A Graphical Proof Theory of Logical Time

Logical time is a partial order over events in distributed systems, constraining which events precede others. Special interest has been given to series-parallel orders since they correspond to formulas constructed via the two operations for “series” and “parallel” composition. For this reason, series-parallel orders have received attention from proof theory, leading to pomset logic, the logic BV , and their extensions. However, logical time does not always form a series-parallel order; indeed, ubiquitous structures in distributed systems are beyond current proof theoretic methods. In this paper, we explore how this restriction can be lifted. We design new logics that work directly on graphs instead of formulas, we develop their proof theory, and we show that our logics are conservative extensions of the logic BV .


Introduction
This paper is about the design of graphical proof systems that have an expressive power beyond logics constrained by the syntax of formulas. We explain, in this introduction, our original observations that compelled us to explore this untapped region in logic, whilst exploring a proof theory of logical time. Logical time is a partial order constraining events. It is employed because some events have to precede others (for instance due to read/write dependencies) and because synchronising clocks is infeasible in distributed systems. A special case is given by series-parallel orders [39, 18,6] which gave rise to a family of non-commutative logics, including pomset logic [36] and BV [19]. A series-parallel order is a partial order that can be constructed by composing smaller components, in series (◁) and in parallel ( ). For example, the following two series-parallel orders a b c d and Modelling logical time, where the causal relation "happens before" constrains events in a distributed system in such a way that they certainly precede others [29], has always been a core motivation for BV [8]. Implication can be a sound tool for reasoning about behavioural preorders between processes such as simulation [11,26,24,25], and it can be used to devise powerful notions of multiparty compatibility and subtyping in session types [23].
However, all the above-mentioned work is limited by the series-parallel restriction, that is, on orders which can be decomposed using series and parallel compositions. Yet, not all patterns of causality can be represented by series-parallel orders. Consider for example read-write dependencies, where shapes such as can occur. In this diagram there are two data write operations where one is dependent on two previous read operations, while the other is dependent on a single read. The diagram in (1) above is not a series-parallel order and indeed is out-of-scope of BV and pomset logic. In contrast, the diagram is in scope of general process models, such as Pratt's original pomsets and event structures [33,34]. Clearly, such non-series-parallel structures arise ubiquitously. As a running example, let us consider consumer-producer queues which are used to preserve message order when composing asynchronous distributed components, as depicted in Figure 1. In these diagrams, the labels on the nodes represent the events of enqueueing or dequeueing four particular messages (a, b, c, and d). Then, an enqueue event must happen before its corresponding dequeue event. Secondly, if one enqueue event happens before another enqueue event, then the corresponding dequeue events occur in the same order. This preserves a first-in-first-out order for messages exchanged using the queue. Finally, queues with capacity n have the restriction that the message i must be dequeued before message i + n is enqueued. Causality patterns for consumer-producer n-queues, where n is the bound on the number of messages that can be enqueued. Nodes labelled by ex and dx respectively represent the enqueuing and dequeuing of the message x. In all three queue types we only represent the first four messages (a, b, c, and d) inserted into the queue.
The edges in the diagrams in Figure 1 should not be seen as the possible paths when "executing" the system, but they describe the dependencies that have to be met before executing each event. If there are no edges between a given pair of events and if all dependencies of these two events are met, then they can be executed in any order. In other words, they describe the Hasse diagram of the induced partial order. In this way, one can see that dependencies are relaxed from left to right in Figure 1, permitting more concurrency.
The first example is a 1-queue, allowing at most one message to be enqueued at any time. The result is a simple total order over events, alternating between enqueue and dequeue events, represented by the formula e a ◁ d a ◁ e b ◁ d b ◁ e c ◁ d c ◁ e d ◁ d d . This helps explain why some verification frameworks for distributed systems, e.g. [15,12], assume 1-queues as a simplified approximation of queues. The 2-queue, allowing two messages to be buffered, is also a series-parallel order, represented by e a ◁ (e b d a ) ◁ (e c d b ) ◁ (e d d c ) ◁ d d . Hence we can use BV or pomset logic to reason about these queues, for example proving that the 1-queue implies the 2-queue, serving as a proof that a 2-queue can simulate behaviours of a 1-queue. 1 In the third example, up to three messages can be buffered in the queue. However, this 3-queue is not a series-parallel order, 2 nor is any other n-queue with n > 2. Therefore the existing proof-theoretical tools cannot be used to reason about them. This is the main motivation for this paper. Can we design proof systems that work directly on graphs that are not bound by the series-parallel restriction, i.e., go beyond formulas?
This problem turned out to be more challenging than expected. A first step was communicated in [1], which devised a minimal logic, called GS, over general undirected graphs, where the symmetric edges generalise the multiplicative connectives of linear logic, without the presence of directed edges representing logical time. The proof theory of GS requires deep inference, where inference rules may be applied deep in any context, not just at the root connective of a formula as for example in the sequent calculus. For developing a structural proof theory in the absence of formulas, we employed the notion of modular decomposition from graph theory, which enables any graph to be represented as a tree of prime graphs.
The main contribution of this paper is a graphical logic that generalizes both, GS and BV. Following the recent discovery that BV and pomset logic are not the same [32], we also made the observation that there is not one canonical choice about what is the "right" logic. We present here two proof systems GV and GV sl , that both (i) are analytic (i.e., suitable for proof search) and (ii) obey cut elimination. In this respect, we can indeed speak of a proper proof theory. We present our logics using open deduction, which is a deep inference formalism. Furthermore, we show (iii) that both logics, GV and GV sl , are conservative extensions of GS and of BV, and (iv) that the provability problem in GV and in GV sl is NP-complete.
Structure of the paper. We begin by recalling some preliminary notions about graphs and their modular decomposition in Section 2 and some preliminaries about deep inference and open deduction in Section 3. In Section 4 we introduce the inference rules for our proof systems and show some of their properties. Then, in Section 5 and Section 6 we prove cut elimination and some of its consequences, including the results on conservativity and complexity mentioned above.

