Alessio Guglielmi's Research / Deep Inference and the Calculus of Structures

Deep Inference and the Calculus of Structures
Quantum Bio-Cryptography for Creative Nano-Security

This page tries to be a comprehensive account of the ongoing research on deep inference. Deep inference started as a personal project, but it is now growing fast and I'm struggling to keep pace with all the developments. Some of them I do not understand in detail. Please keep in mind that what I wrote below can be very subjective and not reflect the opinions of the people involved in this research. If you disagree or find inaccuracies, please let me know! If you are an expert of proof theory and want to quickly understand what deep inference is about, go to Deep Inference in One Minute.

Latest News 29 October 2008The workshop Structures and Deduction 2009 will be part of ESSLLI 2009. 24 July 2008ANR two-year project Démosthène is approved (750.000 EUR). 18 June 2008There is a workshop on Deep Inference, its Algebra, Geometry and Syntax in Nancy, France. 17–21 December 2007Paola Bruscoli and Lutz Straßburger teach the course Introduction to Deep Inference and Proof Nets. There is a Deep Inference Christmas Meeting 2007 in Dresden, on 19/12. Week of 19 November 2007There is a meeting on Deep Inference and the Essence of Proofs in Bern. 21 June 2007There is a Small Workshop on Theory and Application of Deep Inference in Palaiseau. 14 March 2007There are PhD and postdoc positions available in France, with François Lamarche and Lutz Straßburger, on deep inference and related matters.

Contents

1Introduction

Deep inference is a new methodology in proof theory, which is the discipline that studies mathematical proofs. Deep inference is about designing proof systems and formalisms with excellent properties of analyticity, proof complexity and semantics of proofs. With deep inference, we express logics that elude the traditional methods of proof theory, in particular we represent modal logics and logics related to process algebras. Deep inference provides beautiful and simple proof systems and formalisms.

The following picture illustrates a point of view on deep inference: most of traditional proof theory adopts a methodology that we call shallow inference. The methodologies of shallow and deep inference inspire the design of formalisms, which, in turn, express proof systems for several logics. At the bottom of the picture, solid arrows signal the existence of widely established proof systems in the given formalisms, while dashed arrows stand for a shabbier relationship. Some formalisms in deep inference are under development (dashed border).

We will now explore the concepts in the picture in more detail, and we will summarise the results we obtained so far in deep inference, and the current research themes.

1.1Proof Systems

A formal proof consists in breaking down a mathematical argument into small inference steps and connecting them together. Its validity can then be checked by local inspection of the inference steps, as dictated by syntactic inference rules. Checking a proof is a mechanical procedure, and this is much of its value: human fallacies are ruled out, and computers can be employed for checking and also discovering proofs.

A logic can be considered a class of theorems. Normally, mathematics is performed in first-order classical logic, but, in many cases, we use stronger or weaker systems. In computer science, we use many logics, some of which are rather far apart from classical logic.

A set of inference rules is called a (proof) system. For any given logic, several different proof systems exist, which prove the same theorems. However, they might greatly differ for several properties, notably:

• The efficiency in representing proofs: Some systems make available much smaller proofs than others do (these aspects are studied in proof complexity);
• Supporting the automatic discovery of proofs: Some systems immediately entail efficient proof search algorithms, some do not, others do with some effort; the key property for efficient proof search is called analyticity.
• The ability of expressing proofs that are mathematically natural, and to do so without unnecessary syntactic artefacts (bureaucracy): One of the main research problems in proof theory is to find a good correspondence between proofs and their meaning. In particular, the problem of proof identity is prominent; it consists in finding non-trivial notions of proof equivalence, supported by appropriate semantics of proofs and proof systems.

In our research, these three properties of proof systems play a crucial role.

1.2Formalisms

proof systems can be classified according to the style they adhere to. These classes are not always mathematically defined, but, in practice, they are recognisable. We say that proof systems belong to different formalisms. The most important traditional formalisms are the Frege-Hilbert formalism, the sequent calculus and natural deduction, and there are many others; the sequent calculus^* is special because it supports analyticity.

Different formalisms allow for proof systems with different properties. For example, systems in the natural deduction formalism do not lend themselves easily to proof search. Systems with excellent proof search properties, due to analyticity, can be designed in the sequent calculus^*; however, they produce proofs which are exponentially bigger than those in natural deduction, on certain classes of theorems.

Often, formalisms entail interesting notions of normalisation inside their proof systems. For example, natural deduction systems possess a notion of normalisation that corresponds closely to a very important notion of computation, in what we call the Curry-Howard correspondence: normalising proofs in certain systems for intuitionistic logic corresponds to computing in the simply-typed lambda-calculus.

It is sometimes very challenging to design proof systems for a given logic in a given formalism: for example, many variations of modal logic, which are easily expressible in Frege-Hilbert systems, only find awkward presentations in sequent-calculus systems. Often, the reason for adopting the sequent calculus^* is for getting analyticity; however, in some cases, with great effort, one only gets sequent-calculus systems that are not analytic.

For many logics, like modal logic, it might be necessary to adopt non-traditional formalisms, which allow for more freedom in designing inference rules. For example, many modal logic variations can be expressed with analytic proof systems in a formalism called hypersequents, which is a generalisation of the sequent calculus^*.

1.3Shallow and Deep Inference

Formalisms, to a very large extent, dictate the design of inference rules. For example, natural deduction prescribes that, for every connective, two rules are given: one that introduces it and one that eliminates it.

In all traditional formalisms, and in the modern ones derived from them, a methodology that we call shallow inference is adopted. `Shallow´ inference rules operate on connectives that appear in close proximity to the root of formulae, when we consider them as trees. For example, the introduction and elimination rules of natural deduction operate on root connectives of formulae. The sequent calculus usually goes one level deeper than natural deduction, and in some cases two levels deeper. The hypersequents formalism, which is derived from the sequent calculus, does the same.

Shallow inference is a very natural methodology, because it is about generating proofs by a straightforward structural induction on the formulae they prove. However, shallow inference is not optimal regarding the three properties of proof systems mentioned above, and in particular:

• it seems incapable of providing analytic formalisms that are efficient regarding proof complexity;
• formalisms tend to have great bureaucracy, i.e., syntax makes for convoluted representations of mathematical arguments.

Shallow inference also has difficulties in coping with modal logic: modal logic theories can be defined in Frege-Hilbert systems, but obtaining analyticity for them (in sequent systems) can be very difficult, and in some cases unachieved. It is equally difficult or impossible to express proof systems for some logics involving non-commutativity, linearity and other features typical of computer-science concurrency languages, like process algebras.

All these seemingly heterogeneous difficulties can largely be ascribed to one cause: the adoption of the shallow-inference methodology. In fact, and very roughly, all shallow-inference formalisms require the adoption of a so-called meta, or structural, level for organising the `pieces´ of formulae obtained by the structural induction they adopt. For example, the sequent calculus organises these pieces into sets and trees. Because of the historical development of proof theory, the meta level, in a sense, coincides with the algebraic structure of classical logic. This becomes an increasingly serious impediment the more the logics one wants to express depart from classical logic.

