Proof nets through the lens of graph theory: a compilation of remarks

12/23/2019
by   Lê Thành Dũng Nguyên, et al.
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This document is intended to eventually gather a few small remarks on the theory of proof nets and correctness criteria that I have never published. For now the only written part is about Retoré's pomset logic.

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1 On pomset logic

Retoré’s pomset logic [Ret97a] is an extension of MLL+Mix with a binary connective denoted by ‘’ whose particularity is to be non-commutative and self-dual. Its system of proof nets extends the MLL+Mix correctness criterion – “there is no (undirected) cycle using at most one premise edge of each – by considering directed cycles, which can only visit both premises of a if the left one comes before the right one in the cycle.

In this note, we first study the complexity of deciding the correctness of a pomset proof structure. We obtain a coNP-hardness result that morally excludes some possibilities for a sequentialization theorem – this explains why it is difficult to obtain a sequent calculus for pomset logic. Finally, we discuss Slavnov’s recent proposal for such a sequent calculus [Sla19].

1.1 Proof net correctness for pomset logic is coNP-complete

In [Ret97a], Retoré presents pomset proof nets as “RB-graphs”, that is, as digraphs (i.e. directed graphs) equipped with perfect matchings (see also [Ret03] for the MLL+Mix case). The advantage is that the correctness criterion can then be stated as a combinatorial property in the vocabulary of mainstream graph theory: it is the absence of alternating circuits. We recall these notions below.

Definition 1.

A digraph consists of a finite set of vertices and a set of arcs . An circuit of length is a -indexed sequence without repetitions111In the paper [GLMM13], the definition of “circuit” includes this prohibition on vertex repetitions. This seems to be common in the graph theory literature. In the same way we shall use “path” to refer to elementary paths. such that for all , .

A perfect matching of a digraph is a subset of arcs such that:

  • any vertex has exactly one outgoing arc in and exactly one incoming arc in (i.e. there is exactly one pair such that and );

  • for all , – morally, consists of undirected edges.

An alternating circuit is a circuit such that for all , exactly one of and is in (so that the other one is in ). Note that this forces the length of the circuit to be even.

We claim that there is a converse reduction to Retoré’s RB-graphs.

Theorem 2.

The existence of an alternating circuit for a perfect matching in a digraph reduces to the incorrectness of a pomset proof structure in linear time.

Proof sketch.

This has been done for alternating cycles in undirected graphs and MLL+Mix proof structures in our previous work [Ngu18]; in short, the idea is to represent edges by -links if they are in the matching, and by -links if they are outside the matching.. To extend that reduction (called “proofification” in [Ngu18]), we use a gadget with two axiom links and one -link to encode directed arcs whose reverse arc is not in the digraph. See Figure 1 for an example which should be enough to infer the whole construction. ∎

The rest of this section will prove that:

Theorem 3.

Detecting alternating circuits in digraphs is NP-complete.

Corollary 4.

Deciding the correctness of a proof structure in pomset logic is coNP-complete.

(The latter is immediate from the two previous theorems.) This contradicts the polynomial time claim of [Ret97a, Proposition 5] whose reliance on a “standard breadth-first search algorithm” is faulty for subtle reasons.

Figure 1: A digraph equipped with a perfect matching and its translation into a pomset proof structure. Note that is a directed arc from to , while the other undirected edges in the figure represent pairs of directed arcs of the form .

To prove Theorem 3, we invoke a result of Gourvès et al. [GLMM13] in the theory of edge-colored graphs (or rather, in this case, arc-colored digraphs).

Definition 5.

An -arc-colored digraph is a digraph equipped with a mapping fromn the arcs to a finite set of colors. A properly colored circuit is a circuit (without vertex repetitions) in which two consecutive edges always have different colors.

Theorem 6 ([Glmm13]).

Deciding whether a 2-arc-colored digraph contains a properly colored path between two given vertices is -complete, even when the input is restricted to digraphs with no properly colored circuit.

Corollary 7.

Finding a properly colored circuit in a 2-arc-colored digraph is -complete.

Proof sketch.

By reduction from the previous problem. To any 2-arc-colored digraph with designated source and designated target , glue an acyclic gadget to and to add a properly arc-colored path from to with any starting color and any ending color. If the original digraph had no properly colored circuit, then the new one admits a properly colored circuit if and only if there was a properly colored path from to in the original, whose concatenation with a new path from to results in a cycle. ∎

Finally, we have to reduce the above problem to that of Theorem 3. To do so, we can adapt a reduction from 2-edge-colored graphs to perfect matchings in undirected graphs, attributed to Edmonds in [Man95, Lemma 1.1]. See Figure 2 for the idea.

