Proofgraphs for Minimal Implicational Logic
Abstract
It is wellknown that the size of propositional classical proofs can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective nonnormal proofs. The aim of this work is to study how to reduce the weight of propositional deductions. We present the formalism of proofgraphs for purely implicational logic, which are graphs of a specific shape that are intended to capture the logical structure of a deduction. The advantage of this formalism is that formulas can be shared in the reduced proof.
In the present paper we give a precise definition of proofgraphs for the minimal implicational logic, together with a normalization procedure for these proofgraphs. In contrast to standard treelike formalisms, our normalization does not increase the number of nodes, when applied to the corresponding minimal proofgraph representations.
M. AyalaRincón E. Bonelli and I. Mackie (Eds): Developments in Computational Models (DCM 2013) EPTCS 144, 2014, pp. Proofgraphs for Minimal Implicational Logic–LABEL:LastPage, doi:10.4204/EPTCS.144.2 © M. QuispeCruz & E.H. Haeusler & L. Gordeev This work is licensed under the Creative Commons Attribution License.
Proofgraphs for Minimal Implicational Logic
Marcela QuispeCruz \IfArrayPackageLoaded  











1 Introduction
The use of proofgraphs, instead of trees or lists, for representing proofs is getting popular among prooftheoreticians. Proofgraphs serve as a way to provide a better symmetry to the semantics of proofs [11] and a way to study complexity of propositional proofs and to provide more efficient theorem provers, concerning size of propositional proofs. In [3], one can find a complexity analysis of the size of Frege systems, Natural Deduction systems and Sequent Calculus concerning their treelike and listlike representation. This leads to improvement in the size of the listbased proofs compared to treelike proofs, which is based on the observation that the hypotheses occur only once in the lists and more than once in the trees. Thus sharing formulas helps to reduce the size of proofs. There are related works, e.g. [2], that use graphs for representing proofs, pointing out that proofgraphs offer a better way to facilitate the visualisation and understanding of proofs in the underlying logic.
On the other hand [5], [4] and [9] show that the use of Directed Acyclic Graphs (DAGs) together with mechanisms of unification/substitution in proof representations has compacting/compressing factor equivalent to cutintroduction. And, obviously, graphs can save space by means of reference, instead of plain copying. This paper shows yet another advantage of using graphs for representing proofs. We show that using “mixed” graph representations of formulas and inferences in Natural Deduction in the purely implicational minimal logic one can obtain a (weak) normalization theorem that, in fact, is a strong normalization theorem. Moreover the corresponding normalization procedure does not exceed the size of the input, which sharply contrasts to the wellknown exponential speedup of standard normalization. The choice of purely implicational minimal logic () is motivated by the fact that the computational complexity of the validity of is PSPACEcomplete and can polynomially simulate classical, intuitionistic and full minimal logic [12] as well as any propositional logic with a Natural Deduction system satisfying the subformula property [10].
In a more general context, this work has been conducted as part of a bigger treetograph proof compressing research project. The purpose of such proof compression is:

To construct small (if possible, minimal) graphlike representations of standard treelike proofs in a given proof system and – in the propositional case – investigate the corresponding short graphlike theorem provers.

To find short (say, polynomialsize) graphlike analogous of standard treelike proof theoretic operations like e.g. normalization in Natural Deduction and/or cutelimination in Sequent Calculus.
Note that the present work fulfills both conditions with regard to the mimpgraph representation (see below) of chosen Natural Deduction and the corresponding notion of formulaminimality (see Theorems 1 and 2).
Back to the proof normalization, recall the following properties of a given structural deductive system (Natural Deduction, Sequent Calculus, etc):

Normal form: To each derivation of from there is a normal derivation of from .

Normalization: To each derivation of from there is a normal derivation of from , obtained by a particular strategy of reductions application.