Preliminaries on Graphs and their Modular Decomposition
⌢⟩ is a set V G of vertices equipped with an irreflexive and symmetric binary edge relation In a mixed graph we define the following additional edge-relations:

22:4 A Graphical Proof Theory of Logical Time
We say that a graph G is n-colour (for n ≤ 3) if at most n of the three edge relations ⌣ (white edges), ↷ (green edges) and ⌢ (red edges) are non-empty. In particular, we say that G is a complete graph if The empty graph is denoted as ∅.
In the following we may omit the index/superscript G when it is clear from the context. When drawing a graph we draw v w whenever v⌢w, we draw v w whenever v↷w, and we draw no edge at all whenever v⌣w.
is an induced subgraph of G, and we say that G is H-free whenever H ⊑ G does not hold.
As shown previously [1,9], graphs can be used as operators to compose graphs, playing a similar role to connectives in formulas. In developing our proof systems, we use the notion of modular decomposition [16,27,22,30,31,13,7] in order to assign to each graph a tree-like structure where leaves are single vertex graphs, and modules are sub-trees. 3 and all its modules are trivial.
▶ Notation 4. In this paper we identify a module M of a graph G and its induced subgraph G| M . We introduce the following notation for relevant prime graphs on 2, 3 and 4 vertices, respectively: We use the following notation for the composition via (prime) graphs with two vertices: Prime graphs play a special role in modular decomposition and can be considered as generalized (non-decomposable) logic connectives in the sense of [17,10,3].
▶ Theorem 5 ([16, 27, 13]). Let G be a graph with at least two vertices. Then there are some non-empty graphs H 1 , . . . , H n and a prime graph P such that G = P H 1 , . . . , H n . ▶ Corollary 6. Each graph G admits a modular decomposition by means of prime graphs and single-vertex graphs. Such a decomposition is unique modulo associativity of ⊗, and ◁, and prime graphs isomorphisms.
The representation on the right above relies on modular decomposition to reduce the number of edges to draw: the vertices inside a rectangle belong to a same module. Therefore an edge touching a rectangle represents a bundle of edges (of the same type) connecting with each vertex in the module. We can associate a modular decomposition tree to each graph in the same way we associate an abstract syntax tree to a formula. Below is a tree representation of the modular decomposition of the graph above and its formula-like denotation, which uses the prime graphs from (3) and the composition notation from Definition 2.
Indeed, prime graphs can be thought as primitive operators or connectives for graphical logic (see Section 9 of [2]), and the notion of module can be seen as a generalization of subformulas.
In this paper, we use the graphical representation on the right of (4) and the formula-like representation on the right of (5) interchangeably.
▶ Definition 8. A prime graph P is a connector of a graph G (or P -connector if we want to specify the prime graph P ) if there is an occurrence of the graph P in the modular decomposition of G.
For example, in the graph in Example 7 above, we have two connectors ⊗, one ◁, one N and one P 4 .
We assume graphs are labelled. That is, each vertex v of a graph G carries a label ℓ G (v) selected from a label set L. In particular in this paper we assume L = A ∪ A ⊥ where A is a set of atoms {a, b, c, . . .} and A ⊥ is the set of negated atoms {a ⊥ , b ⊥ , c ⊥ , . . .}.

