Pomset logic: a logical and grammatical alternative to the Lambek calculus

01/07/2020 ∙ by Christian Retoré, et al. ∙ 0

Thirty years ago, I introduced a non commutative variant of classical linear logic, called POMSET LOGIC, issued from a particular denotational semantics or categorical interpretation of linear logic known as coherence spaces. In addition to the multiplicative connectives of linear logic, pomset logic includes a non-commutative connective, "<" called BEFORE, which is associative and self-dual: (A<B)^=A^ < B^ (observe that there is no swapping), and pomset logic handles Partially Ordered MultiSETs of formulas. This classical calculus enjoys a proof net calculus, cut-elimination, denotational semantics, but had no sequent calculus, despite my many attempts and the study of closely related deductive systems like the calculus of structures. At the same period, Alain Lecomte introduced me to Lambek calculus and grammars. We defined a grammatical formalism based on pomset logic, with partial proof nets as the deductive systems for parsing-as-deduction, with a lexicon mapping words to partial proof nets. The study of pomset logic and of its grammatical applications has been out of the limelight for several years, in part because computational linguists were not too keen on proof nets. However, recently Sergey Slavnov found a sequent calculus for pomset logic, and reopened the study of pomset logic. In this paper we shall present pomset logic including both published and unpublished material. Just as for Lambek calculus, Pomset logic also is a non commutative variant of linear logic — although Lambek calculus appeared 30 years before linear logic ! — and as in Lambek calculus it may be used as a grammar. Apart from this the two calculi are quite different, but perhaps the algebraic presentation we give here, with terms and the semantic correctness criterion, is closer to Lambek's view.

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1 Presentation

Lambek’s syntactic calculus as Lambek [16] used to call his logic, was in keeping with his preference for algebra (confirmed with his move from categorial grammars to pregroup grammars, which are not a logical system.) Up to the invention of linear logic in the late 80s, Lambek calculus was a rather isolated logical system, despite some study of frame semantics, which are typical of substructural logics.

Linear logic [7] arose from the study of the denotational semantics of system F, itself arising from the study of ordinals. [6] For interpreting systems F (second order lambda calculus) with variable types, one needed to refine the categorical interpretation of simply typed lambda calculus with Cartesian Closed Categories. In order to quantify over types Girard considered the category of coherence spaces (first called qualitative domains) with stable maps (which preserve directed joins and pullbacks). A finer study of coherence spaces led Girard to discompose the arrow type construction in to two steps: one is to contract several object of type into one (modality/exponential !) and the other one being linear implication (noted ) which rather corresponds to a change of state than to a consequence relation.

Linear logic was first viewed as a proof system (sequent calculus or proof nets) which is well interpreted by coherence spaces. The initial article [7] also included the definition of phase semantics, that resembles frame semantics developed for the Lambek calculus. It was not long before the connection between linear logic and Lambek calculus was found: after some early remarks by Girard, Yetter [50] observed the connection at the semantic level, while Abrusci [1] explored the syntactic, proof theoretical connection, while [34] explored proof nets and completed the insight of [46]. Basically Lambek calculus is non commutative intuitionistic multiplicative logic, the order between the two restrictions, intuitionistic and non commutative, being independent. An important remark, that I discussed with Lamarche in [15], says that non commutativity requires linearity in order to get a proper logical calculus.

Around 1988, my PhD advisor Jean-Yves Girard pointed to my attention a binary non commutative connective in coherence spaces. In coherence spaces, this connective has intriguing properties:

  • is self dual , without swapping the two components — by we mean that there is a pair of canonical invertible linear maps between and .