Strong Normalization: To each derivation of from there is a normal derivation of from . This normal form can be obtained by applying reductions to the original derivation in any ordering.
The strong normalization property for a natural deduction system is usually proved by the socalled semantical method:

Define a property on derivations in the Natural Deduction system;

Prove that this property implies strong normalization, that is , where means that is strongly normalizable;

Prove that .
There are wellknown examples of this property : (1) Prawitz’s “strong validity”; (2) Tait’s “convertibility”; (3) Jervell’s “regularity”; (4) Leivant’s “stability”; (5) MartinLöf’s “computability”; (6) Girard’s “candidate de reducibilité”. Note that such semantical method is inconstructive and even in the case of purely implicational fragment of minimal logic it provides no combinatorial insight into the nature of strong normalization. Another, more constructive strategy would be to show that there is a worst sequence of reductions always producing a normal derivation. Let us call it a syntactic method of proving the strong normalization theorem. This method is used in the present paper.
Other methods use assignments of rather complicated measures to derivations such that arbitrary reductions decrease the measure, which by standard inductive arguments yields a desired proof of the strong normalization. In this paper we show how to represent derivations in a graphlike form and how to reduce (eliminating maximal formulas) these representations such that a normalization theorem can be proved by counting the number of maximal formulas in the original derivation. The strong normalization will be a direct consequence of such normalization, since any reduction decreases the corresponding measure of derivation complexity. The underlying intuition comes from the fact that our graph representations use only one node for any two identical formulas occurring in the original Natural Deduction derivation (see Theorem 1 for a more precise description).
In [6] another approach to represent Natural Deduction using graphs is proposed. It reports a graphrepresentation of Natural Deduction, in Gentzen as well as Fitch’s style. In fact the proofs are represented as hypergraphs, or boxedgraphs, with possibility of sharing subproofs. It is developed not only for the implicational fragment, although the representation of linear logic proofs is related as further work. Our approach is different from [6], in that we include graphrepresentations of formulas in the proofs. The fact that our normalization procedure leads to strongnormalization is a consequence of sharing subformulas, and hence subproofs, in our proof graph representations. It is unclear whether a similar result is available using [6].
2 Mimpgraphs
Mimpgraphs are special directed graphs whose nodes and edges are assigned with labels. Moreover we distinguish between formula nodes and rule nodes. The formula nodes are labelled with formulas as being encoded/represented by their principal connectives (in particular, atoms) and the rule nodes are labelled with the names of the inference rules (I and E). Both logic connectives proper and inference names may be indexed, in order to achieve a 1–1 correspondence between formulas (inferences) and their representations (names). Since formulas are uniquely determined by the representations in question, i.e. formula node labels, in the sequel we’ll sometimes identify both; to emphasize the difference we’ll refer to the formers as formula graphs, i.e. the ones whose formula node labels are formulas, instead of principal connectives. The edges are labelled with tokens that identify the connections between the respective rule nodes and formula nodes. Note that formulas may occur only once in the mimpgraph. Subformulas are indicated by outgoing edges with labels (left) and (right), see Figure 1.
The rule nodes, like in Natural Deduction, require the correct number of premises. The premises are indicated by ingoing edges and there are edges from the rule nodes to the conclusion formulas. The righthand side of Figure 1 shows the rule nodes I (implication introduction) and E (implication elimination). Note that the discharging of hypotheses may be vacuous. This case in a mimpgraph is represented by a disconnected graph, where the discharged formula node is not linked to the conclusion of the rule by any directed path.
In the rule nodes, formulas are reused, which is indicated by putting several arrows towards it, hence the number of ingoing/outgoing edges with label (premise), (major premise), (minor premise) and (conclusion) coming or going to a formula node could be arbitrarily large. To make all this a bit more intuitive we give an example of a mimpgraph in Figure 2, which can be seen as a derivation of from . Indices of discarded hypotheses are replaced by additional edges assigned with the label: (discharge). This reusing of formulas is necessary. We remind the reader that some valid implicational formulas, such as (see Figure 3), need to use twice a subformula in a Natural Deduction proof, in this case the subformula is used twice. Because of this, the edges , , and in Figure 3 are indexed in a unique way.
The formula nodes in the graph (Figure 2) are labelled with propositional letters , and , the connective ; the rule nodes are labelled with E and I. The underlying idea is that there is an inferential order between rule nodes that provides the corresponding derivability order; the formula node labelled linked to the delimiter node by an edge labelled is the root node and the conclusion of the proof represented by the graph. Besides, the node linked to the delimiter node by the edge labelled (hypothesis) in the graph is representing the premise .
We want to emphasize that the mimpgraphs put together information on formulagraphs and rule nodes. To make it more transparent we can use bicolored graphs. In this way formula nodes and edges between them are painted red, whereas inference nodes and edges between them and adjacent premises and/or conclusions are painted green. So nodes of types and (propositions) together with adjacent edges are red, whereas nodes labelled I and E together with adjacent edges are green.
Now we give a formal definition of mimpgraphs.
Definition 1.
L is the union of the four sets of labels types:

RLabels is the set of inference labels: {IE,

FLabels is the set of formula labels: {} and the propositional letters ,

ELabels is the set of edge labels: { (left), (right), (final conclusion), (hypothesis) (premise) (minor premise) (major premise) (conclusion) (discharge),

DLabels is the set of delimiter labels: .
Definition 2.
A mimpgraph is a directed graph V, E, L, , where: V is a set of nodes, E is a set of edges, L is a set of labels, V, L, V, where is the source and the target, is a labeling function from V to RFLabels, is a labeling function from E to ELabels.
Mimpgraphs are defined recursively as follows:
 Basis

If is a formula graph with root node ^{1}^{1}1We will use the terms , and to represent the principal connective of the formula , and respectively., then the graph is defined as with the delimiter nodes and and the edges and is a mimpgraph.
 E

If and are mimpgraphs, and the graph (intermediate step) obtained by ^{2}^{2}2By definition equalizes the nodes of with the nodes of that have the same label, and equalizes edges with the same source, target and label into one. contains the edge and the two nodes and linked to the delimiter node , then the graph is defined as with

the removal of the ingoing edges in the node which were generated in the intermediate step (see Figure 4, dotted area in );

a rule node E at the top level;

the edges: E, , ,E, E and , where is a fresh (new) index considering all edges of kind , and ingoing and/or outgoing the formulanodes , and ,
is a mimpgraph (see Figure 4).

 I

If is a mimpgraph and contains a node linked to the delimiter node and the node linked to the delimiter node , then the graph is defined as with

the removal of the edges: ;

a rule node I at the top level;

a formula node linked to the delimiter node by an edge ;

the edges: , , I), I, , and I, , where is a fresh index concerning ingoing and outgoing edges of type and of the formulanodes , and ,
is a mimpgraph (see Figure 5; the node is discharged).

 Iv

^{3}^{3}3the “v” stands for “vacuous”, this case of the rule I discharges a
hypothesis vacuously. This means that has no ingoing Hypedge
If is a mimpgraph, and is a formula graph with root node , and contains a node linked to the delimiter node , then the graph is defined as with

the removal of the edge: ;

a rule node I at the top level;

a formula node linked to the delimiter node by an edge ;

the edges: , , I) and I, , where is an index under the same conditions of the previous case,
is a mimpgraph (see Figure 6).

Lemma1 enables us to prove that a given graph is a mimpgraph without explicitly supplying a construction. Among others it says that we have to check that each node of is of one of the possible types that generate the Basis, E, I and Iv construction cases of Definition 2.
Definition 3 (Inferential Ordering).
Let be a mimpgraph. An inferential order on nodes of is a partial ordering of the rule nodes of , such that, , iff, and are rule nodes, and there is a formula node , such that, n f and is and is , or , is and is , or, is and is .
In order to avoid overloading of indexes, we will omit whenever is possible, the indexing of edges of kind , , , and , remembering that the coherence of indexing is established by the kind of rulenode to which they are linked.
Lemma 1.
is a mimpgraph if and only if the following hold:

There exists a wellfounded (hence acyclic) inferential order on all rule nodes of the mimpgraph^{4}^{4}4We can extend this “green” inferential order to the full “mixed” order by adding new “red” relations corresponding to arrows and between formula nodes. Note that may contain cycles (see Figure 2). However all recursive definitions and inductive proofs to follow are based on the wellfounded “green” order , hence being legitimate..

