The True Concurrency of Herbrand’s Theorem

Herbrand’s theorem, widely regarded as a cornerstone of proof theory, exposes some of the constructive content of classical logic. In its simplest form, it reduces the validity of a ﬁrst-order purely existential formula to that of a ﬁnite disjunction. In the general case, it reduces ﬁrst-order validity to propositional validity, by understanding the structure of the assignment of ﬁrst-order terms to existential quantiﬁers, and the causal dependency between quantiﬁers. In this paper, we show that Herbrand’s theorem in its general form can be elegantly stated and proved as a theorem in the framework of concurrent games, a denotational semantics designed to faithfully represent causality and independence in concurrent systems, thereby exposing the concurrency underlying the computational content of classical proofs. The causal structure of concurrent strategies, paired with annotations by ﬁrst-order terms, is used to specify the dependency between quantiﬁers implicit in proofs. Furthermore concurrent strategies can be composed, yielding a compositional proof of Herbrand’s theorem, simply by interpreting classical sequent proofs in a well-chosen denotational model.


Introduction
"What more do we know when we have proved a theorem by restricted means than if we merely know it is true?" Kreisel's question is the driving force for much modern Proof Theory.This paper is concerned with Herbrand's Theorem, perhaps the earliest result in that direction.It is a simple consequence of completeness and compactness in first-order logic.So it is an example of information being extracted from the bare fact of provability.Usually by contrast one thinks in terms of extracting information from the proofs themselves, typically -as in Kohlenbach's proof mining -via some form of functional interpretation.This has the advantage that information is extracted compositionally in the spirit of functional programming.Specifically information for A and A → B can be composed to give information for B; or, in terms of the sequent calculus, we can interpret the cut rule.It seems to be folklore that there is a problem for Herbrand's Theorem.That is made precise in Kohlenbach [17] which shows that one cannot hope directly to use collections of Herbrand terms for A and A → B to give a collection for B. That leaves the possibility of making some richer data compositional, realised indirectly in Gerhardy and Kohlenbach [11] with data provided by Shoenfield's version [30] of Gödel's Dialectica Interpretation [14].Now functional interpretations make no pretence to be faithful to the structure of proofs as encapsulated in systems like the sequent calculus: they explore in a sequential order terms proposed by a proof as witnesses for existential quantifiers, but this order is certainly not intrinsic to the proof.Thus it is compelling to seek some compositional form of Herbrand's Theorem faithful to the structure of proofs and to the dependency between terms; for cut-free proofs, Miller's expansion trees [24] capture precisely this "Herbrand content" (the information pertaining to quantifier instantiations), but not compositionally.
In this paper, we provide such a compositional form of Herbrand's theorem, presented as a game semantics for first-order classical logic.Our games have two players, both playing on the quantifiers of a formula ϕ. ∃loïse, playing the existential quantifiers, defends the validity of ϕ. ∀bélard, playing the universal quantifiers, attempts to falsify it.This understanding of formulas as games is folklore in mathematical logic and computer science.However, like functional interpretations, such games are usually sequential [7,19].In contrast, our model captures the exact dependence and independence between quantifiers.To achieve that we build on concurrent/asynchronous games [23,27,4], which marry game semantics with the so-called true concurrency approach to models of concurrent systems, and avoid interleavings.So in a formal sense, our model highlights a parallelism inherent to classical proofs.In essence, our strategies are close to expansion trees enriched with an explicit acyclicity witness.
The computational content of classical logic is a longstanding active topic, with a wealth of related works, and it is hard to do it justice in this short introduction.There are, roughly speaking, two families of approaches.On the one hand, some (including the functional interpretations mentioned above) extract from proofs a sequential procedure, e.g.via translation to sequential calculi or by annotating a proof to sequentialize or determinize its behaviour under cut reduction [13,8].Other than that cited above, influential developments in this "polarized" approach include work by Berardi [2], Coquand [7], Parigot [26], Krivine [18], and others.Polarization yields better-behaved dynamics and a non-degenerate equational theory, but distorts the intent of the proof by an added unintended sequentiality.On the other hand, some works avoid polarization -including, of course, Gentzen's Hauptsatz [10].This causes issues, notably unrestricted cut reduction yields a degenerate equational theory [13] and enjoys only weak, rather than strong, normalization [8].Nevertheless, witness extraction