  • is non commutative

  • is associative ;

  • it lies in between the commutative conjunction and disjunction there is a canonical linear map from to an one from to 111Coherence spaces validate the mix rule ;

I designed a proof net calculus with this connective, in which a sequent, that is the conclusion of a proof, is a partially ordered multiset of formulas. This proof net calculus enjoys cut-elimination and a sound and faithful (coherence) semantics 222In the sense that having an interpretation is the same as being syntactically correct cf. theorem 8 preserved under cut elimination. I proposed a version of sequent calculus that easily translates into those proof nets and enjoys cut-elimination as well. [32] However despite many attempts by me and others (Sylvain Pogodalla, Lutz Straßburger) over many years we did not find a sequent calculus that would be complete w.r.t. the proof nets. Later on, Alessio Guglielmi, soon joined by Lutz Straßburger, designed the calculus of structures, a term calculus more flexible than sequent calculus (deep inference) with the before connective [9, 10, 11], a system that is quite close to dicograph rewriting [40, 41], described in section 3.2. They tried to prove that one of their systems called BV was equivalent to pomset logic and they did not succeed. As a reviewer of my habilitation [42] Lambek wrote:

He constructs a model of linear logic using graphs, which is new to me. His most original contribution is probably the new binary connective which he has added to his non commutative version of linear logic, although I did not find where it is treated in the sequent calculus. (J. Lambek, Dec. 3, 2001)

I deliberately omitted my work on sequent calculus in my habilitation manuscript, because none of the sequent calculi I experimented with was complete w.r.t. pomset proof nets which are ”perfect”, i.e. enjoy all the expected proof theoretical properties. In addition, by that time, I did not yet have a counter example to my proposal of a sequent calculus, the one in picture 5 of section 6 was found ten years later with Lutz Straßburger.

However, very recently, Sergey Slavnov found a sequent calculus that is complete w.r.t. pomset proof nets. [48] The structure of the decorated sequents that Slavnov uses is rather complex 333A decorated sequent according to Slavnov is a multiset of pomset formulas with binary relations between sequences of length of formulas from ; those relations are such that whenever the two sequences and have no common elements and entails for any permutation of – those relations correspond top the existence of disjoint paths in the proof nets from to . and the connective is viewed as the identification of two dual connectives one being more like a and the other more like a . As this work is not mine I shall not say much about it, but Slavnov’s work really sheds new light on pomset logic. Given the complexity of this sequent calculus it is pleasant to have some simple sequent calculus and a rewriting system for describing most useful proof nets e.g. the one used for grammatical purposes.

Pomset logic and the Lambek calculus systems share some properties:

  • They both are linear calculi;

  • They both handle non commutative connective(s) and structured sequents;

  • They both have a sequent calculus;

  • They both enjoy cut-elimination;

  • They both have a complete sequent calculus (regarding pomset logic the complete sequent calculus is quite new);

  • They both can be used as a grammatical system.

However Lambek calculus and pomset logic are quite different in many respects:

  • Lambek calculus is naturally an intuitionistic calculus while pomset logic is naturally a classical calculus — although in both cases variants of the other kind can be defined.

  • Lambek calculus is a restriction of the usual multiplicative linear logic according to which the connectives are no longer commutative, while pomset logic is an extension of usual commutative multiplicative linear logic with a non commutative connective.

  • Lambek calculus deals with totally ordered multisets of hypotheses while pomset logic deals with partially ordered multisets of formulas. As grammatical systems, pomset logic allows relatively free word order, while Lambek calculus only deals with linear word orders.

  • Lambek calculus has an elegant truth-value interpretation within the subsets of a monoid (frame semantics, phase semantics), while there is not such a notion for pomset logic.

  • Lambek calculus has no simple concrete interpretation of proofs up to cut elimination (denotational semantics) while coherence semantics faithfully interprets the proofs of pomset logic.

This list shows that those two comparable systems also have many differences. However, the presentation of Pomset logic provided by the present article make Lambek calculus and pomset logic rather close on an abstract level. As he told many of us, Lambek did not like standard graphical or geometrical presentation of linear logic like proof nets. He told me several times that moving from geometry to algebra has been a great progress in mathematics and solved many issues, notably in geometry, and that proof net study was going the other way round. I guess this is related to what he said about theorem 8.