Deep inference is based on a very simple idea: formalisms adopt the same algebraic structure of any given logic for keeping organised the pieces of proofs; in other words, there is no meta level. This means that inference rules must be able to operate at any level (`deeply´) inside formula trees.

The idea is simple, but the adoption of deep inference poses two difficult challenges:

• How do we design inference rules? The absence of a given meta level removes any kind of constraint on inference rules; this freedom is beneficial but entails an unlimited choice of rules, which is daunting, and can also be dangerous (we could create monsters).
• How do we prove normalisation? The greatest part of proof theory's technical body is about normalisation, i.e., transformation of proofs into ones with interesting properties, for example, analyticity. Adopting deep inference breaks at the core all existing normalisation techniques, so a theory of normalisation must be developed from scratch.

1.4Main Results in Deep Inference

We developed the proof theory of a deep-inference formalism that we call the calculus of structures. This is the simplest formalism conceivable in deep inference, because inference rules behave as rewrite rules in term rewriting systems. The calculus of structures is a milestone in the development of deep inference, because of its simplicity and its resemblance to traditional formalisms. Moreover, the normalisation techniques of the calculus of structures are applicable also to other formalisms in deep inference, currently under development.

We achieved the following results for the calculus of structures:

• Classical, intuitionistic, linear and several modal logics are expressed in analytic systems.
• Mixed commutative/non-commutative linear logics BV and NEL are expressed in analytic systems, and we proved that these logics cannot be expressed analytically in shallow inference; these logics have a close correspondence to process algebras like Milner's CCS.
• General and powerful normalisation techniques have been developed for the classical, linear, BV and NEL logics; new normalisation notions, in addition to the traditional cut elimination, have been achieved.
• All proof systems (apart from minor exceptions) are entirely made of local inference rules; a local inference rule is one whose computational complexity is constant. Locality is a difficult property to achieve, and it is not achievable in sequent-calculus systems for classical logic.
• All these proof systems are extremely modular; this means that the interdependency of inference rules, regarding normalisation and other properties, is very low.
• Many systems have been implemented, thanks to techniques that yield inference rules that improve proof search efficiency without sacrificing proof-theoretic cleanliness.
• All the proof systems are simple, in the sense that their inference rules are small and natural.

The calculus of structures generalises most shallow-inference formalisms, in particular the sequent calculus. This means that every proof in shallow-inference formalisms can be `mimicked´ in the calculus of structures, by preserving complexity and without losing any structural property.

It might be desirable, for example for constraining proof search and reducing non-determinism, to reach a compromise between the sequent calculus and the calculus of structures. A new deep-inference formalism, called deep sequents, has been developed, especially targeting modal logics. In deep sequents (and so, in the calculus of structures), modal logics like B and K5, which do not enjoy analytic presentations in the sequent calculus, find simple analytic systems.

The cirquent calculus, a new formalism recently developed by Giorgi Japaridze, benefits from a deep-inference presentation.

The calculus of structures promoted the discovery of a new class of proof nets for classical and linear logic. Proof nets are not proof systems, because they cannot be checked by local inspection, but they play a crucial role in understanding the semantics of proofs, which is one of the most active research areas in proof theory, in close connection with theoretical computer science.

1.5Perspective

Designing syntax is an eminently semantic activity: the only way to avoid creating monsters is having clear semantic objectives and guidance. The ambition of deep inference is to provide a unifying, simple syntax of proofs, which, above all, reflects the meaning of proofs. The semantics of proofs depends heavily on syntax, because proofs, more than formulae, are objects that are built, they are constructions, not just statements. Because of lack of adequate syntax, at present, we are basically unable to answer questions like Are two given proofs the same? (This is the so-called Hilbert's 24th problem.) The only answers we can give right now are either trivial (and uninteresting) or excessively technical. One of the main objectives of deep inference is to provide a simple answer to this question.

• Current research topics and open problems in deep inference

1.6History

Deep inference comes from linear logic and process algebras; more specifically, it comes from seeing proofs as concurrent processes. The first development has been the definition of the calculus of structures and a cut elimination proof for the logic BV, which was studied for being the logical counterpart of the core of the process algebra CCS. We realised that the techniques developed for BV had a much wider applicability, so we broadly developed the calculus of structures and studied its many novel normalisation properties. The initial phase of development took place in Dresden, from 1999 to 2003; now, deep inference is developed in several laboratories around the world. The recent results on modal and intuitionistic logics, proof nets and semantics, and implementations, complete the establishing of deep inference as a solid and comprehensive methodology in proof theory.

• Proof theory group at the International Centre for Computational Logic

2Frequently Asked Questions and Useful Remarks for Referees

Please do not hesitate to ask questions, either by email to me, or, even better, by using our mailing list.

• FAQ-URR page:

1. General
   1. What is deep inference?
   2. What is a structure?
   3. Isn't the word `structure´ reserved for semantics?
   4. Proof theory is famous for producing horrible proofs of its own theorems; are you doing any better?
   5. Does the calculus of structures automatically introduce a mix rule in any system?
   6. In proof search, the nondeterminism induced by deep inference is enormous, how can I cope with this?
   7. The good properties of the proofs in the calculus of structures are obtained by an extensive use of hidden equalities; are you cheating?
   8. Aren't proof nets top-down symmetric objects, not differently from proofs in the calculus of structures?
2. The Calculus of Structures Vs. the Sequent Calculus
   1. Isn't the calculus of structures just a trivial notational variation of the sequent calculus?
   2. Can you do in the calculus of structures anything you can do in the sequent calculus?
   3. There is more bureaucracy in the calculus of structures than in the sequent calculus, and this is why you get more properties: simply because you have more stuff to work with, right?
3. Cut Elimination and the Subformula Property
   1. You make a big deal of having the cut rule in atomic form; but can't you do the same in the sequent calculus?
   2. What's the point in proving again cut elimination for classical and linear logic?
   3. By the way, what's the point in dedicating an entire paper to such an easy statement as cut elimination for propositional classical logic in the calculus of structures?
   4. Your `cut-free´ systems do not have the subformula property! Are you crazy?
4. Term Rewriting
   1. Isn't a system in the calculus of structures just a term rewriting system (modulo an equational theory)?
   2. If so, why not using the term rewriting terminology and conventions?
   3. But then what about rewriting logic?
5. Philosophy
   1. Do the inference rules in the calculus of structures have an associated theory of meaning, like the inference rules in the sequent calculus and natural deduction?

3Papers, Lectures and Theses

Abstracts ON or OFF (Javascript required).