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Figure 1 An expansion tree and winning Σ-strategy for DF.
Figure 2 A partially ordered winning Σ-strategy.
The arena DF ∃ .
Example 2. Consider the formula ψ = ∃x¬P(x)∨P(f(x)) (where where c is some constant symbol. A similar disjunction property holds for general formulas, though it is harder to state.A common way to do so is by reduction to the above: a formula ϕ is converted to prenex normal form and universally quantified variables are replaced with new function symbols added to Σ, in a process called Herbrandization (dual to Skolemization).For instance, the drinker's formula (DF): ∃x∀y¬P(x) ∨ P(y), yields by Herbrandization the formula ψ of Example 2.
Instead, to avoid prenexification and Skolemization and the corresponding distortion of the formula, one may adopt a representation of proofs that displays the instantiation of existential quantifiers with finitely many witnesses while staying structurally faithful to the original formula.To that end Miller proposes expansion trees [24].They can be introduced via a game-theoretic metaphor, reminiscent of [7].Two players, ∃loïse and ∀bélard, debate the validity of a formula.On a formula ∀xϕ, ∀bélard provides a fresh variable x and the game keeps going on ϕ.On ∃xϕ, ∃loïse provides a term t, possibly containing variables previously introduced by ∀bélard.∃loïse, though, has a special power: at any time she can backtrack to a previous existential position, and propose a new term.Figure 1 (left) shows an expansion tree for DF.It may be read from top to bottom, and from left to right: ∃loïse plays c, then ∀bélard introduces y, then ∃loïse backtracks (we jump to the right branch) and plays y, and finally ∀bélard introduces z. ∃loïse wins: the disjunction of the leaves is a tautology.
However the metaphor has limits, it suggests a sequential ordering between branches, which expansion trees do not have in reality: the order is only implicit in the term annotations.Besides, the natural ordering between quantifiers induced by terms is not always sequential.It is, of course, always acyclic -on expansion trees this is ensured by an acyclicity correctness criterion, whose necessity is made obvious by the (incorrect) expansion tree of Figure 3 "proving" a falsehood.This acyclicity entails the existence of a sequentialization, but committing to one is an arbitrary choice not forced by the proof.
A partial order is much more faithful to the proof.In this paper, we show that expansion trees can be made compositional modulo a change of perspective: rather than derived we consider this order primitive, and only later decorate it with term annotations.For instance, we display in Figure 2 the formal object, called a (sequential) winning Σ-strategy, matching in our framework the expansion tree for DF.Another winning Σ-strategy, displayed in Figure 2, illustrates that this order is not always naturally sequential.By lack of space we do not define expansion trees here, though they are captured in essence by our strategies.

Expansion trees as winning Σ-strategies
We now introduce our formulation of expansion trees as Σ-strategies.Although our definitions look superficially very different from Miller's, the only fundamental difference is the explicit display of the dependency between quantifiers.Σ-strategies will be certain partial orders, with elements either "∀ events" or "∃ events".Events will carry terms, in a way that respects causal dependency.Σ-strategies will play on games representing the formulas.The first component of a game is its arena, that specifies the causal ordering between quantifiers.A configuration of an arena (or any partial order) is a down-closed set of events.We write C ∞ (A) for the set of configurations of A, and C (A) for the set of finite configurations.

Definition 3. An arena is
The arena only describes the moves available to both players; it says nothing about terms or winning.Similarly to expansion trees where only ∃loïse can replicate her moves, our arenas will at first be biased towards ∃loïse: each ∃ move exists in as many copies as she might desire, whereas ∀ events are a priori not copied.Figure 4 shows the ∃-biased arena DF ∃ for DF.The order is drawn from top to bottom.Although only ∃loïse can replicate her moves, the universal quantifier is also copied as it depends on the existential quantifier.
Strategies on an arena A will be certain augmentations of prefixes of A. They carry causal dependency between quantifiers induced by term annotations, but not the terms themselves.
For any partial order A and a 1 , a 2 ∈ |A|, we write a 1 A a 2 (or a 1 a 2 if A is clear from the context) if a 1 < A a 2 with no other event in between -this notation was used implicitly in Figures 1 and 2. We call immediate causal dependency.

Definition 4.
A strategy σ on arena A, written σ : A, is a partial order (|σ|, ≤ σ ) with |σ| ⊆ |A|, such that for all a ∈ |σ|, [a] σ is finite (an elementary event structure); subject to: These strategies are essentially the receptive ingenuous strategies of Melliès and Mimram [23], though their formulation, with a direct handle on causality, is closer to Rideau and Winskel's later concurrent strategies [27].Receptivity means that ∃loïse cannot refuse to acknowledge a move by ∀bélard, and courtesy that the only new causal constraints that she can enforce with respect to the game is that some existential quantifiers depend on some universal quantifiers.Ignoring terms, Figure 2 (right) displays a strategy on the arena of Figure 4 -in Figure 2 we also show via dotted lines the immediate dependency of the arena.
Let us now add terms, and define Σ-strategies.