It seems that this ingenious argument avoids the complicated long trip condition of Girard. It constitutes a significant original contribution to the subject. (J. Lambek,Dec 3 2001)

This paper is a mix (!) of easy to access published work, [5, 34, 4, 37, 38, 35, 43] research reports and more confidential publications [32, 31, 5, 18, 34, 31, 33, 18, 19, 15, 20, 40, 41, 39, 42, 30] unpublished material between 1990 and 2020, that are all presented in the same and rather new unified perspective; the presented material can be divided into three topics:

proof nets

handsome proof nets both for MLL Lambek calculus and pomset logic, and other work on proof nets [5, 15, 34],

combinatorics

(di)cographs and sp orders [31, 32, 35, 4, 40, 41, 39, 42, 43, 42, 43, 30],

coherence semantics

[32, 33, 38, 42],

grammatical applications

of pomset logic to computational linguistics [18, 19, 20, 40, 41, 42].

The contents of the present article is divided into six sections as follows:

  1. We first present results on series parallel partial orders, cographs and dicographs that subsumes those two notions and present dicograph either as sp pomset of formulas or as dicographs of atoms, and explain the guidelines for finding a sequent calculus. This combinatorial part is a prerequisite for the subsequent sections.

  2. We then present proofs in an algebraic manner, with deduction rules as term rewriting.

  3. Proof nets without links, the so called handsome proof nets, are presented as well as the cut elimination for them.

  4. The semantics of proof nets, preserved under cut elimination and equivalent to their syntactic correctness is then presented.

  5. Then the sequentialisation ”the quest” of a complete sequent calculus is discussed and we provide an example of a proof net that does not derive from any simple sequent calculus.

  6. Finally we explain how one can design grammars by associating words with partial proof nets of pomset logic.

2 Structured sequents as dicographs of formulas

2.1 Looking for structured sequents

The formulas we consider are defined from atoms (propositional variables or their negation) by means of the usual commutative multiplicative connectives and together with the new non commutative connective (before)— the three of them are associative.

It is assumed that formulas are always in negative normal form: negation only apply to propositional variables; this is possible and standard when negation is involutive and satisfies the De Morgan laws:

We want to deal with series parallel partial orders of formulas: corresponds to parallel composition of partial orders (disjoint union) and corresponds to the series composition of partial orders (every formula in the first partial order is lesser than every formula in the second partial order ). Thus, a formula written with and corresponds to a partial order between its atoms. Unsurprisingly, we firstly need to study a bit partial orders defined with series and parallel composition.

However, what about the multiplciative, conjunction namely the connecitve? It is commutative, but it is distinct from . In order to include in this view, where formulas are binary relations on their atoms, we consider, the more general class of irreflexive binary relations that are obtained by parallel composition, series composition and symmetric series compositions, which basically consists in adding the relations of and the ones of . The relations that are defined using , , are called directed cographs or dicographs for short.

If only and are used the relations obtained are cographs. They have already been quite useful for studying MLL, see e.g. theorem 4 thereafter.

Before defining pomset logic, we need a presentation of directed cographs.

2.2 Directed cographs or dicographs

An irreflexive relation may be viewed as a graph with vertices and with both directed edges and undirected edges but without loops. Given an irreflexive relation let us call its directed part (its arcs) and its symmetric part (its edges) . It is convenient to note for the edge or pair of arcs in and to denote for in when is not in .

We consider the class of dicographs, dicographs for short, which is the smallest class of binary irreflexive relations containing the empty relation on the singleton sets and closed under the following operations defined on two cographs with disjoint domains and yielding a binary relation on

  • symmetric series composition

  • directed series composition

  • parallel composition

Whenever there are no directed edges (a.k.a. arcs) the dicograph is a cograph ( is not used). Cographs are characterised by the absence of as many people (re)discovered including us [35], see e.g. [14].

Whenever there are only directed edges (a.k.a. arcs) the dicograph is an sp order ( is not used) — as rediscovered in [32], see e.g. [24]

Let us call this class the class of dicographs.

We characterised the class of directed dicographs as follows [4, 40, 41]:

Theorem 1

An irreflexive binary relation is a dicographs if and only if:

  • is N-free ( is an sp order).

  • is -free ( is a cograph).

  • Weak transitivity:
    forall in the domain of
    if and then and
    if and then

A dicograph can be described with a term in which each element of the domain appears exactly once. This term is written with the three binary operators , and and for a given dicograph this term is unique up to the associativity of the three operators, and to the commutativity of the first two, namely and .