The following material is broad in scope; if you are new to deep inference and the calculus of structures, start from here (in the suggested order):

• Deep Inference and Its Normal Form of Derivations
  Kai Brünnler
  AbstractWe see a notion of normal derivation for the calculus of structures, which is based on a factorisation of derivations and which is more general than the traditional notion of cut-free proof in this formalism.
  Pdf29 March 2006
  CiE 2006, LNCS 3988, pp. 65–74

• 

  Deep Inference and Symmetry in Classical Proofs
  Kai Brünnler
  AbstractIn this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems.
  PdfFebruary 2004
  PhD thesis, successfully defended on 22 September 2003, published by Logos Verlag
  You can buy this book at Logos-Verlag and at Amazon

• Linear Logic and Noncommutativity in the Calculus of Structures
  Lutz Straßburger
  AbstractIn this thesis I study several deductive systems for linear logic, its fragments, and some noncommutative extensions. All systems will be designed within the calculus of structures, which is a proof theoretical formalism for specifying logical systems, in the tradition of Hilbert's formalism, natural deduction, and the sequent calculus. Systems in the calculus of structures are based on two simple principles: deep inference and top-down symmetry. Together they have remarkable consequences for the properties of the logical systems. For example, for linear logic it is possible to design a deductive system, in which all rules are local. In particular, the contraction rule is reduced to an atomic version, and there is no global promotion rule. I will also show an extension of multiplicative exponential linear logic by a noncommutative, self-dual connective which is not representable in the sequent calculus. All systems enjoy the cut elimination property. Moreover, this can be proved independently from the sequent calculus via techniques that are based on the new top-down symmetry. Furthermore, for all systems, I will present several decomposition theorems which constitute a new type of normal form for derivations.
  PdfPdf in booklet format25 July 2003
  PhD thesis, successfully defended on 24 July 2003

• Nondeterminism and Language Design in Deep Inference
  Ozan Kahramanogullari
  AbstractThis thesis studies the design of deep-inference deductive systems. In the systems with deep inference, in contrast to traditional proof-theoretic systems, inference rules can be applied at any depth inside logical expressions. Deep applicability of inference rules provides a rich combinatorial analysis of proofs. Deep inference also makes it possible to design deductive systems that are tailored for computer science applications and otherwise provably not expressible.
  By applying the inference rules deeply, logical expressions can be manipulated starting from their sub-expressions. This way, we can simulate analytic proofs in traditional deductive formalisms. Furthermore, we can also construct much shorter analytic proofs than in these other formalisms. However, deep applicability of inference rules causes much greater nondeterminism in proof construction.
  This thesis attacks the problem of dealing with nondeterminism in proof search while preserving the shorter proofs that are available thanks to deep inference. By redesigning the deep inference deductive systems, some redundant applications of the inference rules are prevented. By introducing a new technique which reduces nondeterminism, it becomes possible to obtain a more immediate access to shorter proofs, without breaking certain proof theoretical properties such as cut-elimination. Different implementations presented in this thesis allow to perform experiments on the techniques that we developed and observe the performance improvements. Within a computation-as-proof-search perspective, we use deep-inference deductive systems to develop a common proof-theoretic language to the two fields of planning and concurrency.
  Pdf11 September 2006
  PhD thesis, successfully defended on 21 December 2006

There are three slides presentations, mostly on the syntactic aspects of deep inference, that might help; please keep in mind that many of the conjectures stated here have been proved, in the meantime:

• Deep InferenceAlessio GuglielmiTalk13 March 2003

• Proof Theory with Deep InferenceAlessio GuglielmiCourse at the Summer School and Workshop on Proof Theory, Computation and Complexity28 June 2003

• Deep Inference in Theorem ProvingOzan KahramanogullariTalk8 May 2007

In the rest of the section, all papers I know of are listed according to their subject, in no particular order.

3.1Classical and Intuitionistic Logic

So far, for classical logic in the calculus of structures we achieved:

• the cut rule trivially reduces to atomic form;
• one can show cut elimination for the propositional fragment by the simplest argument to date;
• the propositional fragment is fully local, including contraction;
• first order classical logic can be entirely made finitary;
• cut elimination and decomposition theorems are proved.

We can present intuitionistic logic in the calculus of structures with a fully local, cut-free system. The logic of bunched implications BI can be presented in the calculus of structures. Japaridze's cirquent calculus benefits from a deep-inference presentation, in particular in the case of propositional logic.

The following papers exist, in addition to Deep Inference and Its Normal Form of Derivations and Deep Inference and Symmetry in Classical Proofs, mentioned above:

• An Algorithmic Interpretation of a Deep Inference System
  Kai Brünnler and Richard McKinley
  AbstractWe set out to find something that corresponds to deep inference in the same way that the lambda-calculus corresponds to natural deduction. Starting from natural deduction for the conjunction-implication fragment of intuitionistic logic we design a corresponding deep inference system together with reduction rules on proofs that allow a fine-grained simulation of beta-reduction.
  Pdf22 September 2008
  LPAR 08, LNCS 5330, pp. 482–496

• Cirquent Calculus Deepened
  Giorgi Japaridze
  AbstractCirquent calculus is a new proof-theoretic and semantic framework, whose main distinguishing feature is being based on circuit-style structures (called cirquents), as opposed to the more traditional approaches that deal with tree-like objects such as formulas, sequents or hypersequents. Among its advantages are greater efficiency, flexibility and expressiveness. This paper presents a detailed elaboration of a deep-inference cirquent logic, which is naturally and inherently resource conscious. It shows that classical logic, both syntactically and semantically, can be seen to be just a special, conservative fragment of this more general and, in a sense, more basic logic — the logic of resources in the form of cirquent calculus. The reader will find various arguments in favor of switching to the new framework, such as arguments showing the insufficiency of the expressive power of linear logic or other formula-based approaches to developing resource logics, exponential improvements over the traditional approaches in both representational and proof complexities offered by cirquent calculus (including the existence of polynomial size cut-, substitution- and extension-free cirquent calculus proofs for the notoriously hard pigeonhole principle), and more. Among the main purposes of this paper is to provide an introductory-style starting point for what, as the author wishes to hope, might have a chance to become a new line of research in proof theory — a proof theory based on circuits instead of formulas.
  Pdf1 April 2008
  Journal of Logic and Computation, 18 (6) 2008, pp. 983–1028

• Normalisation Control in Deep Inference via Atomic Flows
  Alessio Guglielmi and Tom Gundersen
  AbstractWe introduce `atomic flows´: they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations. This allows us to prove, for propositional logic, a new and very general normalisation theorem, which contains cut elimination as a special case. We operate in deep inference, which is more general than other syntactic paradigms, and where normalisation is more difficult to control. We argue that atomic flows are a significant technical advance for normalisation theory, because 1) the technique they support is largely independent of syntax; 2) indeed, it is largely independent of logical inference rules; 3) they constitute a powerful geometric formalism, which is more intuitive than syntax.
  Pdf31 March 2008
  Logical Methods in Computer Science 4 (1:9) 2008, pp. 1–36

