Sequentialization for full N-Graphs via sub-N-Graphs

Since proof-nets for MLL- were introduced by Girard (1987), several studies have appeared dealing with its soundness proof. Bellin & Van de Wiele (1995) produced an elegant proof based on properties of subnets (empires and kingdoms) and Robinson (2003) proposed a straightforward generalization of this presentation for proof-nets from sequent calculus for classical logic. In 2014 it was presented an extension of these studies to obtain a proof of the sequentialization theorem for the fragment of N-Graphs with conjunction, disjunction and negation connectives, via the notion of sub-N-Graphs. N-Graphs is a symmetric natural deduction calculus with multiple conclusions that adopts Danos-Regnier's criterion and has defocussing switchable links. In this paper, we present a sequentization for full propositional classical N-Graphs, showing how to find a split node in the middle of the proof even with a global rule for discharging hypothesis.

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1 Introduction

Since proof-nets for MLL were introduced by Girard girard:1987 , several studies have been made on its soundness proof. The first correctness criterion defined for proof-nets was given with the definition of the no shorttrip condition: Girard used trips to define empires and proved that if all terminal formulas in a proof-net are conclusions of times links, then there is at least one terminal formula which splits . After Danos–Regnier’s work danos:1989 it has become possible to define empires using their newly defined DR graphs and, with this new notion of empires, Girard proved sequentialization for proof-nets with quantifiers girard:1991 . Another important advance was achieved by the introduction of a new type of subnets, namely kingdoms. Once the notion of kingdoms was introduced, Bellin & Van de Wiele produced an elegant proof of the sequentialization theorem using simple properties of subnets bellin:1995 .

A straightforward generalization of this proof was obtained by Robinson robinson:2003 . He pointed out that Danos–Regnier’s technique relies only on the format of the rules and does not depend on the logic involved. So he devised a proof system based on the classical sequent calculus and applied the characterization of subnets and the proof of sequentialization for MLL to his proof-nets for classical logic. His proof followed the model defined by Bellin & Van de Wiele bellin:1995 .

However, this generalization does not cover the existence of so called switchable links with one premise and more than one conclusion, and also the absence of axiom links. In such systems subnets are not necessarily closed under hereditary premises. So, if a subnet contains a formula occurrence and is above111We say that is above when is a hereditary premise of . in the proof-net, then may not be in this subnet. Other works in linear logic related to these issues are Lafont’s interaction nets (which do not have axiom links) lafont:1995 and the system of Blute et al, which contains such switchable links blute:1996 and inspired the proof-nets for classical logic proposed by Führman & Pym fuhrmann:2007 . Hughes also proposed a graphical proof system for classical logic where proofs are combinatorial rather than syntactic: a proof of is a homomorphism between a coloured graph and a graph associated with hughes:2006 . McKinley, on the other hand, proposed the expansion nets, a system that focus on canonical representation of cut-free proofs mckinley:2010 .

Here we present an extended version of a previous work carvalho:2014 to perform the sequentialization for N-Graphs, a multiple conclusion calculi inspired by the proof-nets for the propositional classical logic developed by de Oliveira oliveira:2001:phd ; oliveira:2001 , but with a switchable defocussing link and without axiom links. One of the main results of this paper, besides giving a new soundness proof for N-Graphs, is the definition of a generalized method to make surgical cuts in proofs for classical logic. This comes with the fact that the presence of the split node in an N-Graph can occur essentially anywhere in the proof, unlike proof-nets where the split node is always a terminal formula.

The need to identify the split node is at the heart of our proof of the sequentialization. In order to achieve that we define the north, the south and the whole empires of a formula occurrence . The first one corresponds to the empires notion of Girard’s and Robinson’s proof-nets. The second one is the largest sub-N-Graph which has as a premise (defined due to the presence of elimination rules in N-Graphs). The last one is the union of the previously defined and it induces a strict ordering over the graph nodes, which will be fundamental to find the split node.

The deduction system defined by N-Graphs has a improper inference rule for the introduction of “”: it allows discharging hypotheses. In this paper we also include this connective and show how to find a split node in the middle of the proof even in the presence of a global rule for the introduction of “”.