The dual of a dicograph on is defined as follows: points are given a superscript, and or , , , , .

Two points and of are said to be equivalent w.r.t. a relation whenever for all with one as and . There are three kinds of equivalent points:

  • Two points and in a dicograph are said to be freely equivalent in a dicograph (notation ) whenever the term can be written (using associativity of and and the commutativity of ) . In other words, , , .

  • Two points and in a dicograph are said to be arc equivalent in a dicograph (notation ) whenever the term can be written (using associativity of , and and the commutativity of and ) . In other words, , , .

  • Two points and in a dicograph are said to be edge equivalent in a dicograph (notation ) whenever the term can be written (using associativity of and and the commutativity of ) . In other words, , , .

2.3 Dicograph inclusion and (un)folding

The order on a multiset of formulas, can be viewed as a set of contraints. Hence, when a sequent is derivable with an sp order it is also derivable with a sub sp order — we named this structural rule entropy [32]. Most of the transformations of a dicograph into a smaller (w.r.t. inclusion) dicograph preserve provability. Hence we need to characterise the inclusion of a dicograph into another and possibly to view the inclusion as a computational process that can be performed step by step. Fortunately,in [4] we characterised the inclusion of a dicograph in another dicograph by a rewriting relation:

Theorem 2

A dicograph is included into a dicograph if and only if the term rewrites to the term using the rules of figure 1 — up to the associativity of , and , and to the commutativity of and .

Figure 1: A complete rewriting system for dicograph inclusion. Beware that the first rule marked with a is wrong when the rewriting rule is viewed as a linear implication on formulas: although all other rewriting rules are correct when viewed as linear implications.

2.4 Folding and unfolding pomset logic sequents

A structured sequent of pomset logic (resp. of MLL) is a multiset of formulas of pomset logic (resp. of MLL) with the connectives endowed with a dicograph.

On such sequents one may define “folding” and “unfolding” which transform a dicograph of formulas into another dicograph of formulas by combining two equivalent formulas and of the dicograph into one formula (folding) or by splitting one compound formula into its two immediate subformulas and with and equivalent in the dicograph. More formally:

Folding

Given a multiset of formulas endowed with a dicograph ,

if in rewrite into — in the multiset, the two formulas and have been replaced with a single .

if in rewrite into — in the multiset, the two formulas and have been replaced with a single formula .

if in rewrite into — in the multiset, the two formulas and have been replaced with a single formula .

Unfolding

is the opposite:

turn into — in the multiset, the formula has been replaced with two formulas and with

turn into — in the multiset, the formula has been replaced with two formulas and with

turn into — in the multiset, the formula has been replaced with two formulas and with

2.5 A sequent calculus attempt with sp pomset of formulas

If we want to extend multiplicative linear logic with a non commutative multiplicative self dual connective (rather than to restrict existing connective to be non commutative), and want to handle partially ordered multisets of formulas, with corresponding to ”the subformula is smaller than the subformula ”.

That way one may think of an order on computations: a cut between and reduces to two smaller cuts and with the cut on being prior to the cut on , while a cut between and reduces to two smaller cuts and with the cut on being in parallel with the cut on . This makes sense when linear logic proofs are viewed as programs and cut-elimination as computation.

Doing so one may obtain a sequent calculus using partially ordered multisets of formulas as in [32] but if one wants a sequent with several conclusions that are partially ordered to be equivalent to a sequent with a unique conclusion, one has to only consider sp partial orders of formulas, as defined in subsection 2.2 with parallel composition noted and series composition noted .

If we want all formulas in the sequent to be ordered the calculus should handle right handed sequents i.e. be classical.444Lambek calculus is intuitionistic and when it is turned into a classical systems, formulas are endowed with a cyclic order,[50, 1, 15], i.e. a ternary relation which is not an order and which is quite complicated when partial — see the ”seaweeds” in [3].

As seen above, we can represent this sp partially ordered multiset of formulas endowed with an sp order by an sp term whose points are the formulas and such a term is unique up to the commutativity of and the associativity of and .