22:6
A Graphical Proof Theory of Logical Time For this purpose we assume negation to be involutive, that is, The dual graph operation above flips labels on vertices, exchanges ⌢ (red) and ⌣ (white) symmetric edges, but preserves ↷ (green) directed edges.
dual of the graph from Example 7 and is represented as follows: In this work we are mainly interested in how vertices are labelled, and not in the identity of the underlying vertex. For this reason we rely on the notion of graph isomorphism.
▶ Definition 11. Let G and G ′ be two graphs. We say that G and

▶ Definition 12. A web is a graph such that ↷ is transitive and such that each connector occurring in its modular decomposition is a 2-colour prime graph. A relation web is a web such that each connector is in
It is worth noticing that the above graphs may be characterised by ruling out the presence of specific induced subgraphs (see, e.g., [37,19,14]). In particular, cographs are P 4 -free undirected graphs, series-parallel orders are N-free directed graphs with ↷ being transitive, and relation webs are T GR -free, P 4 -free, and N-free graphs with transitive ↷. Note that requiring ↷ to be transitive is equivalent to requiring that a graph is T W GG -free, T G GG -free and T R GG -free. Moreover, graphs with no 3-colour connectors are by definition T GR -free and T RG -free, however the converse does not hold.
▶ Example 13. The two graphs below on the left are not webs since they are three-colour prime graphs, while the complete graph on the right is a web since its modular decomposition only contains 2-colour connectors. In drawing webs we may omit directed edges whenever they are derivable from the drawn ones via the transitivity of ↷. This means, we can represent the web u ◁ v ◁ w as u v w, in which we implicitly assume the directed edge from u to w. For example, the web Q 1 in Figure 1 can be represented as a single thread e a d a e b d b e c d c e d d d .
▶ Remark 15. The operation of composing graphs via a graph (Definition 2) is closed with respect to the sets of cographs, series-parallel orders, relation webs and webs. That is, if all H 1 , . . . , H n and G are cographs (resp. series-parallel orders, relation webs, webs), then so is The correspondence between cographs and formulas containing only connectives for conjunction and disjunction is well-known since the 60s [16,27]. We here recall the correspondence between relation webs and the formulas for the non-commutative logic BV from [19,35].
The binary connectives , ◁ and ⊗ are called par, seq and tensor. Par and tensor correspond to disjunction and conjunction. We associate to each formula ϕ a relation web [[ϕ]] defined as follows: We identify an atom a (resp. its negation a ⊥ ) with a single vertex graph labelled by the same atom a (resp. a ⊥ ).
We will make use of this result later in Section 4, when formulating the proof system for BV as a proof system on relation webs rather than a proof system on formulas.

Preliminaries on Open Deduction
Open deduction [20] is a proof formalism based on deep inference [5]. It has originally been defined for formulas, but it is abstract enough such that it can equally well be used for graphs, as already done in [2]. We begin here by defining the notion of context, which is a graph with a hole into which another graph can be plugged, generalising the similar notion of context for formulas. Note that the notion of context is strongly connected to the one of module: ▶ Definition 19. An inference system S is a set of inference rules (as for example shown in Figure 2). A derivation D in S with premise G and conclusion H is denoted H and is defined inductively as follows: Every graph G is a (trivial) derivation (also denoted G) with premise G and conclusion G.