• The Logic of Bunched Implications: A Memoir
  Benjamin Robert Horsfall
  AbstractThis is a study of the semantics and proof theory of the logic of bunched implications (BI), which is promoted as a logic of (computational) resources, and is a foundational component of separation logic, an approach to program analysis. BI combines an additive, or intuitionistic, fragment with a multiplicative fragment. The additive fragment has full use of the structural rules of weakening and contraction, and the multiplicative fragment has none. Thus it contains two conjunctive and two implicative connectives. At various points, we illustrate a resource view of BI based upon the Kripke resource semantics. Our first original contribution is the formulation of a proof system for BI in the newly developed proof-theoretical formalism of the calculus of structures. The calculus of structures is distinguished by its employment of deep inference, but we already see deep inference in a limited form in the established proof theory for BI. We show that our system is sound with respect to the elementary Kripke resource semantics for BI, and complete with respect to the partially-defined monoid (PDM) semantics. Our second contribution is the development from a semantic standpoint of preliminary ideas for a hybrid logic of bunched implications (HBI). We give a Kripke semantics for HBI in which nominal propositional atoms can be seen as names for resources, rather than as names for locations, as is the case with related proposals for BI-Loc and for intuitionistic hybrid logic. The cost of this approach is the loss of intuitionistic monotonicity in the semantics. But this is perhaps not such a grave loss, given that our guiding analogy is of states of models with resources, rather than with states of knowledge, as is standard for intuitionistic logic.
  PdfJune 2007
  MSc thesis, successfully defended in August 2007

• A Characterisation of Medial as Rewriting Rule
  Lutz Straßburger
  AbstractMedial is an inference rule scheme that appears in various deductive systems based on deep inference. In this paper we investigate the properties of medial as rewriting rule independently from logic. We present a graph theoretical criterion for checking whether there exists a medial rewriting path between two formulas. Finally, we return to logic and apply our criterion for giving a combinatorial proof for a decomposition theorem, i.e., proof theoretical statement about syntax.
  Pdf13 April 2007
  RTA 2007, LNCS 4533, pp. 344–358

• A Local System for Intuitionistic Logic
  Alwen Tiu
  AbstractThis paper presents systems for first-order intuitionistic logic and several of its extensions in which all the propositional rules are local, in the sense that, in applying the rules of the system, one needs only a fixed amount of information about the logical expressions involved. The main source of non-locality is the contraction rules. We show that the contraction rules can be restricted to the atomic ones, provided we employ deep-inference, i.e., to allow rules to apply anywhere inside logical expressions. We further show that the use of deep inference allows for modular extensions of intuitionistic logic to Dummett's intermediate logic LC, Gödel logic and classical logic. We present the systems in the calculus of structures, a proof theoretic formalism which supports deep-inference. Cut elimination for these systems are proved indirectly by simulating the cut-free sequent systems, or the hypersequent systems in the cases of Dummett's LC and Gödel logic, in the cut free systems in the calculus of structures.

  • Conference version pdf9 August 2006
    LPAR 2006, LNAI 4246, pp. 242–256
  • Full paper pdf9 August 2006

• Locality for Classical Logic
  Kai Brünnler
  AbstractIn this paper we will see deductive systems for classical propositional and predicate logic in the calculus of structures. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they drop the restriction that rules only apply to the main connective of a formula: their rules apply anywhere deeply inside a formula. This allows to observe very clearly the symmetry between identity axiom and the cut rule. This symmetry allows to reduce the cut rule to atomic form in a way which is dual to reducing the identity axiom to atomic form. We also reduce weakening and even contraction to atomic form. This leads to inference rules that are local: they do not require the inspection of expressions of arbitrary size.
  Pdf10 March 2006
  Notre Dame Journal of Formal Logic, Vol. 47, No. 4, pp. 557–580, 2006
  Review by Sara Negri
  Note about the review: All the concerns of the reviewer are addressed in the paper Two Restrictions on Contraction (which is among the references of the reviewed paper).

• Cut Elimination Inside a Deep Inference System for Classical Predicate Logic
  Kai Brünnler
  AbstractDeep inference is a natural generalisation of the one-sided sequent calculus where rules are allowed to apply deeply inside formulas, much like rewrite rules in term rewriting. This freedom in applying inference rules allows to express logical systems that are difficult or impossible to express in the cut-free sequent calculus and it also allows for a more fine-grained analysis of derivations than the sequent calculus. However, the same freedom also makes it harder to carry out this analysis, in particular it is harder to design cut elimination procedures. In this paper we see a cut elimination procedure for a deep inference system for classical predicate logic. As a consequence we derive Herbrand's Theorem, which we express as a factorisation of derivations.
  Pdf9 March 2005
  Studia Logica, Vol. 82, No. 1, pp. 51–71, 2006

• A First Order System with Finite Choice of Premises
  Kai Brünnler and Alessio Guglielmi
  AbstractWe present an inference system for classical first order logic in which each inference rule, including the cut, only has a finite set of premises to choose from. The main conceptual contribution of this paper is the possibility of separating different sources of infinite choice, which happen to be entangled in the traditional cut rule.
  Pdf1 December 2003
  Presented at First Order Logic 75 under the title A Finitary System for First Order Logic; appeared in Hendricks et al., editor, First-Order Logic Revisited, Logos Verlag, Berlin, 2004, pp. 59–74

• Two Restrictions on Contraction
  Kai Brünnler
  AbstractI show two simple limitations of sequent systems with multiplicative context treatment: contraction can neither be restricted to atoms nor to the bottom of a proof tree.
  Pdf24 November 2003
  Logic Journal of the Interest Group in Pure and Applied Logics, Vol. 11 No. 5, pp. 525–529

• Consistency Without Cut Elimination
  Kai Brünnler and Alessio Guglielmi
  AbstractWe show how to get consistency for first order classical logic, in a purely syntactic way, without going through cut elimination. The procedure is very simple and it also shows how finitariness is actually a triviality (contrarily to what one would guess from textbooks).
  Pdf10 September 2003It is contained in A First Order System with Finite Choice of Premises

• Atomic Cut Elimination for Classical Logic
  Kai Brünnler
  AbstractSystem SKS is a set of rules for classical propositional logic presented in the calculus of structures. Like sequent systems and unlike natural deduction systems, it has an explicit cut rule, which is admissible. In contrast to sequent systems, the cut rule can easily be restricted to atoms. This allows for a very simple cut elimination procedure based on plugging in parts of a proof, like normalisation in natural deduction and unlike cut elimination in the sequent calculus. It should thus be a good common starting point for investigations into both proof search as computation and proof normalisation as computation.
  Pdf10 April 2003
  CSL 2003, LNCS 2803, pp. 86–97

• A Local System for Classical Logic
  Kai Brünnler and Alwen Fernanto Tiu
  AbstractThe calculus of structures is a framework for specifying logical systems, which is similar to the one-sided sequent calculus but more general. We present a system of inference rules for propositional classical logic in this new framework and prove cut elimination for it. The system also enjoys a decomposition theorem for derivations that is not available in the sequent calculus. The main novelty of our system is that all the rules are local: contraction, in particular, is reduced to atomic form. This should be interesting for distributed proof-search and also for complexity theory, since the computational cost of applying each rule is bounded.
  Pdf2 October 2001Now replaced by Locality for Classical Logic
  LPAR 2001, LNAI 2250, pp. 347–361

• Current research topics and open problems in classical, intuitionistic and intermediate logics

3.2Proof Complexity

The basic proof complexity properties of propositional logic in the calculus of structures are known. Deep inference is as powerful as Frege systems, and more powerful than Gentzen systems, in the restriction to analytic systems.