2 N-Graphs

Proposed by de Oliveira oliveira:2001:phd ; oliveira:2001 , N-Graphs is a symmetric natural deduction (ND) calculus with the presence of structural rules, similar to the sequent calculus. It is a multiple conclusion proof system for classical logic where proofs are built in the form of directed graphs (“digraphs”). Several studies have been developed on N-Graphs since its first publication in 2001 oliveira:2001:phd , like Alves’ development on the geometric perspective and cycle treatment towards the normalization of the system alves:2005 and Cruz’s definition of intuitionistic N-Graphs cruz:2013 . A normalization algorithm was presented for classical N-Graphs alves:2011 , along with the subformula and separation properties alves:2009 . More recently a linear time proof checking algorithm was proposed andrade:2013 . N-Graphs also inspired a natural deduction system for the logic of lattices restall:2005 .

2.1 Proof-Graphs

The system is defined somewhat like the proof-nets. There is the concept of proof-graphs, which are all graphs constructed with the valid links where each node is the premise and conclusion of at most one link, and the concept of N-Graphs, which are the correct proof-graphs, i.e. the proof-graphs that represent valid proofs. These constructions are analogous to the definition of proof-structure and proof-net, respectively.

The links represent atomic steps in a derivation. Focussing links are the ones with two premises and one conclusion, as illustrated by Fig. 1 (, , , and contraction). The defocussing links are the ones with one premise and two conclusions, as shown in Fig. 1 (, , , and expansion). All other links are called simple links and have only one premise and one conclusion (Fig. 2).

There are two kinds of edges (“solid” and “meta”) and the second one are labeled with an “m” (). The solid indegree (outdegree) of a vertex is the number of solid edges oriented towards (away from) it. The meta indegree and outdegree are defined analogously. The set of vertices with indegree (outdegree) equal to zero is the set of premises (conclusions) of the proof-graph , and is represented by (). The set of vertices with solid indegree equal to zero and meta indegree equal to one is the set of canceled hypothesis of ().

Figure 1: Focussing and defocussing links.
Figure 2: Simple links.

A logical link represents a derivation in ND and we have introduction and elimination links for each connective; () acts as the law of the excluded middle. A structural link expresses the application of a structural rule as it is done in sequent calculus: it enables weakening a proof (, , and ), duplicating premises (expansion link) and grouping conclusions in equivalence classes (contraction link). There is no link to emulate the interchange rule because in a proof-graph the order of the premises is not important for the application of derivation rules.

The axioms are represented by proof-graphs with one vertex and no edges. Then, a single node labeled by is already a valid derivation: it represents an axiom in sequent calculus (). So here it makes no sense to talk about the smallest subgraph having as a conclusion: it would be trivially the vertex labeled by . Therefore the notion of kingdoms, as defined and used by Bellin & Van de Wiele bellin:1995 for their sequentialization, is useless for N-Graphs.

In Fig. 3 there are three proof-graphs. The first one is an invalid “proof” for . The others are correct derivations for and (contraction and expansion edges are dotted).

Figure 3: Proof-graphs with cycles.

2.1.1 Meta-edge and the scope of the hypothesis

Besides expansion and contraction links there is the link. Both Ungar and Gentzen systems are formulated in such a way that when the connective is introduced, it may eliminate an arbitrary number of premises (including zero). In N-Graphs this introduction is made in a more controlled way, which also complicates the task of identifying inadequate proof-graphs. For example, the first proof in Fig. 4 is not correct, but the second one is.

Figure 4: Meta edge: an invalid application on the left for and a sound one on the right for .

2.2 Soundness criteria

Similar to Danos-Regnier criterion danos:1989 , we define the following subgraphs associated to a proof-graph.

Definition 1 (Switching)

Given a proof-graph , a switching graph associated with is a spanning subgraph222A spanning subgraph is a subgraph of containing all the vertices of . of in which the following edges are removed: one of the two edges of every expansion link and one of the two edges of every contraction link.