22:8 A Graphical Proof Theory of Logical Time
Every instance of a rule G r H in S is a derivation with premise G and conclusion H.
If D 1 is a derivation with premise G 1 and conclusion H 1 , and D 2 is a derivation with premise G 2 and conclusion H 2 , and H 1 ∼ f G 2 , then the composition of D 1 and D 2 is a derivation D 2 • f D 1 denoted as below on the left.
If f is clear from context we may simply write D 2 • D 1 and denote it as above on the right in order to ease readability. However, even if the isomorphism f is not written, we always assume it is part of the derivation and explicitly given. If G is a graph with n vertices and D 1 , . . . , D n are derivations with premise G i and conclusion H i for each i ∈ {1, . . . , n}, then G D 1 , . . . , D n is a derivation with premise G G 1 , . . . , G n and conclusion G H 1 , . . . , H n denoted as below on the left (if G = ⊗ or G = ◁ or G = we may write the derivations as below on the right).
H is a derivation and C[□] a context, then we have a derivation C .

Proof Systems on Webs
In this section, we recall the proof systems BV from [19] (that we here formulate as a proof system on relation webs) and GS from [2] (which is a proof system on undirected graphs), and we introduce two new proof system on webs using the rules from Figure 2 4 Note that in [2,1] the system GS is defined with the rule However it is easy to show that s.sw is derivable using s .
For all rules we assume P, Q and R prime webs with Following the deep inference tradition, the rules come in pairs, a down-rule (marked with ↓) and an up-rule (marked with ↑), which are dual to each other. The proof systems in (8) above only contain the down-rules, as the up-rules are admissible, which we will show in the next section. 5 The rules ai are called atomic interactions and allow to derive the atomic implication a ⊸ a. The rules s (called switch), q BV ↓ and q BV ↑ are the rules obtained by adapting the standard proof system BV on formulas to relation webs. When generalising s from relation webs to webs, this rule admits two formulations s and s ⊗ , which are respectively called switch par and switch tensor. Intuitively, s allows to move a connected component The rules p are called prime graph rules. 7 Each instance of p↓ creates disjunctions of the modules in a prime connector R and the corresponding modules in an occurrence of its dual 5 The rules sl and qm are in fact down-rules, but we omitted the ↓ as we do not need the corresponding up-rules in this paper. 6 From inspecting the rules, one can observe that s and s⊗ are dual to each other. And indeed in simpler systems like BV and GS, the s⊗ is admissible like all other "up-rules". However, for GV and GV sl , this is no longer the case (see Example 44 in the appendix). 7 Differently from [1,2] in this paper we drop the side condition |V P | ≥ 4 in p-rules. This choice restrains the set of rules required to simulate the general i-rules (see Lemma 24). Note that this makes the s-rule also a special case of p↓.  .............................................. .............................................. ................................................

Figure 3
Proof of Q 1 ⊸ Q 2 from the introduction. This is a correct proof in BV, in GV, and in GV sl . R ⊥ and puts them in conjunction. The rules q and qm are called q-rules and generalise the two rules q BV ↓ and q BV ↑ from BV to general prime webs with an empty set of ⌢-edges. The rule sl is called slice and is yet another generalisation the q BV ↓-rule. It formalises the idea that during proof search we aim at slicing a directed graph into a "before" and an "after" part by introducing additional ↷-edges. This mimics a longstanding idea in distributed systems: a partial order is a representation of many possible linear orders in which concurrent events may occur [29]. For example, below is a proof of a small web provable in GV sl but not provable in GV.
Other examples of derivations are given in Figure 3, Figure 4, Figure 5, and Figure 6.
▶ Remark 20. The rules p↓, p↑, q↓ and q↑ are crucial to obtain a proof system containing only atomic interactions (see Proposition 28). These rules gather two dual connectors by merging their set of edges, therefore obtaining a complete graph. In particular, the formulation of the rule q↓ exploits the fact that if Q is a prime web such that Q ↷ ̸ = ∅, then either Q or Q ⊥ is a complete graph. This will be crucial for proving the results in this paper. However, as Example 13 shows, this fact is not true for general 3-colour prime graphs. The question whether the results on webs, that are presented in this paper, can be generalized to arbitrary graphs is an open problem.
Let us now show that GV and GV sl do indeed extend BV beyond the restriction of relation webs (i.e., formulas). A and B be relation webs. if
The i↑ rule is the deep inference equivalent to the cut rule from sequent calculus and proving its admissibility is equivalent to proving the cut-elimination result. We recall some admissibility results for rules in BV and GS from the literature where we notice that the admissibility of cuts can be achieved by proving the admissibility of up-rules.
where ⊗ m is the complete undirected graph with m vertices, and conclude by induction hypothesis.
If P ↷ ̸ = ∅, we conclude similarly to the previous case by using q↓ instead of p↓ and therefore replacing the ⊗ m with C = P (if Proof. This follows in exactly the same way as in any other deep inference system [21]. ◀