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The True Concurrency of Herbrand's Theorem Definition 5. A Σ-strategy on arena A is a strategy σ : A, with a labelling function . Rather than having ∀ moves introduce fresh variables, we consider them as variables themselves.Hence, the ∃ moves carry terms having as free variables the ∀ moves in their causal history.For instance the diagram of Figure 1 (right) is meant formally to denote the one on the right (where superscripts are the terms given by λ).In the sequel we omit the (redundant) annotation of ∀bélard's events.
Besides the fact that they are not assumed finite, Σ-strategies are more general than expansion trees: they have an explicit causal ordering, which may be more constraining than that given by the terms.A Σ-strategy σ : A is minimal iff whenever a 1 σ a 2 such that a 1 ∈ fv(λ σ (a 2 )), then a 1 A a 2 as well.In a minimal Σ-strategy σ : A, the ordering ≤ σ is actually redundant and can be uniquely recovered from λ σ and ≤ A .Now, we adjoin winning conditions to arenas and define winning Σ-strategies.As in expansion trees, we aim to capture that the substitution (by terms from the strategies) of the expansion of the original formula is a tautology.
expressing winning conditions, where QF ∞ Σ (x) denotes the infinitary quantifier-free formulasobtained from QF Σ (x) by adding infinitary connectives i∈I ϕ i and i∈I ϕ i , with I countable.
For a game interpreting a formula ϕ, the winning conditions associate configurations of the arena ϕ with the propositional part of the corresponding expansion of ϕ.For instance: recalling that the arena for DF appears in Figure 4.In the second clause, (the true formula) comes from ∀bélard not having played ∀ 6 yet, yielding victory to ∃loïse on that copy.The winning conditions yield syntactic, uninterpreted formulas: we keep the second formula as-is although it is equivalent to .Finally, we can define winning strategies.

Constructions on games and Herbrand's theorem
To complete our statement of Herbrand's theorem with Σ-strategies, it remains to set the interpretation of formulas as games.To that end we introduce a few constructions on games, first at the level of arenas and then enriched with winning conditions.We write ∅ for the empty arena.If A is an arena, A ⊥ is its dual, with same events and causality but polarity reversed.We review some other constructions.
Definition 8.The simple parallel composition A 1 A 2 of A 1 and A 2 has as events the tagged disjoint union {1} × |A 1 | {2} × |A 2 |, as causal order that given by (i, a) ≤ A1 A2 (j, a ) iff i = j and a ≤ Ai a , and, as polarity pol A1 A2 ((i, a)) = pol Ai (a).

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This construction has a general counterpart i∈I A i with I at most countable, defined likewise.In particular we will later use the uniform countably infinite parallel composition ω A. Another important construction is prefixing.
Now, let us enrich these with winning, yielding the constructions on games used for interpreting formulas.Importantly, the inductive interpretation of formulas requires us to consider formulas with free variables.For V a finite set, a V-game is defined as a game A (Def. 6), except that winning may also depend on V: . We now define all our constructions, on V-games rather than games.The duality (−) ⊥ extends to V-games, simply by negating the winning conditions: for all x ∈ C ∞ (A), The of arenas gives rise to two constructions, ⊗ and `, on V-games: Definition 10.For A and B V-games, we define two V-games with arena A B and winning conditions Note the implicit renaming so that ) respectively -we will often keep such renamings implicit.
Observe that ⊗ and `are De Morgan duals, i.e., (A ⊗ B) ⊥ = A ⊥ `B⊥ .We write these operations ⊗ and `rather than ∧ and ∨, because they behave more like the connectives of linear logic [12] than those of classical logic; for each V the ⊗ and `will form the basis of a * -autonomous structure and hence a model of multiplicative linear logic (see Section 3).
To interpret classical logic however, we will need replication.
Definition 11.For V-game A, we define the V-games !A, ?A with arena ω A and winning: ) is an infinite conjunction (resp.disjunction), it simplifies to a finite one when x visits finitely many copies (with cofinitely many copies of W A (∅)).
Next we show how V-games support quantifiers.

Definition 12.
Let A a (V {x})-game, we define the V-game ∀x.A and its dual ∃x.A with arenas ∀.A and ∃.A respectively, with W ∀x.A (∅) = , W ∃x.A (∅) = ⊥, and: formulas, reflecting the bias towards ∃loïse.This is indeed compatible with the examples given previously.We can now state our concurrent version of Herbrand's theorem.
Besides the game-theoretic language, the difference with expansion trees is superficial: on ϕ, expansion trees essentially coincide with the minimal top-winning Σ-strategies σ : ϕ ∃ .The effort to change view point, from a syntactic construction to a (game) semantic one, will however pay off now, when we show how to compose Σ-strategies.

