas and even on the formulas of the pure multiplicative fragment mll, may be obtained from the np-hardness of mll⊤ and Propositions 2.1 and 2.2. Indeed, since the values of μ, ρ are discrete, it is just as hard to approximate them as it is to compute them exactly. But iterating the presumed polynomial-time heuristic would yield a close approximation of ρ computable in polynomial time.

Similar reasoning may also be used to obtain Theorem 3.2 when one considers formulas of the form

A^{#} = ((A ⊗ (d^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} d^{⊥})) ⊗ ⊥)^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ⊤,

where d is the distinguished propositional atom and A does not contain d. It is the case that ρ(A^{#}) = 1 if A is provable, else 0.

Furthermore, Theorem 3.2 may be strengthened to provable formulas of depth at most 2, by proof-theoretic techniques of Mints and Kanovitch.

In the simple version of the proof game, the reasoning outlined above cannot be directly lifted to mall, where the additive connectives & and ⊕ are allowed as well. The main obstruction is the exponential size of entire mall proof tree. A part of this difficulty has already been mentioned in the remark just after Proposition 2.1. However, instead of simply building the entire proof tree, one could sample into the tree of possible plays. In this context it makes sense to ask that a proof search heuristic be computable in randomized polynomial time, bpp.

Using such a heuristic, one could simulate a single play of the simple version of the proof game against a random opponent. Considering this play a single data point, one could repeatedly sample a polynomial number of times. The law of averages yields a bound on the likelihood of sampling error. The result is that such a heuristic may be used a polynomial number of times to closely approximate the optimal score, which is known to be pspace-hard [21,22].

Theorem 3.3 Let ϵ ≥ 0. If bpp≠pspace, then there are no bpp-computable heuristics for the simple linear logic proof game on mall formulas such that the expected score resulting from proponent’s use of the heuristic throughout a play of the game is within factor 1 + ϵ from the optimal score.

Theorem 3.3 relies on recent complexity-theoretic techniques for proving pspace lower bounds on optimization problems [5,12,23,28,8,7]. In contrast, the following theorem may be obtained directly from the pspace-hardness of mall and Proposition 2.2.

Theorem 3.4 Let ϵ ≥ 0. Every ϵ-heuristic H for the advanced linear logic proof game on provable mall formulas is pspace-hard. That is, for any language L ∈ pspace and its associated characteristic function χ_{L}, there exist polynomial-time computable functions f_{1}, f_{2} such that χ_{L} = f_{2} ∘ H ∘ f_{1}.

Note: pspace-hardness of approximation was defined in [7] and [8] in terms of Turing-reducibility, but their results as well as ours satisfy the more restrictive notion of many-one reducibility used here. It is possible that some approximation problems exist that are pspace-hard with respect to Turing reductions, but not many-one reducibility. For such problems many-one reducibility would be too restrictive, but for all the problems we have studied many-one reducibility is more convenient.

4 Further Work

We do not know any significant positive results on proof search heuristics in linear logic. It is not known whether any problem related to linear logic proof search heuristics is maxsnp-complete in the sense of [25,24]. Optimization problems in maxsnp, such as max-sat, the problem of maximizing the number of satisfiable clauses in a 3-CNF, have a nontrivial approximation threshold c > 1 in the sense that they are polynomial-time approximable within any factor c′ > c, but np-hard to approximate within any factor 1 < c′ < c.

Another potential direction for future research is the consideration of average case complexity, that is, the possibility of proof search heuristics that on “most” sequents provide a good next proof rule instance, but on some sequents choose a rather poor proof rule instance. Our results above only show that it is hard to build a heuristic that is “never too bad,” while one may be as interested in heuristics that are “usually good.”

Acknowledgements: We would like to thank Madhu Sudan for directing us to [8,7], and Sanjeev Arora and Joan Feigenbaum for their informative conversations and email correspondence and for their help with the literature. Scedrov would like to thank Stanford University and SRI International for hosting him during his visit in the spring of 1995. Patrick Lincoln would like to thank the University of Pennsylvania for hosting him during his visit in the spring of 1996. All the authors would like to thank the Isaac Newton Institute for Mathematical Sciences in Cambridge, England for hosting them at various times in the fall of 1995.

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