Cut elimination
The i↑ rule in (10) is the deep inference equivalent to the cut rule. We show in this section, that this rule is admissible for our systems, i.e., every web provable with cut is also provable without.
▶ Theorem 26. The rule i↑ is admissible for GV and for GV sl .
The main consequence of this theorem is the transitivity of the consequence relation defined by implication A ⊸ B = A ⊥ B is transitive. A, B, and C be any webs. If A ⊸ B and B ⊸ C are provable in GV (resp. GV sl ), then so is A ⊸ C.

▶ Corollary 27. Let
Proof. Given proofs of A ⊥ B and B ⊥ C, we also have a proof of (A ⊥ B) ⊗ (B ⊥ C). Applying s twice gives us A ⊥ (B ⊗ B ⊥ ) C, and then we can obtain A ⊥ C, via i↑. ◀ To prove Theorem 26, we rely on methods developed for deep inference in general [38,21,19,4] and for graphical systems in particular [1]. There are two steps: where GX ∈ {GV, GV sl } and G is an arbitrary web. The first step is decomposing the general i↑ (which introduces an arbitrary web when going up in a proof) into smaller steps that are easier to control, and it is achieved with Lemma 25 and the following proposition, that is an immediate consequence of Lemma 24.
It remains to show the second step in (11), i.e., the following theorem.
▶ Theorem 29. The rules ai↑, p↑, and q↑ are admissible for GV and GV sl .
Proving this theorem follows the same outline as in other deep inference systems, via splitting and context reduction stated below.
▶ Lemma 30 (Tensor Splitting). Let GX ∈ {GV, GV sl }, let A and B be nonempty webs, and let G be an arbitrary web. If ⊢ GX G (A ⊗ B), then there are webs K A and K B such that there are derivations Note that the well-known splitting tensor lemma for linear logic proof-nets [10] states the existence of a splitting tensor, whereas Lemma 30 (and any splitting tensor lemma in deep inference in general) says that every tensor can be made splitting by first reducing the context. These three lemmas are already enough to prove the admissibility of ai↑ for GV and GV sl : Assume we have a proof of C[a ⊥ ⊗ a]. By the three lemmas above, we get webs K, K a ⊥ , K a and derivations With the derivations D ∅ , D K , D 1 , D 2 and an instance of ai↓, we can now obtain a proof of C[∅], as desired.