Definition 2 (Meta-switching, virtual edge)

Given a proof-graph , a meta-switching graph associated with is a switching of in which every link with meta-edge is replaced by one of the following edges: the one from to or an edge from to , which is defined as virtual edge.

Definition 3 (N-Graph derivation)

A proof-graph is a N-Graph derivation (or N-Graph for short) iff every meta-switching graph associated with is acyclic and connected.

The focussing and defocussing links may also be classified according to their semantics. The links

, , , and expansion are called conjunctive. The disjunctive links are: , , , and contraction. Here contraction and expansion draw attention: their geometry contradicts their semantic and they are switchable (the ones that have one of its edge removed in every meta-switching). Although focussing, the contraction has a disjunctive semantic; and the expansion is a conjunctive link, even though defocussing. This means the formula occurences this links connect in a proof-graph must be already connected some other way in order to the proof to be sound. So the second and third proof-graphs in Fig. 3 are N-Graphs, but the first one is not because its cycle is not valid.

The also plays an important role in the soundness criteria. The premise of the link () and the canceled hypothesis () need to be already connected some other way in the proof for it to be sound. Thus the meta-switching must choose to connect to or . In the first proof-graph of Fig. 4 the conclusion of is , so this formula already carries a dependency on and the meta-edge removes it from the set of premises. However, there is another occurrence of , which is used by the link to obtain a “proof” of .

The soundness criteria captures this when the meta-switching choses the virtual edge, which links and , and the result is not a tree. It does not occur with the other proof-graph of Fig. 4: all the two meta-switchings are acyclic and connected.

Soundness and completeness of the system were proved through a mapping between N-Graphs and (sequent calculus for classical logic) oliveira:2001:phd ; oliveira:2001 and in Section 6 we give a new proof of sequentialization.

3 Sub-N-Graphs

Definition 4 (sub-N-Graph)

We say that is a subproof-graph of a proof-graph if is a subgraph of and is a proof-graph. If a vertex labeled by a formula occurrence is such that (), then is an upper (lower) door of . If is also a N-Graph, then it is a sub-N-Graph.

Let and be sub-N-Graphs of a N-Graph .

Lemma 1 (Union bellin:1995 )

is a N-Graph iff .

Once is a N-Graph, their meta-switchings do not have cycles and so any subgraph of may not have a cyclic meta-switching. Then we must prove only the connectedness of all meta-switchings associated with . If , then any meta-switching associated with is not connected. Now let and be the restriction of to . Since is a N-Graph, there is a path between and in . For the same reason, there is a path between and in . Thus there is a path between and in once . ∎

Lemma 2 (Intersection bellin:1995 )

If , then is a N-Graph.

As in the previous lemma, it is sufficient to prove the connectivity of . Since , let . If is the only vertex present in , then it is connected and so is a N-Graph (axiom). Otherwise, let be any other vertex in , be a meta-switching of and be an extension of for . Once and are sub-N-Graphs, there are a path between and in and a path in . If , then is a meta-switching for and has a cycle. So and and are connected. ∎

Definition 5 (North, south and whole empires)

Let be a formula occurrence in a N-Graph . The north (south) empire of , represented by () is the largest sub-N-Graph of having as a lower (upper) door. The whole empire of () is the union of and .

If we prove that and exist, then it is immediate the existence of by lemma 1. In the following section we give two equivalent constructions of empires and prove some properties.

4 North and south empires

4.1 Constructions and existence

Definition 6 ( and )

Let be a formula occurrence in a N-Graph and an associated meta-switching of . If is a premise of a link with a conclusion and the edge belongs to , then remove this edge and is the component that contains and is the other one (if is premise of a disjunctive defocussing link different from , then has two components). If is not premise of any link in , then is ( is empty). is defined analogously: if is a conclusion of a link with a premise and the edge belongs to , then remove it and is the component which has and is the other one (if is conclusion of a conjunctive focussing link, then has two components). If is not conclusion of any link in , then is equal to ( is empty).

As the virtual edge added in some meta-switchings connects two conclusions of the link, we consider the main conclusion of the link as the conclusion of this virtual edge for the purpose of deciding if we must remove the edge to construct or .