Compositional Herbrand's theorem
Unlike expansion trees, strategies can be composed.Whereas Theorem 13 above could be deduced via the connection with expansion trees, that proof would intrinsically rely on the admissibility of cut in the sequent calculus.Instead, we will give an alternative proof of Herbrand's theorem where the witnesses are obtained truly compositionally from any sequent proof, without first eliminating cuts.In other words, strategies will come naturally from the interpretation of the classical sequent calculus in a semantic model.
That (3) implies (2) is more subtle: first, one may restrict a winning σ : ϕ to ϕ ∃ to obtain a finite top-winning strategy.However, this top-winning strategy may not be finite.Yet, it follows by compactness that there is always a finite top-winning sub-strategy that may be effectively computed from σ. See the Appendix C for details.
The proof that (1) implies (3) is our main contribution: a winning strategy will be computed from a proof using our denotational model of classical proofs.

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Our source sequent calculus (Figure 6) is fairly standard, one-sided, with rules presented in the multiplicative style.A notable variation is that sequents carry a set V of free variables, that may appear freely in formulas.The introduction rule for ∀ introduces a fresh variable, whereas the introduction rule for ∃ provides a term whose free variables must be in V.
What mathematical structure is required to interpret this sequent calculus?Ignoring the V annotations, the first group is nothing but Multiplicative Linear Logic (MLL).Propositional (V-)MLL can be interpreted in a * -autonomous category [3].Accordingly, in Section 3, we first construct a * -autonomous category Ga of games and winning Σ-strategies.Then, in Section 4, we build the structure required for the interpretation of quantifiers, still ignoring contraction and weakening.For each set of variables V we construct a * -autonomous category V-Ga, with a fibred structure to link the V-Ga together for distinct Vs and suitable structure to deal with quantifiers, obtaining a model of first-order MLL.Finally in Section 5 we complete the interpretation by adding the exponential modalities from linear logic to the interpretation of quantifiers, and get from that an interpretation of contraction and weakening.

A * -autonomous category
The following theorem, on cut reduction for MLL, is folklore.
There is a set of reduction rules on MLL sequent proofs, written MLL , such that for any proof π of a sequent Γ, there is a cut-free π of Γ such that π * MLL π .
The reduction MLL comprises logical reductions, reducing a cut on a formula ϕ/ϕ ⊥ , between two proofs starting with the introduction rule for the main connective of ϕ/ϕ ⊥ ; and structural reductions, consisting in commutations between rules so as to reach the logical steps.We assume some familiarity with this process.
In this section we aim to give an interpretation of MLL proofs, which should be invariant under cut-elimination.Categorical logic tells us that this is essentially the same as producing a * -autonomous category.We opt here for the equivalent formulation by Cockett and Seely as a symmetric linearly distributive category with negation [6].

Definition 16.
A symmetric linearly distributive category is a category C with two symmetric monoidal structures (⊗, 1) and (`, ⊥) which distribute: there is a natural δ A,B,C : A⊗(B `C) C → (A⊗B)`C, the linear distribution, subject to coherence conditions [6].A symmetric linearly distributive category with negation also has a function (−) ⊥ on objects and families of maps Note also the degenerate case of a compact closed category, which is a symmetric linearly distributive category where the monoidal structures (⊗, 1) and (`, ⊥) coincide.
Abusing terminology, we will refer to symmetric linearly distributive categories with negation by the shorter * -autonomous categories.This should not create any confusion in the light of their equivalence [6].If C a * -autonomous category comes with a choice of P(t 1 , . . ., t n ) (an object of C) for all closed literal, then this interpretation can be extended to all closed quantifier-free formulas following Figure 5.For all such ϕ, we have ϕ ⊥ = ϕ ⊥ .
The interpretation of MLL proofs in a * -autonomous category C is standard [29]: a proof π of a MLL sequent ϕ 1 , . . ., ϕ n is interpreted as a morphism π : Furthermore, the categorical laws make this interpretation invariant under cut reduction.
¤( ( 8 So a proof has the same denotation as its cut-free form obtained by Theorem 15.In the rest of this section we construct a concrete * -autonomous category of games and winning Σ-strategies; supporting the interpretation of MLL.This is done in three stages: first we focus on composition of Σ-strategies (without winning), then we extend this to a compact closed category.Finally, adding back winning, we split into two ⊗ and `, and prove * -autonomy.