We know how the tensor rule and the cut rule behave w.r.t. formulas. The only aspect that deserves some tuning, is the order on the formulas after applying those binary rules. Our choice is guided by two independent criteria:

  1. The resulting partial order should be an sp order.

  2. This sequent calculus should enjoy cut elimination:

    • If there is a cut between and with coming immediately from a rule from and and coming immediately from a rule from one should be able to locally turn those rules into two consecutive cuts, one between and and then one with .

    • If there is a cut between and with coming immediately from a rule from and coming immediately from a rule from one should be able to locally turn those rules into two consecutive cuts, one between and and then one with .

A simple sequent calculus is provided in figure 2. 555An alternative solution to have on sp orders is to have rule between two minimum in their order component, and to have cut between two formulas one of which is isolated in its ordered sequent. This alternative is trickier and up to our recent investigation does not enjoy better properties than the version given above in figure 2

A property of this calculus is that cuts can be part of the order on conclusions. Indeed, one may define a cut as a formula that never is used as a premise of a logical rules. That way, the order can be viewed as an order on computation. A cut reduces into two cuts that are : , while cut reduces into two cuts that are : — beware that is a cut and that a operation on dicographs is different from the connective, which combines formulas. When one of the two premises of the cut is an axiom, this axiom and the cut are simply removed from the proof as usual, and this is possible because cut only applies when the two premises are isolated in the sp order. An alternative proof of cut elimination can be obtained from the cut elimination theorem for proof nets with links or without to be defined in sections 3.2 or 7.1, as established in [32, 37, 41] — because the reduction of a proof net coming from a sequent calculus proof also comes from a sequent calculus proof. Thus one has:

Theorem 3

Sequent calculus of figure 2 enjoys cut-elimination.

⊢Γ  ⊢Δ⊢Γ^¡Δdimix      ⊢Γ⊢Γ’ entropy (Γ’ sub sp order of Γ)

⊢a, a^⟂

⊢A^⅋Γ  ⊢B^⅋Δ⊢Γ^⅋(A⊗B) ^⅋Δ⊗ / cut when A=B^⟂

⊢Γ[A^⅋B] ⊢Γ[A⅋B] ⅋ when A∼⋅ ⋅B      ⊢Γ[A^¡B] ⊢Γ[A¡B] ¡ when A∼B

Figure 2: Sequent calculus on sp pomset or formulas; called sp-pomset sequent calculus

3 Pomset logic and MLL as (di)cograph rewriting:

Before we define a deductive system for pomset logic, let us revisit (as we did in [39, 43]) the deductive system of Multiplicative Linear Logic (MLL). Those results are highly inspired from proof nets, but once they are established they can be presented before proof nets are defined.

In this section a sequent is simply a dicograph of atoms which as explained above can be viewed using folding of section 2.4 as a dicograph of formulas or as an sp order between formulas depending on how many folding transformations and which one are performed.

Regarding, multiplicative linear logic (MLL), observe that is the largest cograph or even the largest dicograph w.r.t. inclusion that could be derived in MLL — indeed there cannot be any tensor link nor any before connection between the two dual occurrences of atoms issued from the same axiom link, for two reasons: first in sequent calculus they cannot lie in different sequents and therefore they cannot be conjoined by or ; second, as explained in subsection 3.2, in the proof net this would result in a prohibited (ae) cycle. Observe that , the largest derivable cograph in MLL is acutally derivable in MLL, hence in any extension of MLL:

3.1 Standard multiplicative linear logic as cograph rewriting

In [39] we considered an alternative way to derive theorems of usual multiplicative linear logic MLL, by considering a formula as a binary relation, and more precisely as a cograph over its atoms, by viewing as and as . As there is no connective in linear logic the series composition is not used, and there is no sp order on conclusions.

Because of the chapeau of the present section 3 any sequent of MLL can be viewed is a cograph on atoms that is included into . Because of theorem 1, rewrites to using the rules of figure 1 that concern and i.e. , and . Observe that when viewed as a linear implication (considering the rules involving those two connectives), the first line is an incorrect linear implication, while is derivable in MLL and in MLL+MIX where the rule MIX is the one studied in [5], which also is derivable with :

Actually all tautologies of multiplicative linear logic MLL can be derived using from an axiom , and all tautologies of linear logic enriched with the MIX rule, MLL+MIX, can be derived by and (MIX).