F S C D 2 0 2 2
In order to prove admissibility of p↑ and of q↑ in a similar way, we need two more variants of the splitting lemma. The first (Lemma 33) deals with prime webs in general, and the second (Lemma 34) deals with complete webs, i.e., webs in which for any two vertices x, y, we either have x⌢y or x↷y or x↶y. These lemmas are now sufficient to complete the proof of rule elimination (Theorem 29).
Proof of Theorem 29. For ai↑, this has been shown above. For p↑, we proceed similarly as in [1,2] for GS. For q↑, assume ⊢ GX C Q M 1 , . . . , M n ⊗ Q ⊥ N 1 , . . . , N n . By Lemma 32 there is a web K, such that for any web X there are the derivations Since Q ⌢ = ∅, we have that Q ⊥ is a complete web, and we can apply Lemma 34 to D Y giving us K 1 , . . . , K n and the derivations above right (for all i ∈ {1, . . . , n}). We conclude with ◀ Observe that Lemma 30 is just a special case of Lemma 34 and also a special case of Lemma 33, as cases (A) and (B) of Lemma 33 collapse both to Lemma 30 if P = ⊗. Therefore, it only remains to prove Lemma 31, Lemma 32, Lemma 33, and Lemma 34, which is done by induction on the size of the web to be proved. Even though the general pattern of the overall argument to prove Theorem 26 is similar to other deep inference systems, the details are much more involved. Some of the problems have already been observed for the system GS in [1]. For example, Lemma 32, Lemma 33 and Lemma 34 are not proved one after the other, but simultaneously, by mutual induction on the size (defined above) of the conclusion. An additional difficulty for GV and GV sl is the explosion of the cases to consider. When proving Lemma 33, we have to do a case analysis on the last rule applied to the proof of G P M 1 , . . . , M n . More details on the splitting proof can be found in Appendix A.
▶ Remark 36. The s ⊗ is usually admissible in a deep inference system, as it is an up-rule (and dual to s ). This was also the case for GS [1]. However, when we consider mixed graphs, the rule is no longer admissible. Or, put differently, cut elimination does not hold if we remove the rule from the system (see Appendix B).
▶ Example 37. To illustrate the dynamics of splitting, consider the alternative proofs in Figure 5 which prove the same web as found in the conclusion of the proof on the left of Figure 4. The proofs in Figure 5 can be obtained by applying splitting to the first proof in Figure 4. For the proof on the left of Figure 5, apply Lemma 34 to the leftmost disjoint graph, which ensures the existence of the derivation found below the instance of the q↓ rule, that does not change the sub-graph selected. Similarly, the second proof in Figure 5 is obtained by applying splitting to the connected sub-graph labelled with a, c and e ⊥ .  Figure 5 Alternative proofs of the web proved by the left derivation in Figure 4.

Analyticity, Conservativity, Complexity
In this section, we discuss some consequences of the cut-elimination result obtained in the previous section. The most important is that GV and GV sl are both analytic. Since deep inference systems in general, and our graphical proof systems in particular, do not have a "subformula property" in the traditional sense, we present here a formal notion of what we mean by analyticity of a graphical proof system. For this, we start from the notion of connector given in Definition 8.

▶ Definition 38. A subconnector of G is a prime web that is an induced subgraph of a connector in G. A connector (or subconnector) P is called proper if
▶ Theorem 39 (Analyticity). If ⊢ GX G for GX ∈ {GV, GV sl }, then G admits an analytic proof in GX.
Proof Sketch. This is proved similarly to the same result for GS given in [2], by taking into account the additional rules in GV and GV sl . When going bottom-up in a derivation (as in proof search), the only two rules that can introduce a new proper connector that is not a subconnector of the conclusion are s (when the N in Figure 2 is empty) and q↓ (when L i in Figure 2 is empty for some i ∈ {1, . . . , |V Q |}). Since the premise of every proof is ∅, all these connectors have to be destroyed eventually. This can happen via the rules ai↓ or s ⊗ (one immediate submodule of the connector is removed), or via the rule p↓ or q↓ (the connector itself is destroyed, the submodules remain). It can now be shown via rule permutations that the rule instance that creates the connector and the rule instance that destroys it can be brought together, such that we can rearrange the derivation in a way that the offending connector does not occur. ◀ ▶ Corollary 40. Let GX ∈ {GV, GV sl } and let G be an undirected graph. If ⊢ GX G, then G admits a derivation D in GX such that every connector of a web in D is a subconnector of G.
Proof Sketch. For proper connectors this has been shown in Theorem 39. Hence we only need to consider P -connectors that are directed prime graphs. The only way to create such a new P -connector is via s , q↓,or qm. Then P is either destroyed by destroying all its modules, or it is destroyed after merging via qm (or q↓ if P = ◁) with another P -connector. In both cases we use a similar rule permutation argument as in the proof of Theorem 39 to eliminate the occurrence of the P -connector. ◀ As a consequence of Theorem 39, we know that, in an analytic proof, any proper connector occurring in the conclusion of a q↓ also occurs in its premise. This condition is not required by definition for the rule q↓ (see Figure 2) where the side condition only asks that L i J i is non-empty: the rule might introduce bottom-up a new connector whenever a L i is empty. The inclusion of the additional restriction L i ̸ = ∅ for all i ∈ {1, . . . , n} for q↓ in the definition of the rule would require a more complex formulation of the splitting statement.
To better understand that subtlety consider the example in Figure 6. The proof on the left is obtained by applying splitting to the leftmost connected component of the conclusion; while the proof on the right has a shape obtained by applying splitting to the rightmost connected component. In the latter proof, the bottommost q↓ introduces a new connector establishing relations between the modules of the leftmost and central components of the conclusion. As indicated in the figure, this is achieved by introducing empty modules in different places in both webs affected by that instance of the q↓ rule, making use of the weak side condition mentioned above. The application of a restricted version of the q↓ would force the merge of the two webs to establish additional relations which later would prevent a successful proof. However, observe that this proof violates analyticity.
In contrast, the proof on the left is analytic. Even though that proof also appeals to the same weak side conditions and introduces in the premise of the qm-rule instance a new connector that is not a subconnector of the conclusion, this new connector is not proper because it contains no ⌢-edge. This is the reason for restricting our analyticity result to proper connectors.
Nonetheless, this restricted form of analyticity is enough to show that both, GV and GV sl , are conservative extensions of GS and of BV.
▶ Corollary 43. Provability in GV and in GV sl is NP-complete.
Proof. It has been shown before that provability in BV is NP-complete [28]. Hence, by Theorem 42 we obtain NP-hardness of provability in GV and GV sl . For containment in NP, observe that for any rule instance H r G in GV sl we have that ∥H∥ < ∥G∥. For a web G we . Therefore any derivation in GV sl (or in GV) of a web G has at most length O(|V G | 3 ). ◀