Composition of Σ-strategies
We construct a category Ar Σ having arenas as objects, and as morphisms from A to B the Σ-strategies σ : A ⊥ B, also written σ : + G G C will be computed in two stages: first, the interaction τ σ is obtained as the most general partial-order-with-terms satisfying the constraints given by both σ and τ -Figure 7 displays such an interaction.Then, we will obtain the composition τ σ by hiding events in B. In the example of Figure 7 we get the single annotated event ∃ f(g(c),h(c)) 5 . We fix some definitions on terms and substitutions.
) for the substitution operation.Substitutions form a category S, which is cartesian: the empty set ∅ is terminal, and the product of V 1 and V 2 is their disjoint union Consider first the closed interaction of two Σ-strategies σ : A and τ : A ⊥ .As they disagree on the polarities on A we drop themτ σ will be a neutral Σ-strategy on a neutral arena: Definition 18.A neutral arena is an arena, without polarities.Neutral strategies σ : A, are defined as in Definition 4 without (2), (3).Neutral Σ-strategies additionally have λ σ : (s ∈ |σ|) → Tm Σ ([s] σ ), and are idempotent: for all a ∈ |a|, λ σ (a)[λ σ ] = λ σ (a).
Forgetting polarities, every Σ-strategy is a neutral one.Given σ and τ , τ σ is a minimal strengthening of σ and τ , regarding both the causal structure and term annotations, i.e., a meet for the partial order (idempotence above is required for it to be antisymmetric): , and for all x ∈ C (|σ|), λ τ x subsumes λ σ x (regarded as substitutions x S → x).
Ignoring terms, any two σ and τ have a meet σ ∧ τ ; this is a simplification of the pullback in the category of event structures, exploiting the absence of conflict [31].The partial order (|σ ∧ τ |, ≤ σ∧τ ) has events all common moves of σ and τ with a causal history compatible with both ≤ σ and ≤ τ , and for ≤ σ∧τ the minimal causal order compatible with both.
However, two neutral Σ-strategies do not necessarily have a meet for (see Example 45 in Appendix A).Hence, we focus on the meets occurring from compositions of Σ-strategies and show that for σ : A and τ : A ⊥ dual Σ-strategies the meet does exists: Variables appearing in λ τ σ cannot be events in B -they must be negative in A ⊥ C.So we can define τ σ = (τ σ) ∩ (A C) the restriction of τ σ to A C, with same causal order and term annotation.The pair (|τ σ|, ≤ τ σ ) is a strategy, as an instance of the constructions in [4], and this extends to terms so that τ σ : A ⊥ C is a Σ-strategy, the composition of σ and τ .Because interaction is defined as a meet for , it follows that it is compatible with it, i.e., if σ σ , then τ σ τ σ .This is preserved by projection, and hence τ σ τ σ as well.This compatibility of composition with will be used later on, together with the easy fact that is more constrained on Σ-strategies: Lemma 21.For σ, σ : A Σ-strategies, if σ σ , then λ σ (s) = λ σ (s) for all s ∈ |σ|.
To complete our category, we also define the copycat strategy.

Definition 22. For an arena A, the copycat Σ-strategy c c
The proof of categorical laws are variations on construction of the bicategory in [4].

Proposition 23.
There is a poset-enriched category Ar Σ with arenas as objects, and Σ-strategies as morphisms.

Compact closed structure
We show that Ar Σ is compact closed.The tensor product of arenas A and B is A B. For Σ-strategies σ 1 : for the obvious renaming.It is not difficult to prove:

Proposition 24. Simple parallel composition yields an enriched functor : Ar
For the compact closed structure, we elaborate the renaming used above.We write f : A ∼ = B for an isomorphism of arenas, preserving and reflecting all structure.
We use this to define the structural morphisms for the symmetric monoidal structure of Ar Σ .For instance, the iso α A,B,C : The other structural morphisms arise similarly.Coherence and naturality then follows from the key copycat lemma: As a corollary we get coherence for the structural morphisms (following from those on isomorphisms), and naturality.For all A we get η A : ∅ Ar Σ + G G A ⊥ A and A : A A ⊥ Ar Σ + G G ∅ as the obvious renamings of copycat.Checking the law for compact closed categories is a variation of the idempotence of copycat.Overall: Proposition 27.Ar Σ is a poset-enriched compact closed category.