Figure 2: A 2-arc-colored digraph and its translation into a digraph with a perfect matching.

1.2 A sequent calculus?

1.2.1 A complexity-theoretic obstacle to sequentialization

Here’s a vague argument (future work: attempt to relate this formally to e.g. Alwen Tiu’s non-existence theorem [Tiu06] on “shallow systems” for BV).

Suppose we had a sequentialization theorem, i.e. some kind of inductive characterization of pomset proof nets (perhaps presented as a sequent calculus, but not necessarily). Then one could look for the last inference rule of a sequentialization by enumerating all splitting possibilities. Suppose further that this enumeration can be done in polynomial time (this is typically the case for MLL+Mix: naively, for each terminal -link, compute the connected components without it). Then as soon as one finds a legitimate splitting, one could simply recurse on the obtained subnets (the sub-proof-structures of a correct pomset proof structure are correct themselves). This would yield a polynomial time sequentialization algorithm, which would also decide correctness (if at some point there is no possible last rule, the proof structure is incorrect), thus contradicting our coNP-hardness result unless P=NP!

Moral of the story: ordinary sequent calculi won’t work.

1.2.2 A pedestrian reconstruction of Slavnov’s calculus

First step: sequentialization

The notion of “inductive characterization of proof nets” is broader than that of sequent calculus, so let’s tackle that first. Given the obstacle that we noticed above, one idea is to incorporate some kind of NP-complete condition – why not directly mention the existence or not of feasible222I.e. conforming to the constraints involved in the correctness criterion. The usual term “switching path” is not adequate here since this criterion is not defined via switchings… circuits/paths? – into the inductive construction rules.

Theorem 8.

The set of correct pomset proof nets is inductively generated by the axiom, Mix, and rules of MLL+Mix, as well as the rule: if and are two distinct conclusions of a proof net, and there is no directed feasible path from to , then adding a -link with premises and conclusion yields a proof net.

Proof.

We must show that any proof net can be obtained from smaller ones by these rules:

  • If it contains a terminal -link, then this is obvious.

  • If it contains a terminal -link with premises , then any feasible path from would yield a feasible circuit and contradict the correctness criterion, so the precondition for the -rule is satisfied and this -link can be taken as the last rule.

  • If all its terminal links are -links, then we apply the sequentialization theorem for MLL+Mix to the proof net obtained by replacing every by a – if the original net was correct, so is its “commutative projection”. This gives us the existence of a splitting -link in the projection, which can be lifted to the original net.

Conversely, it is straightforward that every proof structure generated this way is correct. ∎

Remark 9.

This rule for kind of says: “move along, no deep combinatorics to be found here, I give up”. Imagine the analogous rule for : given a proof net with conclusions , add an -link if there is no feasible path from to nor from to . This would also be correct but less satisfactory than the -rule we all know and love; the latter relies in the end on a highly non-trivial “structure from acyclicity” property which manifests in graph theory as Kotzig’s theorem on unique perfect matchings (see also other examples in [Ngu19]).

Second step: designing decorated sequents

Can we turn this into a generalized sequent calculus? There is a formal and trivial solution: take proof structures as generalized sequents, the rules in the previous theorem as inference rules, and the root of a derivation tree as this derivation’s translation into a proof net (its desequentialization). There are two inconveniences with this:

  • A generalized sequent contains too much information; ideally one would not want much more than the formula being proved.

  • Computing some of the relevant information, namely the non-reachability condition between conclusions for the -rule, is coNP-complete, which impedes the possibility of deciding the correctness of a proof by low-complexity local checks.

A first attempt to repair both issues would be to take

generalized sequent = list of conclusions + reachability binary relation on conclusions

but unfortunately this information cannot be propagated by itself inductively by the inference rules. Say that there are feasible paths from to and from to ; if we add a link, can we deduce that there will be a new feasible path from to ? Since feasible paths need to be elementary – that is, they cannot repeat vertices – this deduction is only valid if the aforementioned paths are disjoint. Hence Slavnov’s solution [Sla19]

generalized sequent = conclusions + reachability between -tuples by disjoint feasible paths

which seems to be enough to make the induction work.

Is this satisfactory?

For now we have only discussed cut-free proofs. Of course one can treat cuts like tensors, but what bothers me is that it is unclear how to define a cut-elimination procedure on the sequent calculus, especially for the reduction of a

cut since is similar to from the point of view of Slavnov’s calculus.