Every node of is of one of the following six types:
 L

is labelled with one of the propositional letters: {p, q, r, … }. has no outgoing edges and .
 F

has label and has exactly two outgoing edges with label and , respectively. has outgoing edges with labels , or ; and it has at most one ingoing edge with label and at most one ingoing edge with label .
 E

has label E and has exactly one outgoing edge (E, , ), where is a node type L or F. has exactly two ingoing edges () and (E), where is a node type L or F. There are two outgoing edges from the node : and .
 I

has label I (or Iv, if discharges an hypothesis vacuously), has one outgoing edge (I, , ), and one (or zero for the case Iv) outgoing edge (I, , ). has exactly one ingoing edge: I, where is a node type L or F. There are two outgoing edges from the node : and .
 H

has label and has exactly one outgoing edge .
 C

has label C and has exactly one ingoing edge .
Proof.
: Argue by induction on the construction of mimpgraph (Definition 2). For every construction case for mimpgraphs we have to check the three properties stated in Lemma. Property (2) is immediate. For property (1), we know from the induction hypothesis that there is an inferential order on rule nodes of the mimpgraph. In the construction cases I, Iv or E, we make the new rule node that is introduced highest in the ordering, which yields an inferential ordering on rule nodes. In the construction case E, when we have two inferential orderings, on and on . Then can be given an inferential ordering by taking the union of and and in addition putting for every rule node such that .
: Argue by induction on the number of rule nodes of . Let be the topological order that is assumed to exist. Let be the rule node that is maximal w.r.t. . Then must be on the top position. When we remove node , including its edges linked (if is of type I) and the node type is linked to the premise of the rule node, we obtain a graph that satisfies the properties listed in Lemma. By induction hypothesis we see that is a mimpgraph. Now we can add the node again, using one of the construction cases for mimpgraphs: Basis if is a L node or F node, E if is an E node, I if is an I node. ∎
It is natural to consider minimal mimpgraphlike representations of given natural deductions. Actually one can try to minimize the number of FLabels and/or RLabels, but for the sake of brevity we consider only the Foption, as it helps to reduce the size under standard normalization (see the next section). To grasp the point note that mimpgraph in Figure 2 (see above) is Fminimal, i.e. its Flabelled nodes refer to pairwise distinct formulas. This observation is summarized by
Theorem 1 (Fminimal representation).
Every standard treelike natural deduction has a uniquely determined (up to graphisomorphism) Fminimal mimplike representation , i.e. such a one that satisfies the following four conditions.

is a mimpgraph whose size does not exceed the size of .

and both have the same (set of) hypotheses and the same conclusion.

There is graph homomorphism that is injective on RLabels.

All FLabels occurring in denote pairwise distinct formulas.
Proof.
Let and be the set of nodes and formulas, respectively, occurring in . Note that determines a fixed surjection that may not be injective (for in , one and the same formula may be assigned to different nodes). In order to obtain take as Rnodes the inferences occurring in assigned with the corresponding “green” RLabels representing inferences’ names (possibly indexed, in order to achieve a 1–1 correspondence between inferences and RLabels, cf. Figure 2). Define basic Fnodes of as formulas from assigned with the corresponding “red” FLabels representing formulas’ principal connectives (possibly indexed, in order to achieve a 1–1 correspondence between formulas and FLabels, cf. Figure 2). So the total number of all basic Fnodes of is the cardinality of the set , while being a mapping from the nodes of onto the basic Fnodes of . To complete the construction of we add, if necessary, the remaining Fnodes labelled by failing “red” representations of subformulas of , , and define the ELabels of (both “green” and “red”), accordingly. Note that by the definition all nodes of have pairwise distinct labels. In particular, every FLabel occurs only once in , which yields the crucial condition 4. ∎
3 Normalization for mimpgraphs
In this section we define the normalization procedure for mimpgraphs. It is based on standard normalization method given by Prawitz. Thus a maximal formula in mimpgraphs is a  followed by a E of the same formula graph (see Definition 4). It is the same notion of maximal formulas that is being used in natural deduction derivations. So a maximal formula occurrence is the consequence of an application of an introduction rule and major premise of an application of an elimination rule. But here we assume that derivations are represented by mimpgraphs. We wish to eliminate such maximal formula by dropping nodes and edges that are involved in the maximal formula. However, it could also happen that between the rule nodes I and E there are several other maximal formulas.
Definition 4.
A maximal formula in a mimpgraph (see Figure 7) is a subgraph of consisting of:

the formula nodes , , , the rule node I and the delimiter node ;

the rule node E at the top level;

the edges: , , I), I, , I, E, E and E;
Definition 5.
(1) For , a ppath in a proofgraph is a sequence of vertices and edges of the form: , such that is a hypothesis formula node, is the conclusion formula node, alternating between a rule node and a formula node. The edges alternate between two types of edges: the first is and the second . (2) A branch is an initial part of a ppath which stops at the conclusion formula node or at the first minor premise whose major premise is the conclusion of a rule node.
Definition 6.
Let a graph obtained by dropping rule nodes in a mimpgraph then, the reordering of is defined as the graph with the following (new) inference order on the rule nodes of .

for a rule node starting with hypothesis.

if the conclusion formula of rule node is premise or major premise of .
Proposition 1.
If a graph is obtained by a reordering by means of the operation defined in Definition 6 then, is a mimpgraph.
Definition 7.
Given a mimpgraph with a maximal formula , eliminating a maximal formula is the following transformation of a mimpgraph, where the maximal formula satisfies the following requirements:

Between the rule nodes I and E there are zero or more maximal formulas with inferential orders within the range of these rule nodes.

There is an edge I, and, the formula node has zero or more ingoing edges.

There is an edge E, and, the formula node is the premise of zero or more of another rule nodes.

If a branch will be separated from the inferential order this branch must be insertable in the following branch, according to the order, i.e. the conclusion of this separated branch is the premise in the following branch.
The elimination of a maximal formula is the following operation on a mimpgraph (see Figure 8, the dotted arrows are representing sets of edges):

If there is no maximal formula between the rule nodes I and E then follow these steps:

If the edge I is the only ingoing edge to and the edge E is the only outgoing edge from then remove the edges to and from the formula node , and the formula node .

Remove the edges to and from the nodes I, E and .

Remove the nodes I, E and .


Otherwise eliminate the maximal formulas between the rule nodes I and E as in the previous step.
Note that the removal of a node I generated by case Iv, in the Definition 2, disconnects the graph meaning that the subgraph hypotheses linked, by the edge , to eliminated node E is no longer connected to the delimiter .
Let us show in Figure 9 an instance of the eliminating a maximal formula in tree form. Note that this example shows the reason why essentially our (weak) normalization theorem is directly a strong normalization theorem. The formula is not a maximal formula before a reduction is applied to eliminate the maximal formula . This possibility of having hidden maximal formulas in Natural Deduction is the main reason to use more sophisticated methods whenever proving strong normalization. In mimpgraphs there is no possibility to hide a maximal formula because all formulas are represented only once in the graph. In this graph is already a maximal formula. We can choose to remove any of the two maximal formulas. If is chosen to be eliminated, by the mimpgraph normalization procedure, its reduction eliminates the too. On the other hand, the choice of to be reduced only eliminates itself. In any case the number of maximal formula decreases.
We shall construct the normalization proof for mimpgraphs. This proof is guided by the normalization measure. That is, the general mechanism from the proof determines that a given mimpgraph should be transformed into a nonredundant mimpgraph by applying of reduction steps and at each reduction step the measure must be decreased. The normalization measure will be the number of maximal formulas in the mimpgraph.
Also note a following important observation concerning Fminimal mimp representations (see Theorem 1). Since Fminimal mimps can have at most one occurrence of hypotheses and/or , every proper reduction step will diminish the size of deduction. Hence the size of the graph (= the number of nodes) can serve as another inductive parameter, provided that the normalization is being applied to Fminimal mimpgraph representations.
Theorem 2 (Normalization).
Every mimpgraph can be reduced to a normal mimpgraph having the same hypotheses and conclusion as . Moreover, for any standard treelike natural deduction , if (the Fminimal mimplike representation of , cf. Theorem 1), then the size of does not exceed the size of , and hence also .
Remark 3.
The second assertion sharply contrasts to the wellknown exponential speedup of standard normalization. Note that the latter is a consequence of the treelike structure of standard deductions having different occurrences of equal hypotheses formulas, whereas all formulas occurring in Fminimal mimplike representations are pairwise distinct.
Proof.
This characteristic of preservation of the premises and conclusions of the derivation is proved naturally. Through an inspection of each elimination of maximal formula is observed that the reduction step (see Definition 7) of the mimpgraph does not change the set of premises and conclusions (indicated by the delimiter nodes and ) of the derivation that is being reduced.
In addition, the demonstration of this theorem has two primary requirements. First, we guarantee that through the elimination of maximal formulas in the mimpgraph, cannot generate more maximal formulas.The second requirement is to guarantee that during the normalization process, the normalization measure adopted is always reduced.
The first requirement is easily verifiable through an inspection of each case in the elimination of maximal formulas. Thus, it is observed that no case produces more maximal formulas. The second requirement is established through the normalization procedure and demonstrated through an analysis of existing cases in the elimination of maximal formulas in mimpgraphs. To support this statement, it is used the notion of normalization measure, we adopt as measure of complexity (induction parameter) the number of maximal formulas . Besides, as already mentioned, working with Fmimimal mimpgraph representations we can use as optional inductive parameter the ordinary size of mimpgraphs. ∎
Normalization Process
We know that a specific mimpgraph can have one or more maximal formulas represented by . Thus, the normalization procedure is described by the following steps:

Choosing a maximal formula represented by .

Identify the respective number of maximal formulas .

Eliminate maximal formula as defined in Definition 7.

In this application one of the following three cases may occur:
 a)

The maximal formula is removed.
 b)

The maximal formula is removed but the formula node is maintained, hence is decreased;
 c)

All maximal formulas are removed.

We repeat this process until the normalization measure is reduced to zero and becomes a normal mimpgraph.
Since the process of the eliminating a maximal formula on mimpgraphs always ends in the elimination of at least one maximal formula, and with the decrease in the number of vertices of the graph, we can say that this normalization theorem is directly a strong normalization theorem.
4 Conclusions and Related works
This representation of a proof in mimpgraph requires fewer nodes than the tree and the list representation of proofs. For the case of lists, it is enough to observe that a subformula of a formula is already in any graph representation of it. If both take part in the proof the size is smaller than in the mentioned representations. The ability to represent any Natural Deduction proof is preserved. Another important advantage of a compact representation of graphs is that it allows to deduce some structural properties of proofgraphs, for example based on a mimpgraph, it is easy to see an upper bound in the length of the reduction sequence to obtain a normal proof. It is the number of maximal formulas.
There is some previous research concerning the use of graphs to represent proofs developed on connections to substructural logics as Linear Logic, see [8] and [7] for example. The main motivation of this just mentioned investigations is to provide a sound way of representing Linear Logic proofs without dealing with unique labeling and complicated rules for relabeling and discharging mechanisms need to represent Linear Logic proofs as trees in Natural Deduction styles as well as in Sequent Calculus. Proofnets were such representations and a syntactical criteria on the possible paths on them were considered as a soundness criteria for a proofgraph to be a proofnet. Proofnets have a cutrule quite similar to the cut in Sequent Calculus. For the Multiplicative fragment of Classical Linear Logic, there is a linear time cutelimination theorem. However, when the additive versions of the connectives are considered, the usual complexity of the cutelimination raises up again. Linear Logic is an important Logic whenever we consider the study of a concurrent computational systems and its semantics strongly uses concurrency theory concepts. Our investigations, on the other hand, is not motivated by prooftheoretical semantics ^{5}^{5}5The name that nowadays it is used to denote the kind of research pioneered by JeanYves Girard. From the purely prooftheoretical point of view, we use graphs to reduce the redundancy in proofs in such a way that we do not allow hidden maximal formulas in our graph representation of a Natural Deduction proof. As a secondary motivation, this is a preliminary step into investigating how a theorem prover based on graphs is more efficient than usual theorem provers. Our proofgraphs represent the formulas themselves in a way that each subformula is a unique node in the graph. Proof nets do not represent formulas in this way.
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