A linearly distributive category with negation
Finally, we reinstate winning conditions.We first note: Proposition 28.There is a (poset-enriched) category Ga Σ with objects the games (Definition 6) on Σ, and morphisms Σ-strategies σ : A ⊥ `B, also written σ : That copycat is winning boils down to the excluded middle.That τ σ : A ⊥ `C is winning if σ : A ⊥ `B and τ : B ⊥ `C are, is as in [5]: It suffices to check winning, which is straightforward.It remains to prove that all structural morphisms from Ar Σ (copycat strategies) are winning, which boils down to the following sufficient conditions to hold: For A, B games, a win-iso f : A → B is an iso This easily entails that all structural morphisms (including linear distributivity) are winning.Finally η A :

A model of first-order MLL
We move on to MLL 1 , i.e., all rules except for contraction and weakening.Before developing the interpretation, we discuss cut elimination.There are three new cut reduction rules, displayed in Figure 8: the new logical reduction (∀/∃), and two for the propagation of cuts past introduction rules for ∀ and ∃.Writing π MLL1 π for the reduction obtained with these new rules together with MLL : Proposition 32.Let π be any MLL 1 proof of V Γ.Then, there is a cut-free proof π of The first rule of Figure 8 requires the introduction of substitution on proofs.In general, for a proof π of V2 Γ and γ : by propagating γ through π, substituting formulas and terms.A degenerate case of this is the substitution of a proof π of V Γ by weakening w V,x : As this leaves the formulas and terms unchanged we leave it implicit in the reduction rulesit is used for instance implicitly in the commutation Cut/∀.
Substitution is key in the cut reduction of quantifiers.However it is best studied independently of quantifiers, in a model of V-MLL (see Figure 6).This is the topic of the next subsection, prior to the interpretation of the introduction rules for quantifiers.

A fibred model of V-MLL
Following [20,28], we expect to model V-MLL and substitution in: Definition 33.Let * -Aut be the category of * -autonomous categories and functors preserving the structure on the nose.A strict S-indexed * -autonomous category is a functor T : S op → * -Aut.
Such definitions (e.g.hyperdoctrines [28]) are usually phrased only up to isomorphism; for simplicity we opt here for a lighter definition.Writing V n = {x 1 , . . ., x n }, we say that T supports Σ if for every predicate symbol P of arity n there is P Vn a chosen object of T (V n ).For t 1 , . . ., t n ∈ Tm Σ (V) we can then set For any finite V, this lets us interpret V-MLL in T (V) as in Section 3.Besides V-MLL in isolation, this also models substitutions.In games the functorial action of T on γ : V 1 → V 2 will correspond to substitution on games A[γ] = T (γ)(A) and strategies σ[γ] = T (γ)(σ).This matches syntactic substitution, as T (γ) preserves the * -autonomous structure.
Let us now introduce the concrete structure.For any finite V, the fibre T (V) is the category Ga Σ V built in Section 3, on the extended signature Σ V. Recall that its objects C S L 2 0 1 8

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The True Concurrency of Herbrand's Theorem are games on the signature Σ V, i.e., the V-games of Section 2.3.Morphisms between V-games A and B are winning (Σ V)-strategies on A ⊥ `B regarded as a game on signature Σ V -also called winning Σ-strategies on the V-game A ⊥ `B.
Finally, for A a V 2 -game and γ : V 1 → V 2 a substitution, the game T (γ)(A) = A[γ] is defined as having arena A, and, for . Likewise, given A and B two V-games and σ : A ⊥ `B, σ[γ] has the same components as σ, but term annotations It is a simple verification to prove: ) is a strict * -autonomous functor preserving the order.

Quantifiers
Finally, we give the interpretation of ∀I and ∃I.For now, we consider a linear interpretation − of formulas defined like − ∃ V except for ∃xϕ V = ∃x.ϕ V .Besides preserving the * -autonomous structure, substitution also propagates through quantifiers, from which we have: This will be used implicitly from now on.The definition of quantifiers on games of Definition 12 extends to functors : (∀x.A) ⊥ `∀x.B plays copycat on the initial ∀, then plays as σ (similarly for ∃ V,x (σ)).Following Lawvere [20], one expects adjunctions ∃ V,x T (w V,x ) ∀ V,x .Unfortunately, this fails -we present this failure later as the non-preservation of Cut/∀ .
We now interpret ∀I and ∃I.First, we give a strategy introducing a witness t.