Definition 7 (Principal meta-switching girard:1987 ; girard:1991 )

Let be a formula occurrence. We say that a meta-switching () is principal for () when it chooses the edges satisfying the following restrictions:

  1. is a contraction link and a premise is the formula occurrence ( is an expansion link and a conclusion is the formula occurrence ): the meta-switching chooses the edge with .

  2. is a contraction link and only one premise belongs to (): the meta-switching links the conclusion with the premise which is not in ().

  3. is an expansion link and only one conclusion belongs to (): () selects the edge which has the conclusion that is not in ().

  4. is a link and only or only belongs to (): () selects the edge which connects to the other formula that is not in ().

  5. is a link and or is the formula occurrence : the meta-switching chooses the edge with .

Lemma 3

The north (south) empire of exists and is given by the two following equivalent conditions:

  1. (), where ranges over all meta-switchings of ;

  2. the smallest set of formula occurrences of closed under the following conditions:

    1. ();

    2. if is a simple link and , then (if and , then );

    3. if is a conjunctive focussing link and , then (if and , then );

    4. if is a disjunctive defocussing link different from and or , then (if and or , then );

    5. if is an expansion link and , then (if and , then );

    6. if is a contraction link and , then (if and , then );

    7. if is a link and , then (if and , then );

    8. if is a simple link, and , then (if , then );

    9. if is a conjunctive focussing link, and or , then (if or , then );

    10. if is a disjunctive defocussing link different from , and , then (if , then );

    11. if is an expansion link, and , then (if , then );

    12. if is a contraction link, and , then (if , then );

    13. if is a link, and , then (if , then ).

We will prove the case for according to bellin:1995 (the case for is similar)

  1. 2 1: we show that 1 is closed under conditions defining 2. Its immediate ( contains for every meta-switching ). If and in all meta-switchings there is an edge , then we conclude (imperialistic lemma girard:1991 ). So the construction is also closed under conditions 2b, 2c, 2d, 2h, 2i and 2j. Conditions 2e, 2l and 2m are also simple. Now suppose that 1 does not respect 2f. Then there is a contraction link such that , but , for or . Consider the first one: for some . Since , then and so . Once is not empty, must be premise of a link whose one conclusion is and . Let be the path between and in . Since , this edge does not belong to (Fig. 5). Consider now a switch like , except that . Note that is in too and (because ). Then we may extend and get a path between and without the edge in : we obtain a cycle in , which is a contradiction. Therefore 1 is closed under 2f. For similar reason, we conclude that 1 is closed under 2k and 2g too.

    Figure 5: If we choose the edge , we get a cycle.
  2. 1 2: let a principal meta-switching for . We will prove 2. 2 , because both contain . But definition 7 ensures that it is impossible to leave once we are in . Since , we conclude that .

Corollary 1

and .

Corollary 2

Let be a premise333Note that it is not valid if is a canceled hypothesis. and a conclusion. Then .

Lemma 4

and are the largest sub-N-Graphs which contains as a lower and upper door, respectively.

The proof uses the same argument presented in bellin:1995 (see Proposition 2). ∎

Fig. 6 illustrates some concepts about empires. For example, in the N-Graph on left, we have (formulas in green), and (formulas in yellow). The formula occurrence in red belongs to both empires. We can see that there is no sub-N-Graph which contains as conclusion (premise) and is larger than (), as both conclusions of a expansion link are needed to add its premise (condition 2e). For the second N-Graph we have the same color scheme for , and here we can not have the conclusion of the contraction link because we need both premises (condition 2l).

Figure 6: N-Graphs for and .

4.2 Nesting lemmas

Lemma 5 (Nesting of empires I girard:1991 )

Let and be distinct formula occurrences in a N-Graph. If and , then .

Lemma 6 (Nesting of empires II girard:1991 )

Let and be distinct formula occurrences in a N-Graph. If and , then .