1.3 Comparison with visible acyclicity and hypercorrectness

Pomset logic comes from trying to extract a syntactic correctness criterion from the coherence space semantics: the interpretation of a proof structure in coherence spaces can be defined by means of so-called experiments, and we want the result of the experiments to be a valid member of the semantics333By which we mean that the set of points obtained by experiments forms a clique.. For MLL proof nets, Retoré showed that this is equivalent to MLL+Mix correctness [Ret97b], and the correctness criterion for pomset proof nets was designed to extend this correspondence (this is discussed in [Ret97a]; in fact the ‘’ connective was first discovered by Girard in coherence spaces).

Pagani has applied a similar methodology to MELL (Multiplicative-Exponential Linear Logic) proof structures: he shows in [Pag06] that the validity of coherence space experiments – using a certain “non-uniform” interpretation of the exponentials – is equivalent to a certain graph-theoretical condition, visible acyclicity, which is weaker than the usual correctness criterion for MELL. This is later extended to differential interaction nets in [Pag12]; since coherence spaces are not a semantics of differential linear logic, the result of [Pag12] is formulated with respect to Ehrhard’s finiteness spaces instead.

A similarity between correctness for pomset proof nets and visible acyclicity is that both involve directed edges and cycles. Thanks to this, it is straightforward to to show that visible acyclicity is NP-hard, by adapting the proof for pomset logic; however we do not know whether, conversely, it is in NP. Actually this hardness proof for visible acyclicity came first, and was presented at the DICE 2018 workshop, before we noticed recently (Nov. 2019) that it also applied to pomset logic.

Let us also mention Tranquili’s hypercorrectness criterion for MALL (Multiplicative-Additive) proof structures, coming from their semantics in hypercoherences [Tra08]. Here again the condition obtained is weaker than the usual correctness criterion – so there are hypercorrect MALL proof structures that are not sequentializable.

References

  • [GLMM13] Laurent Gourvès, Adria Lyra, Carlos A. Martinhon, and Jérôme Monnot. Complexity of trails, paths and circuits in arc-colored digraphs. Discrete Applied Mathematics, 161(6):819–828, April 2013.
  • [Man95] Yannis Manoussakis. Alternating paths in edge-colored complete graphs. Discrete Applied Mathematics, 56(2):297–309, January 1995.
  • [Ngu18] Lê Thành Dung Nguyễn. Unique perfect matchings and proof nets. In Hélène Kirchner, editor, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018), volume 108 of Leibniz International Proceedings in Informatics (LIPIcs), pages 25:1–25:20, Dagstuhl, Germany, 2018. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.
  • [Ngu19] Lê Thành Dũng Nguyễn. Constrained path-finding and structure from acyclicity. arXiv:1901.07028v1 [cs], January 2019. arXiv: 1901.07028.
  • [Pag06] Michele Pagani. Acyclicity and coherence in multiplicative and exponential linear logic. In Pierre-Louis Curien, editor, Proceedings of the Twentieth International Workshop on Computer Science Logic, volume 4207 of Lecture Notes in Computer Science, pages 531–545, Szeged, Hungary, 2006. Springer.
  • [Pag12] Michele Pagani. Visible acyclic differential nets, Part I: Semantics. Annals of Pure and Applied Logic, 163(3):238–265, 2012.
  • [Ret97a] Christian Retoré. Pomset logic: A non-commutative extension of classical linear logic. In Gerhard Goos, Juris Hartmanis, Jan Leeuwen, Philippe Groote, and J. Roger Hindley, editors, Typed Lambda Calculi and Applications, volume 1210, pages 300–318. Springer Berlin Heidelberg, Berlin, Heidelberg, 1997.
  • [Ret97b] Christian Retoré. A semantic characterisation of the correctness of a proof net. Mathematical Structures in Computer Science, 7(5):445–452, October 1997.
  • [Ret03] Christian Retoré. Handsome proof-nets: perfect matchings and cographs. Theoretical Computer Science, 294(3):473–488, February 2003.
  • [Sla19] Sergey Slavnov. On noncommutative extensions of linear logic. Logical Methods in Computer Science, Volume 15, Issue 3, September 2019.
  • [Tiu06] Alwen Tiu. A System of Interaction and Structure II: The Need for Deep Inference. Logical Methods in Computer Science, 2(2):4, April 2006. arXiv: cs/0512036.
  • [Tra08] Paolo Tranquilli. A Characterization of Hypercoherent Semantic Correctness in Multiplicative Additive Linear Logic. In Michael Kaminski and Simone Martini, editors, Computer Science Logic, volume 5213, pages 246–261. Springer Berlin Heidelberg, Berlin, Heidelberg, 2008.