In other words, ∃ t
A plays ∃ annotated with t, then proceeds as copycat on A. We have: Proposition 37. Let A be a V-game, and t ∈ Tm Σ (V).Then, We interpret ∃I by post-composing with ∃ t A (as in Figure 10 without the last step).This validates Cut/∃ , by associativity of composition.
To a strategy σ, the operation interpreting ∀I adds ∀ as new minimal event, and sets it as a dependency for all events whose annotation comprise the distinguished variable x.Definition 38.For σ a (Σ V {x})-strategy on A ⊥ B, the (Σ V)-strategy ∀I x A,B (σ) : Indeed, if ∀bélard does not play (2, ∀) we get a tautology, otherwise the remaining configuration is in σ and so is tautological.This completes the interpretation of MLL 1 .This interpretation leaves ∀/∃ invariant, but fails Cut/∀ .This stems from the fact that the minimal Σ-strategies are not stable under composition (see Example 46 in Appendix A).The interpretation of cut-free proofs yield minimal Σ-strategies.In contrast, in compositions 5:15 interpreting cuts, causality may flow through the syntax tree of the cut formula, and create causal dependencies not reflected in the variables.Hence, cut reduction may weaken the causal structure.
Lemma 40.For σ : A Ar Σ + G G B and τ : B By Lemma 21 these two have the same terms on common events.In fact, ∀I x A,C (τ σ) and ∀I x B,C (τ ) σ also have the same events -they correspond to the same expansion tree, only the acyclicity witness differs.But the variant of with |σ 1 | = |σ 2 | is not a congruence: relaxing causality of σ in τ σ may unlock new events, previously part of causal loops.
As is preserved by all operations on Σ-strategies, we deduce: For MLL 1 , we conjecture that "having the same expansion tree" (i.e., same events and term annotations) is actually a congruence, yielding a * -autonomous hyperdoctrine.As this would not hold in the presence of contraction and weakening, we leave this for future work.

Contraction and weakening
In this section we reinstate ! and ? in the interpretation of quantifiers, i.e., ∀x.ϕ V = !∀x. ϕ V {x} and ∃x ϕ V = ?∃xϕ V x -this is reminiscent of Melliès' discussion on the interaction between quantifiers and exponential modalities in a polarized setting [22].Unlike for MLL 1 , we only aim to map proofs to Σ-strategies on the appropriate game, with no preservation of reduction.We must interpret contraction and weakening, but also revisit the interpretation of rules for quantifiers as the interpretation of formulas has changed.
Weakening is easy: for any game A, any Σ-strategy σ : A + G G 1 is winning; for definiteness, we use the minimal e A : A + G G 1, only closed under receptivity.Contraction is much more subtle.To illustrate the difficulty, we present in Figure 9 two simple instances of the contraction Σ-strategy (without term annotations).The first looks like the usual contraction of AJM games [1].It can be used to interpret the contraction rule on existential formulas, where it has the effect of taking the union of the different witnesses proposed.But in LK, one can also use contraction on a universal formula, which will appeal to a strategy like the second.Any witness proposed by ∀bélard will then have to be propagated to both branches to ensure that we are winning (mimicking the effect of cut reduction).
In order to define this contraction Σ-strategy along with the tools to revisit the introduction rules for quantifiers, we will first study some properties of the exponential modalities.
Recall ! and ?from Definition 11, both based on arena ω A. First, we examine their functorial action.Let σ : Rather than defining directly the contraction, we build co ϕ : ϕ V For ϕ quantifier-free, the empty co ϕ : ϕ V + G G ! ϕ V is winning.We C S L 2 0 1 8

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The True Concurrency of Herbrand's Theorem Figure 11 Some win-isos with exponentials whose lifting are used in the interpretation.11.We get co ϕ∧ψ and co ϕ∨ψ by induction and composition with !
Finally, co ?∃x. ϕ is obtained analogously to the contraction on the right of Figure 9.
Lemma 43.For any (V {x})-game A, there is a winning µ A,x : ∃x.!A Proof.After the unique minimal ∀ move (on the left hand side), the strategy simultaneously plays all the (i, ∃) (on the right hand side) with annotation ∀; then proceeds as c c !A .
We get co ?∃x. ϕ x by induction, post-composition with ?µ ϕ ,x and distribution of ?over !.
Proposition 44.For any ϕ ∈ Form Σ (V), there is a winning co ϕ Combining Proposition 44 with other primitives (including !A + G G A, playing copycat between A and the 0 th copy on the left, closed under receptivity), we get δ ϕ We complete the interpretation in Figure 10, omitting W, which is by post-composition with e A and silently using the isomorphism between winning Σ-strategies from 1 to Γ `A and from Γ ⊥ to A. This concludes the proof of Theorem 14.