[Lemmas 5 and 6] Construct a principal meta-switching for with some additional details:

  1. contraction link whose conclusion belongs to : if the conclusion is not in , then we proceed as we do for a principal meta-switching for (if only one premise is in , choose the other premise);

  2. expansion link whose premise belongs to : if the premise is not in , then we proceed as we do for a principal meta-switching for (if only one conclusion is in , choose the other conclusion);

  3. link whose main conclusion belongs to : if the main conclusion is not in , then we proceed as we do for a principal meta-switching for (if only the premise or only the canceled hypothesis is in , choose the formula which is not in );

  4. if is a premise of a link whose conclusion is in : then we choose the edge .

First suppose . We try to go from to without passing through . Since is principal for and , all formulas in the path from to belong to . But and sometime we leave . By construction 2 of lemma 3, there are only three ways of leaving without passing through : passing through a contraction link whose only one premise belongs to , or passing through an expansion link whose only one conclusion belongs to , or passing through a link whose only the premise or the canceled hypothesis is in ; but, steps 1, 2 and 3 avoid this cases, respectively.

Therefore it is impossible to leave in , unless . This implies . Since and , we conclude .

Now suppose . 1, 2 and 3 ensure we do not have any edges between and 444 represents the set of all formula occurrences which are not in in , except perhaps for . But now and therefore . So . Since and , no formula of belongs to and thus . ∎

From these two previous lemmas we have nesting lemmas 7 and 8 for south empires too (the proofs are similar to the previous ones) and from these four nesting lemmas, it is possible to proof nesting lemmas between north and south (9, 10 and 11).

Lemma 7 (Nesting of empires III girard:1991 )

Let and be distinct formula occurrences in a N-Graph. If and , then .

Lemma 8 (Nesting of empires IV girard:1991 )

Let and be distinct formula occurrences in a N-Graph. If and , then .

Lemma 9 (Nesting of empires V)

Let and be distinct formula occurrences in a N-Graph. If and , then .

Lemma 10 (Nesting of empires VI)

Let and be distinct formula occurrences in a N-Graph. If and , then .

Lemma 11 (Nesting of empires VII)

Let and be distinct formula occurrences in a N-Graph. If and , then .

5 Whole empires

We defined the whole empire of as the union of the north and the south empires of . Now we use the north and south empires properties to find new ones about whole empires.

Lemma 12

is a sub-N-graph.

Once we proved that and are N-graphs (lemma 4) and , we get is a sub-N-graph by lemma 1. ∎

Corollary 3

Let be a premise and a conclusion. Then (by corollary 2).

Lemma 13 (Nesting of whole empires I)

Let and be distinct occurrences. If and , then .

Since and , we get: , , and . We apply nesting of empires lemmas:

  1. if and , then (by lemma 6);

  2. if and , then (by lemma 11);

  3. uniting 1 and 2 and applying the distributive law: ;

  4. if and , then (by lemma 11);

  5. if and , then (by lemma 8);

  6. uniting 4 and 5 and applying the distributive law: ;

  7. uniting 3 and 6, the distributive law: .

Lemma 14 (Nesting of whole empires II)

Let and be distinct occurrences. If and , then .

Once , we have and . For we get or . We will prove the lemma for (the case for south is analogous):

  1. if and , then (by lemma 5);

  2. if and , then (by lemma 9);

  3. if and , then ;

  4. if , then .

Definition 8 ()

Let and be formula occurrences of . We say iff .

It is immediate that is a strict ordering of formula occurrences of which are not premises neither conclusions, since we have for any domain set and any subset of 555 is the power set of , is a poset. Maximal formulas with regard to will split . Given that the whole empires of premises and conclusions are always equal to by corollary 3, we are not interested in these formulas. So they are not in the domain of . One may easily verify that if there are no contraction, extension and links, for all formula of , and so any formula would be maximal. The next three following lemmas show how these links act on .

Lemma 15

Let be a link different from such that there is a formula-occurrence which and . Then .

Once , we have two cases. If , then since , must be a contraction link and its other premise does not belong to (construction 2 in lemma 3). Therefore is a conclusion of a contraction link and this implies (by 2f in lemma 3). So . If , then we will have (by lemma 6): a contradiction. Thus and, by lemma 14, we conclude . The case for is analogous.