Thus, we can define a proof system gMLL for MLL working with sequents as cographs of atoms as follows. Axioms are : (the two dual atoms are connected by an edge in a different relation called for axioms). There is just one deduction rule presented as a rewrite rule (up to commutativity and associativity): .

Let us call this deductive system gMLL (g for graph), then [39, 43] established that cograph rewriting is an alternative proof systems to MLL and MLL+MIX.

Theorem 4

MLL proves a sequent with atoms if and only if gMLL proves the unfolding of (the cograph of atoms corresponding to , that is the of the unforging of each formula in ), i.e. rewrites to using .

MLL+MIX proves a sequent with atoms if and only if gMLL+mix proves the unfolding of , i.e. rewrites to using and .

Proof.  Easy induction on sequent calculus proofs see e.g. [39, 43]. A direct proof by Straßbruger can be found in [49].

The interesting thing is that all proofs can be transformed that way. Unfortunately it if much easier with an inductive definition of proofs like sequent calculus, and, unfortunately for pomset logic, it is hard to prove it directly on a non inductive notion of proof like proof nets.

Proposition 1

The calculi gMLL and gMLL+mix can safely be extended to structured sequents of formulas of MLL (not just atoms), i.e. cographs of MLL formulas with the rules of folding and unfolding with the same results.

Proof.  This is just an easy remark, based on proof nets, which can be viewed as a consequence subsection 7.1.

3.2 Pomset logic as a calculus of dicographs: dicog-RS

Using the above results for MLL suggests defining a deductive system for pomset logic in the same manner. All rewriting rules are correct but : they correspond to proof nets or to sequent calculus derivations (with the sp-pomset sequent calculus of figure 2) and to canonical linear maps in coherence spaces. So it suggest that a rewriting system defined as gMLL+mix in the previous section (but with dicographs instead of cographs) might yield all the proofs we want e.g. all correct proof nets.

Axioms

is a tautology.

Rules

Whenever a dicograph of atoms which is a tautology rewrites to a dicograph (hence with the sames atoms) by any of the 10 rules , , , , , , , , , of figure 1 — i.e. all rules of figure 1 but .

Unfortunately, proving that all proof nets are derivable by rewriting is not simpler than proving that they can be obtained from the sequent calculus. This would entail the equivalence of pomset logic with BV calculus as discussed in [12].

3.3 Cuts

What about the cut rule? For such logical systems based on rewriting systems like gMLL(+MIX), of the dicog-RS view of pomset logic, which does not work with ”logical rules” in the standard sense, there are no binary rules that would combine a and a . So the only view of a cut is simply a tensor which never is inserted inside a formula. A dicograph may be written . Observe that contains the duals of the atoms in , because it is a cut, and that there is one for each , because they are the atoms of minus the well balanced atoms of and , one cannot say that the pair corresponds to some from — necessarily for some pair one is among the and and one is among the and .

However one cannot say that a proof of dicog-RS, i.e. a sequence of derivations yielding a dicograph with cuts (i.e. with a sub dicograph term ) may be turned into a dicog-RS derivation whose final dicographs is restricted to the atoms that are neither in not in . Indeed the atoms in and vanish during the process and none of the rewrite rules is able to do so — furthermore if one looks at step by step cut elimination, it precisely uses the prohibited rewriting rule !

We shall see later that in fact cut elimination holds for proof nets that are dicographs of atoms but without any inductive notion of derivation.

4 Proof nets

This section presents proof structures and nets (the correct proof structures), in an abstract and algebraic manner, without links nor trip conditions: such proof strctures and nets are called handsome proof structures and nets. Basically proof nets consists in a dicograph of atoms representing the conclusion formula, and axioms that are disjoint pairs of dual atoms constituting a partition of the atoms of . The proof net can be viewed as an edge bi-coloured graph: the dicograph is represented by arcs and edges (Red and Regular in the pictures), while the axioms (Blue and Bold in the picture). In such a setting, the correctness criterion expresses some kind of orthogonality between and . A proof net can also be viewed as a term, axioms being denoted by indices used exactly twice on dual atoms.