Conclusion
For LK there is no hope of preserving unrestricted cut reduction without collapsing to a boolean algebra [13].There are non-degenerate models for classical logic with an involutive negation, e.g.Führman and Pym's classical categories [9] with reduction only preserved in a lax sense; but our model does not preserve reduction even in this weaker sense.Besides, our semantics is infinitary: from the structural dilemma in [8] we obtained a proof of some ∃x.ϕ with ϕ quantifier-free (no ∀bélard moves) yielding an infinite Σ-strategy (see Appendix B).
Both phenomena could be avoided by adopting a polarized model, abandoning however our faithfulness to the raw Herbrand content of proofs.It is a fascinating open question whether one can find a non-polarized model of classical first-order logic that remains finitary -this is strongly related to the actively investigated question of finding a strongly normalizing reduction strategy on syntaxes for expansion trees [15,21,16].

A Counter-examples
In this section, we detail a few counter-examples referred to in the main text., have no meet.
Assume they have a meet σ.Necessarily, since e e1 1 e e2 2 σ 1 , σ 2 , then σ must comprise the events-with-annotations e e1 1 and e e2 2 .But we also have for any constant symbol c.Therefore, σ must also include event-with-annotation e t 3 .But t must be an instance of f (e 1 ), f (e 2 ); and must instantiate to f(c) for all constant symbol c.So t must have the form f(e) for some e ∈ [e 3 ], i.e., e ∈ {e 1 , e 2 , e 3 }.It is direct to check that none of those options gives a neutral Σ-strategy that is below both σ 1 and σ 2 for .

Example 46. Consider σ
where we omit the annotation of negative events, forced by Σ-receptivity.
Their composition has ∀ 4 ∃ c 1 , which is not a minimal strategy since c does not have ∀ 4 as a free variable.This counter-example also means that we do not have the adjunction expected from categorical logic ∃ V,x T (w V,x ) ∀ V,x .More precisely, Lemma 40 cannot be strengthened into an equality.Indeed, note that τ = ∀I x (∀2∀31),1 (∃ x 2 ∃ c 3 ).On the other hand, τ σ = ∀ 4 ∃ c 1 , which cannot be of the form ∀I x ∀11,1 -this construction would put no causal link from ∀ 4 to ∃ c 1 , since c does not involve the variable x.The intuition behind this failure is that ∀I x A,B only introduces causal links that follow occurrences of a variable x.However, after composition, we may end up with Σ-strategies that are not minimal, i.e., they have immediate causal links not reflecting directly a syntactic dependency.In other words, in order to get an adjunction as one would expect, only the term information would have to be retained -but our interpretation remembers more.

B Non-finiteness of the interpretation
From the infinitary primitives in the interpretation, it is natural to expect the interpretation to be infinitary.It was surprisingly difficult to find such an example, however one can do so by revisiting standard pathological examples in the proof theory of classical logic, having arbitrarily large normal forms.More precisely, we construct an LK proof of the formula ∃x.whose interpretation is infinite, despite the fact that there is no move by ∀bélard in the game.
Our starting point is the following proof: This proof is referred to in [8] as a structural dilemma.There are two ways to push the Cut beyond contraction, as the two proofs interact, and try to duplicate one another.This is an example of a proof where unrestricted cut reduction does not necessarily terminate; and which has infinitely large cut-free forms.
In order to construct a proof with an infinite interpretation, we will start with this proof, with ϕ = ∀x.⊥ ∨ ∃y., which to shorten notations we will just write as ∀ ∨ ∃.
Omitting details, here is the interpretation of the left branch of (we omit term annotations, which always coincide with the unique predecessor for ∃loïse's moves).

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The True Concurrency of Herbrand's Theorem The second branch of is symmetric, so we do not make it explicit.Now, we interpret the Cut rule and the composition yields below.
It is interesting to note that although has arbitrarily large cut-free forms, the corresponding strategy only plays finitely many ∃loïse moves for every ∀bélard move.However, we are on the right path to finding an infinitary Σ-strategy.
The next step is to set (with s some unary function symbol) the proof 2 below with interpretation We now use these to compute the interpretation of 3 , a cut between and 2 : We are almost there.It suffices now to note that 3 provides a proof of (∃x.=⇒ ∃x. ) ∧ (∃x. =⇒ ∃x. ).These two implications can be composed by cutting 3 against the following proof 4 : Write 5 for the proof of ∃x.∨∀y.⊥ obtained by cutting 3 and 4 .The interpretation of 5 is the composition of 3 and 4 , which triggers the feedback loop causing the infiniteness phenomenon.We display below the corresponding interaction.For the "synchronised" part of formulas, we will use 0 for components resulting from matching dual quantifiers, and for components resulting for matching dual propositional connectives.

Figure 6
Figure 6Rules for the sequent calculus LK.

Figure 8
Figure 8 Additional cut elimination rules for MLL1.

Figure 9
Figure 9 Two examples of contraction.

Figure 10
Figure 10 Interpretation of the remaining rules of LK.