Next lemma is similar, but for expansion link:

Lemma 16

Let be a link different from such that there is a formula-occurrence which and . Then .

The following lemma is the corresponding for link and its use same ideas as above.

Lemma 17

Let be a link such that there is a formula occurrence which or , but . Then .

6 Sequentialization

We saw in Sections 4 and 5 how to define empires for proof-graphs with switchable defocussing links (expansion) and proved some properties. Now we will show a new proof of sequentialization for these proof-graphs. Without loss of generality, we assume as and as , where the formula belongs to the premise or conclusion of the link.

Theorem 1 (Sequentialization)

Given a N-Graph derivation , there is a sequent calculus derivation of in the classical sequent calculus whose occurrences of formulas and are in one-to-one correspondence with the elements of and , respectively.

We proceed by induction on the number of links of .

  1. does not have any link (it has only one vertex labelled with ): this case is immediate. is .

  2. has only one link: since is a N-Graph, then this link is not a contraction, an expansion, neither a . This case is simple, once there is a simple mapping between links and sequent calculus rules, which makes the construction of immediate (completeness proof oliveira:2001:phd ; oliveira:2001 ). For example, in case :

  3. has an initial expansion link (the premise of the link is a premise of ): the induction hypothesis has built a derivation ending with . Then is achieved by left contraction:

  4. has a final contraction link (the conclusion of the link is a conclusion of ): here the induction hypothesis has built a derivation ending with . Hence is obtained by right contraction:

  5. has a final link (the main conclusion of the link is a conclusion of ): here the induction hypothesis has built a derivation ending with . Hence is obtained by :

  6. has more than one link, but no initial expansion link, no final contraction link and no final link: this case is more complicated and is similar to that one in MLL in which all terminal links are . Yet here we have an additional challenge: the split node is in the middle of the proof. Choose a formula occurrence which is maximal with respect to . We claim that . That is, labels the split node.

    Suppose not. Then let be a formula occurrence such that and be a principal meta-switching for . Given that , the path from to in passes through a conclusion of . Let be the last node which belongs to in and the next one in (i.e. ). There are two cases for the edge incident to and :

    1. belongs to a contraction link whose other premise is not in : according to lemma 15 we have , contradicting the maximality of in .

    2. belongs to an expansion link whose other conclusion is not in : we apply lemma 16 and conclude here too. We contradict our choice again.

    3. belongs to a whose is the main conclusion and the other formula is not in : we use lemma 17 and also get .

    Thus . Let be sets of formula occurrences such that: , and . Since is a N-Graph and is a lower door, the induction hypothesis built ending with . Once is a N-Graph and is an upper door, the induction hypothesis made ending with . So is achieved by cut rule:

7 Conclusion

With N-Graphs, the structural links are based on the sequent calculus, but the logical links emulate the rules of ND. Sequent calculus (classical and linear) have only introduction rules. On the other hand, natural deduction and N-Graphs present elimination rules, so we need to adapt the notion of empire from proof-nets to account for multiple-conclusion ND. This was done with south empires.

Another feature of N-Graphs and ND is the presence of improper rules. The introduction of “” has correctness criteria that do not apply only to local formulas, but to the whole deduction. Thus, besides the presence of expansion link, which is defocussing and switchable, the sequentialization proof showed here also to accomplish a global rule in a system that adopts Danos-Regnier’s criteria.

By mapping derivations in sequent calculus to derivations in ND-style in the way we have just described we have been able to formulate a new and rather general method of performing surgical cuts on proofs in multiple conclusion ND, giving rise to subnets for classical logic. In N-Graphs the split nodes, maximal with respect to the ordering induced by the union of the empires, may be located anywhere in the proof, not only as a terminal node representing a conclusion. We show an example in Fig. 6: every initial link is defocussing unswitchable and every final link is focussing unswitchable in both N-Graphs. Their maximal nodes are highlighted and they split the proofs into two correct proofs. This is illustrated in Fig. 7 for the N-Graph on the left (Fig. 6). The same procedure could also be applied to any of the two maximal nodes in the N-Graph on the right.

Figure 7: Example of how to cut using the maximal node.

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