4.1 Handsome pomset proof nets

In fact, proof nets have (almost) been defined above!

A pomset logic handsome proof structure or dicog-PN is a dicograph over a (multi)set of atoms, , i.e. propositional letters and their duals. it is fairly possible that two atoms have the same name, i.e. it is a multiset of atoms. Let us call the binary relation or simply using the notations of the previous section 2.2; observe that no two edges are incident, and that each point is incident to exactly one edge in : the edges constitute a perfect matching of the whole graph with both edges and edges and arcs.

Given two proof structures and whose atoms and axioms are the same, and whose conclusion formulas and only differ because of the associativity of and the commutativity of , the proof structures and are equal — while in proof structures with links they would be different.

Correctness criterion 1

A handsome proof structure is said to be a proof net whenever every elementary circuit (directed cycle) of alternating edges in and in contains a chord — an edge or arc connecting two points of the circuit but not itself nor its reverse in the circuit. In short, every ae circuit contains a chord. Observe that this chord cannot be in , hence it is in , and it can either be an arc or an edge.

Theorem 5 (Nguyên)

Recently it was established that checking whether a proof structure satisfies the above correctness criterion is coNP complete [25].

Figure 3: A proof structure containing a chordless alternate elementary (on the left) and a proof net without any chordless alternate elementary path (on the right).
Theorem 6

Given a proof net if (so ) using rewriting rules of Figure 1 except then is a proof net as well, i.e. all the rewrite rule preserve the correctness criterion on page 1.

Proof.  See [40, 41].

It is easily seen that in general does not preserve correctness:

. Using , rewrites to
, and the proof net contains the chordless æ circuit .

Observe that it does not mean that every correct proof net with axioms can be obtained from by the allowed rewrite rules (all but ) where is . Indeed, since it is known that but one cannot tell that is not used. Indeed, as shown above does not preserve correctness but it may happen:

As indicated in section 2.2 we write for the edge or par of opposite arcs

is correct, and using it rewrites to which is correct as well.

4.2 Cut and cut-elimination

What about the cut rule? This calculus has no rules in the standard sense, in particular no binary rules that would combine a and a . A cut is a tensor which never is inserted inside a formula.

So a cut in this setting simply is a symmetric series composition in a dicograph whose form is . Assume the atoms of are so atoms of are . Cut-elimination consist in suppressing all edges and arcs between two atoms of , all edges and arcs between two atoms of , and all edges with — so the only edges incident to are (call those edges atomic cuts) and with neither in not in . If, in this graph, an atom is in the relation with an in , then the result of cut elimination is the closest point not in nor in reached by an alternating sequence of -edges and elementary cuts starting from – observed that this point is necessarily named , that we call its cut neighbour. To obtain the proof resulting from cut-elimination suppress all the atoms of and as well as the incident arcs and edges and connect every atom to its cut neighbour with a edge.

Theorem 7

Cut elimination preserves the correctness criterion of dicog-PN proof nets and consequently the f dicog-PN proof nets enjoy cut-elimination.

Proof.  The preservation of the absence of chordless æ circuit during cut elimination is proved in [40, 41].

4.3 From sequent calculus and rewrite proofs to dicog-PN

Proofs of the sequent calculus given in figure 2 are easy turn into a dicog-PN proof net inductively. Such a derivation starts with axioms as it is well known, and in any kind of multiplicative linear logic the atoms and that can be traced from the axiom that introduced them to the conclusion sequent, which, after some unfolding can be viewed as a dicograph of atoms . The dicog-PN proof structure corresponding to the sequent calculus proof simply is , and fortunately is a correct proof net.

Proposition 2

A proof of sequent calculus corresponds to a dicog-PN i.e. to a handsome proof structure without chordless alternate elementary path, i.e. into a handsome proof net.

Proof.  By induction on the proof, we showed in shown in [40, 41] that the neither the rules nor the the unfolding can introduce a chordless ae cycle.