The following papers exist:

• Extension without Cut
  Lutz Straßburger
  AbstractIn proof theory one distinguishes sequent proofs with cut and cut-free sequent proofs, while for proof complexity one distinguishes Frege-systems and extended Frege-systems. In this paper we show how deep inference can provide a uniform treatment for both classifications, such that we can define cut-free systems with extension, which is neither possible with Frege-systems, nor with the sequent calculus. We show that the propositional pidgeon-hole principle admits polynomial-size proofs in a cut-free system with extension. We also define cut-free systems with substitution and show that the system with extension p-simulates the system with substitution. This yields a new (and simpler) proof that extended Frege-systems p-simulate Frege-systems with substitution. Finally, we propose a new class of tautologies that have short proofs in extended systems, but might not in Frege systems without extension.
  Pdf12 June 2008

• Proof complexity of the Cut-Free Calculus of Structures
  Emil Jeřábek
  AbstractWe investigate the proof complexity of analytic subsystems of the deep inference proof system SKSg (the calculus of structures). Exploiting the fact that the cut rule (i↑) of SKSg corresponds to the ¬-left rule in the sequent calculus, we establish that the “analytic” system KSg + c↑ has essentially the same complexity as the monotone Gentzen calculus MLK. In particular, KSg + c↑ quasipolynomially simulates SKSg, and admits polynomial-size proofs of some variants of the pigeonhole principle.
  Pdf30 April 2008
  To appear on Journal of Logic and Computation

• On the Proof Complexity of Deep Inference
  Paola Bruscoli and Alessio Guglielmi
  AbstractWe obtain two results about the proof complexity of deep inference: 1) deep-inference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; 2) there are analytic deep-inference proof systems that exhibit an exponential speed-up over analytic Gentzen proof systems that they polynomially simulate.
  Pdf11 February 2008
  To appear on ACM Transactions on Computational Logic

3.3Deep Sequents

A new formalism called `deep sequents´ has been defined, which is especially suitable to modal logics.

The following papers exist:

• Syntactic Cut-Elimination for Common Knowledge
  Kai Brünnler and Thomas Studer
  AbstractWe see a cut-free infinitary sequent system for common knowledge. Its sequents are essentially trees and the inference rules apply deeply inside of these trees. This allows to give a syntactic cut-elimination procedure which yields an upper bound of φ₂0 on the depth of proofs, where φ is the Veblen function
  Pdf14 September 2007
  Presented at Methods for Modalities 5

• Deep Sequent Systems for Modal Logic
  Kai Brünnler
  AbstractWe see a systematic set of cut-free axiomatisations for all the basic normal modal logics formed by some combination the axioms d, t, b, 4, 5. They employ a form of deep inference but otherwise stay very close to Gentzen’s sequent calculus, in particular they enjoy a subformula property in the literal sense. No semantic notions are used inside the proof systems, in particular there is no use of labels. All their rules are invertible and the rules cut, weakening and contraction are admissible. All systems admit a straightforward terminating proof search procedure as well as a syntactic cut elimination procedure.

  • Conference version pdf30 March 2006
    Proceedings of Advances in Modal Logic 2006, pp. 107–119
  • Full paper pdf21 August 2007

3.4Modal Logic

We can present systematically several normal propositional modal logics, including S5, B and K5, for which cut elimination is proved. We also investigated geometric theories, some of which we expressed in the calculus of structures.

The following papers exist:

• Deep Inference for Hybrid Logic
  Lutz Straßburger
  AbstractThis paper describes work in progress on using deep inference for designing a deductive system for hybrid logic. We will see a cut-free system and prove its soundness and completeness. An immediate observation about the system is that there is no need for additional rewrite rules as in Blackburn’s tableaux, nor substitution rules as in Seligman’s sequent system.
  Pdf15 May 2007
  Proceedings of International Workshop on Hybrid Logic 2007 (HyLo 2007), pp. 13–22

• Classical Modal Display Logic in the Calculus of Structures and Minimal and Cut-free Deep Inference Calculi for S5
  Rajeev Goré and Alwen Tiu
  AbstractWe begin by showing how to faithfully encode the Classical Modal Display Logic (CMDL) of Wansing into the Calculus of Structures (CoS) of Guglielmi. Since every CMDL calculus enjoys cut-elimination, we obtain a cut-elimination theorem for all corresponding CoS calculi. We then show how our result leads to a minimal cut-free CoS calculus for modal logic S5. As far as we know, no other existing CoS calculi for S5 enjoy both these properties simultaneously.
  Pdf4 May 2007
  Journal of Logic and Computation, Vol. 17 (4) 2007, pp. 767–794

• A Deep Inference System for the Modal Logic S5
  Phiniki Stouppa
  AbstractWe present a cut-admissible system for the modal logic S5 in a framework that makes explicit and intensive use of deep inference. Deep inference is induced by the methods applied so far in conceptually pure systems for this logic. Thus, the formulation of a system in such a framework is an evolutional process and leads to positive proof theoretical results. The system enjoys systematicity and modularity, two important properties that seek satisfaction from modal systems. Furthermore, it enjoys a simple and direct design: the rules are few and the modal rules are in exact correspondence to the modal axioms.
  Pdf1 March 2006
  Studia Logica, Vol. 85 (2) 2007, pp. 199–214

• Purity Through Unravelling
  Robert Hein and Charles Stewart
  AbstractWe divide attempts to give the structural proof theory of modal logics into two kinds, those pure formulations whose inference rules characterise modality completely by means of manipulations of boxes and diamonds, and those labelled formulations that leverage the use of labels in giving inference rules. The widespread adoption of labelled formulations is driven by their ability to model features of the model theory of modal logic in its proof theory.
  We describe here an approach to the structural proof theory of modal logic that aims to bring under one roof the benefits of both the pure and the labelled formulations. We introduce two proof calculi, one labelled sequent formulation and one pure formulation in the calculus of structures that are shown to be in a systematic correlation, where the latter calculus uses deep inference with shaped modal rules to capture in a pure manner the manipulations that the former calculations mediates through the use of labels.
  We situate this work within a larger investigation into the proof theory of modal logic that solves problems that existed with the earlier investigation based on prefix modal rules. We hold this development provides yet stronger evidence justifying the claim that good, pure proof theory for modal logic needs deep inference.
  Pdf25 June 2005
  Proceedings of Structures and Deduction '05, pp. 126–143