Conclusion
We presented two analytic proof systems, GV and GV sl (Figure 2), that conservatively extend the logic BV to graphs (Theorem 42). The graphs, called webs (Definition 12), feature both undirected and directed edges, which generalise, respectively, the multiplicative conjunction of linear logic, and the non-commutative connective seq of BV. In addition, our systems are conservative extensions of GS (Theorem 41), which features general simple graphs. This is no coincidence, since GS was conceived precisely as the minimal core of graphical logics like GV for which it was already clear that we needed to generalise the tools of proof theory. Indeed, we were able to adapt the deep inference technology developed for GS to prove cut elimination for GV and GV sl (Theorem 26 and Section 5). Differences compared to GS surprised us. Notably, the s ⊗ rule is not admissible in GV (explained further in Appendix B), which in fact simplifies splitting (Lemmas 30-34), since s ⊗ can be used to remove contexts (Lemma 32). A nuance of GV is that rules for directed graphs require weaker side-conditions than similar rules for symmetric graphs. For example, if we restrict q↓ and qm such that all modules in one of the graphs are non-empty, then there is no proof of the web in Fig. 6 -an observation we can use to prove that such a restricted system cannot satisfy cut elimination. This forced us to take additional care defining analyticity Theorem 39, used to establish conservativity.  In this case, we can apply Lemma 34 and conclude by inductive hypothesis. ◀ Sketch of Proof of Splitting Lemma for Complete Webs (Lemma 34). By induction on |V C | using Lemma 33. We know that all connectors and subconnectors of a complete graph are complete graphs, therefore webs. Therefore we can assume C = C ′ C 1 , . . . , C m for some m and complete webs C ′ , C 1 , . . . , C m where C ′ is also prime. We can apply Lemma 33. If Case (A) of Lemma 33 applies, then we conclude by induction hypothesis. Otherwise Case (B) applies and we can assume without of loss of generality that we have K X and K Y such that ⊢ GX K X C 1 M 1 , . . . , M l and ⊢ GX K Y C ′′ M l+1 , . . . , M n for some 1 ≤ l ≤ n, where C ′′ = C ′ ∅, C 2 , . . . , C m . Since C ′′ is a complete web, we can conclude by induction hypothesis. ◀ The Need for s ⊗ in GV ▶ Example 44 (s ⊗ is not admissible.). The rule q BV ↑ (which overlaps with s ⊗ , as shown in Lemma 21) is admissible in BV [19]. Similarly, the rule s ⊗ is admissible in GS [1]. However, in [2] it was shown that s ⊗ is required to prove that GS is a conservative extension of the multiplicative linear logic (MLL). Furthermore, we are able to provide an example showing that s ⊗ is not admissible in any of our systems conservatively extending GS and BV. Consider the following webs.

Proof of Context-Reduction (Lemma 32). If
By cut elimination for GV, we get ⊢ GV B C C. However, ⊢ GV\{s⊗} B C C does not hold.