The above result also yield cut elimination for the sequent calculus. Indeed, proof nets obtained by cut-elimination from a proof net issued from the sequent calculus also are issued from the sequent calculus.

The derivation by dicograph rewriting dicog-RS also only yield correct proof structures.

Proposition 3

Any proof obtained by rewriting from yields a handsome proof structure without chordless alternate elementary path, i.e. into a dicog-PN.

Proof.  Observe that satisfies the criterion, so because of theorem 6, the result is clear.

5 Denotational semantics of pomset logic within coherence spaces

Denotational semantics or categorical interpretation of a logic is the interpretation of a logic in such a way that a proof of is interpreted as a morphism from an object to an object in such a way that whenever reduces to by (the transitive closure) of -reduction or cut-elimination. A proof of (when there is no ) is simply interpreted as a morphism from the terminal object to . More details can be found in [17, 8].

Once the interpretation of propositional variables is defined, the interpretation of complex formulas is defined by induction on the complexity of the formula. The set of morphisms from to is in bijective correspondence with an object written . Morphisms are defined by induction on the proofs and one has to check that the interpretations of a proof before and after one step of cut elimination is unchanged.

For intuitionistic logic, the category is cartesian closed, and for classical logic, at least simply, it is impossible666The fact that cartesian closed categories with involutive negation have at most one morphim between any two object is known as Joyal argument (see e.g. [17]); however there are complicated solutions like Selinger’s control categories [47] for classical deductive systems that ”control” the non determinism of classical cut elimination,like Parigot’s calculus, [26].. Regarding linear logic, a categorical interpetation takes place in a monoidal closed category (with monads for the exponentials of linear logic).

5.1 Coherence spaces

The category of coherence spaces is a concrete category: objects are (countable) sets endowed with a binary relation, and morphimms are linear maps. It interprets the proofs up to cut-elimination or reduction initially propositional intuitionistic logic and propositional linear logic (possibly quantified). Actually, coherence spaces are tightly related to linear logic: indeed, linear logic arose from this particular semantics, invented to model second order lambda calculus i.e. quantified propositional intuitionistic logic [6]. Coherence spaces are themselves inspired from the categorical work on ordinals by Jean-Yves Girard; they are the binary qualitative domains.

A coherence space is a set (possibly infinite) called the web of whose elements are called tokens, endowed with a binary reflexive and symmetric relation called coherence on noted or simply when is clear.

The following notations are common and useful:

iff and
iff or
iff and

A proof of is to be interpreted by a clique of the corresponding coherence spaces , a cliques being a set of pairwise coherent tokens in — we write for and for all . Observe that forall , if then . A linear morphism from to is a morphism mapping cliques of to cliques of such that:

  • Let be a family of pairwise compatible cliques that is to say then .777The morphism is said to be stable when holds more generally for the union of a directed family of cliques of , i.e. .

  • if then .

Due to the removal of structural rules, linear logic has two kinds of conjunction:

Those two rules are equivalent when contraction and weakening are allowed. The multiplicatives (contexts are split, above) and the additives (contexts are duplicated, above). Regarding denotational semantics, the web of the coherence space associated with a formula with a multiplicative connective is the Cartesian product of the webs of and — while it is the disjoint union of the webs of and when is additive.

Negation is a unary connective which is both multiplicative and additive: and iff

One may wonder how many binary multiplicatives there are, i.e. how many different coherence relations one may define on from the coherence relations on and on .

We can limit ourselves to the ones that are covariant functors in both and — indeed there is a negation, hence a contravariant connective in is a covariant connective in . Hence when both components are so are the two couples, and when they are both coherent, so are the two couples.

To define a multiplicative connective, is to define when in function of and , so to fill a nine cell table — however if is assumed to be covariant in both its argument, seven out of the nine cells are filled.

If one wants to be commutative, there are only two possibilities, namely () and ().

However if we don’t ask for the connective to be commutative we have a third connective (and actually a fourth connective which is )

This connective generalises to partial orders. Assume we have an sp order on the formulas can be defined with and — two tuples and of the web are strictly coherent whenever .

Linear implication, which can be defined as is :