• Geometric Theories and Modal Logic in the Calculus of Structures
  Robert Hein
  AbstractMuch of the success of modal logic can be attributed to the adoption of relational semantics. Consequently, modal logic is seen as logic of relational structures, where logical axioms correspond to structural properties. Alex Simpson, in his 1993 PhD thesis, introduced a labelled proof theory for modal logic that that allows cut-elimination for a class of modal logics, which is characterised by so called geometric theories. This includes important and well know logics such as M, B, S4 and S5. This thesis tries to make a bridge between Simpson's result and purely symbolic proof theory. We introduce a method to characterise frame relational properties by means of deep inference in the Calculus of Structures. The results are only partial. Only what we call 3/4-Scott-Lemmon logics are characterised and we only give plausible reason, rather than a proof that the cut-elimination argument can be transferred too.
  Gzipped postscript20 March 2005
  MSc thesis, successfully defended on 23 March 2005

• The Design of Modal Proof Theories: The Case of S5
  Phiniki Stouppa
  AbstractThe sequent calculus does not seem to be capable of supporting cut-admissible formulations for S5. Through a survey on existing cut-admissible systems for this logic, we investigate the solutions proposed to overcome this defect. Accordingly, the systems can be divided into two categories: in those which allow semantic-oriented formulae and those which allow formulae in positions not reachable by the usual systems in the sequent calculus. The first solution is not desirable because it is conceptually impure, that is, these systems express concepts of frame semantics in the language of the logic.
  Consequently, we focus on the systems of the second group for which we define notions related to deep inference—the ability to apply rules deep inside structures—as well as other desirable properties good systems should enjoy. We classify these systems accordingly and examine how these properties are affected in the presence of deep inference. Finally, we present a cut-admissible system for S5 in a formalism which makes explicit use of deep inference, the calculus of structures, and give reasons for its effectiveness in providing good modal formulations.
  Pdf20 October 2004
  MSc thesis, successfully defended on 27 October 2004

• A Systematic Proof Theory for Several Modal Logics
  Charles Stewart and Phiniki Stouppa
  AbstractThe family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a Hilbert-Lewis style and conditions on the accessibility relation on frames.
  By contrast, the usual structural proof theory of modal logic, as given in Gentzen systems, is ad-hoc. While we can formulate several modal logics in the sequent calculus that enjoy cut-elimination, their formalisation arises through system-by-system fine tuning to ensure that the cut-elimination holds, and the correspondence to the formulation in the Hilbert-Lewis systems becomes opaque.
  This paper introduces a systematic presentation for the systems K, D, M, S4, and S5 in the calculus of structures, a structural proof theory that employs deep inference. Because of this, we are able to axiomatise the modal logics in a manner directly analogous to the Hilbert-Lewis axiomatisation. We show that the calculus possesses a cut-elimination property directly analogous to cut-elimination for the sequent calculus for these systems, and we discuss the extension to several other modal logics.
  Postscript2 September 2004
  Proceedings of Advances in Modal Logic 2004, pp. 309–333

• Current research topics and open problems in modal logic

3.5Linear Logic

So far, for linear logic in the calculus of structures it has been achieved:

• the cut rule trivially reduces to atomic form;
• the propositional fragment is fully local, including promotion and contraction;
• cut elimination and decomposition theorems are proved.

The following papers exist, in addition to Linear Logic and Noncommutativity in the Calculus of Structures, mentioned above:

• Some Observations on the Proof Theory of Second Order Propositional Multiplicative Linear Logic
  Lutz Straßburger
  AbstractWe investigate the question of what constitutes a proof when quantifiers and multiplicative units are both present. On the technical level this paper provides two new aspects of the proof theory of MLL2 with units. First, we give a novel proof system in the framework of the calculus of structures. The main feature of the new system is the consequent use of deep inference, which allows us to observe a decomposition which is a version of Herbrand’s theorem that is not visible in the sequent calculus. Second, we show a new notion of proof nets which is independent from any deductive system. We have “sequentialisation” into the calculus of structures as well as into the sequent calculus. Since cut elimination is terminating and confluent, we have a category of MLL2 proof nets. The treatment of the units is such that this category is star-autonomous.
  Pdf26 May 2008

• MELL in the Calculus of Structures
  Lutz Straßburger
  AbstractThe calculus of structures is a new proof-theoretical framework, like natural deduction, the sequent calculus and proof nets, for specifying logical systems syntactically. In a rule in the calculus of structures, the premise as well as the conclusion are structures, which are expressions that share properties of formulae and sequents. In this paper, I study a system for MELL, the multiplicative exponential fragment of linear logic, in the calculus of structures. It has the following features: a local promotion rule, no non-deterministic splitting of the context in the times rule and a modular proof for the cut-elimination theorem. Further, derivations have a new property, called decomposition, that cannot be observed in any other known proof-theoretical framework.
  Pdf24 November 2003
  Theoretical Computer Science, Vol. 309, pp. 213–285

• A Local System for Linear Logic
  Lutz Straßburger
  AbstractIn this paper I will present a deductive system for linear logic, in which all rules are local. In particular, the contraction rule is reduced to an atomic version, and there is no global promotion rule. In order to achieve this, it is necessary to depart from the sequent calculus to the calculus of structures, which is a generalization of the one-sided sequent calculus. In a rule, premise and conclusion are not sequents, but structures, which are expressions that share properties of formulae and sequents.

  • Conference version pdf9 August 2002
    LPAR 2002, LNAI 2514, pp. 388–402
  • Full paper pdf10 June 2002
    Technical Report WV-02-01, Technische Universität Dresden

• Current research topics and open problems in linear logic

3.6Commutative/Non-commutative Linear Logic

We conservatively extend mixed multiplicative and multiplicative exponential linear logic with a self-dual non-commutative operator. The systems so obtained cannot be presented in the sequent calculus, but they enjoy the usual properties of locality, decomposition and cut elimination available in the calculus of structures. We can present Yetter's cyclic linear logic in the calculus of structures and prove cut elimination; interestingly, cyclicity is naturally subsumed by deep inference. New, purely proof-theoretical, techniques are developed for reducing the non-determinism in the calculus of structures.

The following papers exist, in addition to Linear Logic and Noncommutativity in the Calculus of Structures, mentioned above:

• A System of Interaction and Structure IV: The Exponentials
  Alessio Guglielmi and Lutz Straßburger
  AbstractWe study some normalisation properties of the deep-inference proof system NEL, which can be seen both as 1) an extension of multiplicative exponential linear logic (MELL) by a certain non-commutative self-dual logical operator; and 2) an extension of system BV by the exponentials of linear logic. The interest of NEL resides in: 1) its being Turing complete, while the same for MELL is not known, and is widely conjectured not to be the case; 2) its inclusion of a self-dual, non-commutative logical operator that, despite its simplicity, cannot be axiomatised in any analytic sequent calculus system; 3) its ability to model the sequential composition of processes. We present several decomposition results for NEL and, as a consequence of those and via a splitting theorem, cut elimination. We use, for the first time, an induction measure based on flow graphs associated to the exponentials, which captures their rather complex behaviour in the normalisation process. The results are presented in the calculus of structures, which is the first, developed formalism in deep inference.
  Pdf21 December 2007
  Submitted to Mathematical Structures in Computer Science
  This paper was previously titled A Non-commutative Extension of Multiplicative Exponential Linear Logic and is the journal version of A Non-commutative Extension of MELL

• System BV Is NP-Complete
  Ozan Kahramanogullari
  AbstractSystem BV is an extension of multiplicative linear logic (MLL) with the rules mix, nullary mix, and a self-dual, non-commutative logical operator, called seq. While the rules mix and nullary mix extend the deductive system, the operator seq extends the language of MLL. Due to the operator seq, system BV extends the applications of MLL to those where sequential composition is crucial, e.g., concurrency theory. System FBV is an extension of MLL with the rules mix and nullary mix. In this paper, by relying on the fact that system BV is a conservative extension of system FBV, I show that system BV is NP-complete by encoding the 3-Partition problem in FBV. I provide a simple completeness proof of this encoding by resorting to a novel proof theoretical method for reducing the nondeterminism in proof search, which is also of independent interest.

  • Conference version Pdf3 June 2005
    WoLLIC 2005, Electronic Notes in Theoretical Computer Science 143, 2006, pp. 87–99
  • Full paper pdf10 February 2007
    Annals of Pure and Applied Logic, Vol. 152(1–3) 2008, pp. 107–121

• A System of Interaction and Structure
  Alessio Guglielmi
  AbstractThis paper introduces a logical system, called BV, which extends multiplicative linear logic by a non-commutative self-dual logical operator. This extension is particularly challenging for the sequent calculus, and so far it is not achieved therein. It becomes very natural in a new formalism, called the calculus of structures, which is the main contribution of this work. Structures are formulae subject to certain equational laws typical of sequents. The calculus of structures is obtained by generalising the sequent calculus in such a way that a new top-down symmetry of derivations is observed, and it employs inference rules that rewrite inside structures at any depth. These properties, in addition to allowing the design of BV, yield a modular proof of cut elimination.
  Pdf27 January 2007
  ACM Transactions on Computational Logic, Vol. 8 (1:1) 2007, pp. 1–64
  The journal version has been butchered by the editorial process, the preprint (linked here) is fine.

• Reducing Nondeterminism in the Calculus of Structures
  Ozan Kahramanogullari
  AbstractThe calculus of structures is a proof theoretical formalism which generalizes the sequent calculus with the feature of deep inference: In contrast to the sequent calculus, inference rules can be applied at any depth inside a formula, bringing shorter proofs than any other formalisms supporting analytical proofs. However, deep applicability of the inference rules causes greater nondeterminism than in the sequent calculus regarding proof search. In this paper, we introduce a new technique which reduces nondeterminism without breaking proof theoretical properties and provides a more immediate access to shorter proofs. We present this technique on system BV, the smallest technically non-trivial system in the calculus of structures, extending multiplicative linear logic with the rules mix, nullary mix, and a self-dual non-commutative logical operator. Because our technique exploits a scheme common to all the systems in the calculus of structures, we argue that it generalizes to these systems for classical logic, linear logic, and modal logics.
  Pdf23 August 2006
  LPAR 2006, LNAI 4246, pp. 272–286

• A System of Interaction and Structure II: The Need for Deep Inference
  Alwen Tiu
  AbstractThis paper studies properties of the logic BV, which is an extension of multiplicative linear logic (MLL) with a self-dual non-commutative operator. BV is presented in the calculus of structures, a proof theoretic formalism that supports deep inference, in which inference rules can be applied anywhere inside logical expressions. The use of deep inference results in a simple logical system for MLL extended with the self-dual non-commutative operator, which has been to date not known to be expressible in sequent calculus. In this paper, deep inference is shown to be crucial for the logic BV, that is, any restriction on the "depth" of the inference rules of BV would result in a strictly less expressive logical system.
  Pdf3 April 2006
  Logical Methods in Computer Science, Vol. 2 (2:4) 2006, pp. 1–24
  There are pictures that help understanding this paper in Alwen Tiu's MSc thesis Properties of a Logical System in the Calculus of Structures

• Structures for Multiplicative Cyclic Linear Logic: Deepness vs Cyclicity
  Pietro Di Gianantonio
  AbstractThe aim of this work is to give an alternative presentation for the multiplicative fragment of Yetter’s cyclic linear logic. The new presentation is inspired by the calculus of structures, and has the interesting feature of avoiding the cyclic rule. The main point in this work is to show how cyclicity can be substituted by deepness, i.e. the possibility of applying an inference rule at any point of a formula. We finally derive, through a new proof technique, the cut elimination property of the calculus.
  Gzipped postscript15 April 2004
  CSL 2004, LNCS 3210, pp. 130–144

• A Non-commutative Extension of MELL
  Alessio Guglielmi and Lutz Straßburger
  AbstractWe extend multiplicative exponential linear logic (MELL) by a non-commutative, self-dual logical operator. The extended system, called NEL, is defined in the formalism of the calculus of structures, which is a generalisation of the sequent calculus and provides a more refined analysis of proofs. We should then be able to extend the range of applications of MELL, by modelling a broad notion of sequentiality and providing new properties of proofs. We show some proof theoretical results: decomposition and cut elimination. The new operator represents a significant challenge: to get our results we use here for the first time some novel techniques, which constitute a uniform and modular approach to cut elimination, contrary to what is possible in the sequent calculus.
  Pdf9 August 2002
  LPAR 2002, LNCS 2514, pp. 231–246

• The Undecidability of System NEL
  Lutz Straßburger
  AbstractSystem NEL is a conservative extension of multiplicative exponential linear logic (MELL) by a self-dual non-commutative connective called seq which lives between the par and the times. In this paper, I will show that system NEL is undecidable by encoding two counter machines into NEL. Although the encoding is quite simple, the proof of the faithfulness is a little intricate because there is no sequent calculus and no phase semantics available for NEL.

  • Conference version pdf20 August 2003
    Presented at WoLLIC 2003 under the title System NEL Is Undecidable, Electronic Notes in Theoretical Computer Science 84
  • Full paper pdf12 February 2003
    Technical Report WV-03-05, Technische Universität Dresden

• A1-Unification
  Alwen Fernanto Tiu
  Gzipped postscript19 February 2002
  Technical Report WV-01-08, Technische Universität Dresden

• Combining A1- and AC1-Unification Sharing Unit
  Alwen Fernanto Tiu
  Gzipped postscript19 February 2002
  Technical Report WV-01-09, Technische Universität Dresden

• Properties of a Logical System in the Calculus of Structures
  Alwen Fernanto Tiu
  AbstractThe calculus of structures is a new framework for presenting logical systems. It is a generalisation of a traditional framework, the one-sided sequent calculus. One of the main features of the calculus of structures is that the inference rules are deep: they can be applied anywhere inside logical expressions. Rules in the sequent calculus are, in contrast, shallow. A certain logical system in the calculus of structures, called System BV, is studied here. We see that the deep-nesting of rules is a real distinguishing feature between the two frameworks. To this purpose a notion of shallow systems is introduced, such that sequent systems are particular instances. A counterexample, sort of a fractal structure, is then presented to show that there is no shallow system for the logic behind BV, and hence no sequent system for BV. This result contributes to justifying the claim that the calculus of structures is a better logical framework than sequent calculus, for certain logics.
  Pdf12 September 2001Now replaced by A System of Interaction and Structure II: The Need for Deep-Inference
  MSc thesis, successfully defended on 1 August 2001, Technical Report WV-01-06, Technische Universität Dresden

• Non-commutativity and MELL in the Calculus of Structures
  Alessio Guglielmi and Lutz Straßburger
  AbstractWe introduce the calculus of structures: it is more general than the sequent calculus and it allows for cut elimination and the subformula property. We show a simple extension of multiplicative linear logic, by a self-dual non-commutative operator inspired by CCS, that seems not to be expressible in the sequent calculus. Then we show that multiplicative exponential linear logic benefits from its presentation in the calculus of structures, especially because we can replace the ordinary, global promotion rule by a local version. These formal systems, for which we prove cut elimination, outline a range of techniques and properties that were not previously available. Contrarily to what happens in the sequent calculus, the cut elimination proof is modular.
  Pdf28 June 2001
  CSL 2001, LNCS 2142, pp. 54–68

• A Calculus of Order and Interaction
  Alessio Guglielmi
  This paper has been thoroughly rewritten as A System of Interaction and Structure. The introduction of the new paper has not been written under the effect of psychotropic substances. Please forget about this paper.
  Technical Report WV-99-04, Technische Universität Dresden

• Current research topics and open problems in commutative/non-commutative linear logic

3.7Proof Nets, Semantics of Proofs and the War to Bureaucracy

Deep inference and the calculus of structures are influencing the design of a new generation of proof nets. Moreover, they offer new insight for semantics of proofs and categorical proof theory. Finally, they open decisive new perspectives in the fight against bureaucracy.

The following papers exist:

• On the Axiomatisation of Boolean Categories with and without Medial
  Lutz Straßburger
  AbstractIn its most general meaning, a Boolean category is to categories what a Boolean algebra is to posets. In a more specific meaning a Boolean category should provide the abstract algebraic structure underlying the proofs in Boolean Logic, in the same sense as a Cartesian closed category captures the proofs in intuitionistic logic and a *-autonomous category captures the proofs in linear logic. However, recent work has shown that there is no canonical axiomatisation of a Boolean category. In this work, we will see a series (with increasing strength) of possible such axiomatisations, all based on the notion of *-autonomous category. We will particularly focus on the medial map, which has its origin in an inference rule in KS, a cut-free deductive system for Boolean logic in the calculus of structures. Finally, we will present a category proof nets as a particularly well-behaved example of a Boolean category.

  • Conference version pdf21 June 2006
    Accepted at Cl&C '06 (satellite workshop of ICALP '06), under the title `What Could a Boolean Category Be?´
  • Full paper pdf5 October 2007
    Theory and Applications of Categories, Vol. 18 (18) 2007, pp. 536–601

• What Is a Logic, and What Is a Proof?
  Lutz Straßburger
  AbstractI will discuss the two problems of how to define identity between logics and how to define identity between proofs. For the identity of logics, I propose to simply use the notion of preorder equivalence. This might be considered to be folklore, but is exactly what is needed from the viewpoint of the problem of the identity of proofs: If the proofs are considered to be part of the logic, then preorder equivalence becomes equivalence of categories, whose arrows are the proofs. For identifying these, the concept of proof nets is discussed.
  Pdf23 October 2006
  Logica Universalis—Towards a General Theory of Logic, pp. 135–152, Birkhäuser, 2007

• Proof Nets and the Identity of Proofs
  Lutz Straßburger
  AbstractThese are the notes for a 5-lecture-course given at ESSLLI 2006 in Malaga, Spain. The URL of the school is http://esslli2006.lcc.uma.es/. The course is intended to be introductory. That means no prior knowledge of proof nets is required. However, the student should be familiar with the basics of propositional logic, and should have seen formal proofs in some formal deductive system (e.g., sequent calculus, natural deduction, resolution, tableaux, calculus of structures, Frege-Hilbert-systems, ...). It is probably helpful if the student knows already what cut elimination is, but this is not strictly necessary. In these notes, I will introduce the concept of “proof nets” from the viewpoint of the problem of the identity of proofs. I will proceed in a rather informal way. The focus will be more on presenting ideas than on presenting technical details. The goal of the course is to give the student an overview of the theory of proof nets and make the vast amount of literature on the topic easier accessible to the beginner. For introducing the basic concepts of the theory, I will in the first part of the course stick to the unit-free multiplicative fragment of linear logic because of its rather simple notion of proof nets. In the second part of the course we will see proof nets for more sophisticated logics.
  Pdf20 October 2006
  Technical Report 6013, INRIA

• From Proof Nets to the Free *-Autonomous Category
  François Lamarche and Lutz Straßburger
  AbstractIn the first part of this paper we present a theory of proof nets for full multiplicative linear logic, including the two units. It naturally extends the well-known theory of unit-free multiplicative proof nets. A linking is no longer a set of axiom links but a tree in which the axiom links are subtrees. These trees will be identified according to an equivalence relation based on a simple form of graph rewriting. We show the standard results of sequentialization and strong normalization of cut elimination. In the second part of the paper we show that the identifications enforced on proofs are such that the class of two-conclusion proof nets defines the free *-autonomous category.
  Pdf5 October 2006Journal version of On Proof Nets for Multiplicative Linear Logic with Units
  Logical Methods in Computer Science, Vol. 2 (4:3) 2006, pp. 1–44

• Completeness of MLL Proof-Nets w.r.t. Weak Distributivity
  Jean-Baptiste Joinet
  AbstractWe examine ‘weak-distributivity’ as a rewriting rule ^WD→ defined on multiplicative proof-structures (so, in particular, on multiplicative proof-nets: MLL). This rewriting does not preserve the type of proofs-nets, but does nevertheless preserve their correctness. The specific contribution of this paper, is to give a direct proof of completeness for ^WD→: starting from a set of simple generators (proof-nets which are a n-ary ⊗ of ℘-ized axioms), any mono-conclusion MLL proof-net can be reached by ^WD→ rewriting (up to ⊗ and ℘ associativity and commutativity).
  Pdf8 September 2006
  The Journal of Symbolic Logic, Vol. 72 (1) 2007, pp. 159–170
  Invited talk at WoLLIC 2003, under the title `Calculus of Structures and Proof Nets´, Electronic Notes in Theoretical Computer Science 84; a short version appeared in the proceedings of Structures and Deduction '05, pp. 81–94

• Exploring the Gap between Linear and Classical Logic
  François Lamarche
  AbstractThe Medial rule was first devised as a deduction rule in the Calculus of Structures. In this paper we explore it from the point of view of category theory, as additional structure on a *-autonomous category. This gives us some insights on the denotational semantics of classical propositional logic, and allows us to construct new models for it, based on suitable generalizations of the theory of coherence spaces.
  Pdf27 June 2006
  Theory and Applications of Categories, Vol. 18 (17) 2007, pp. 473–535

• Categorical Models of First Order Classical Proofs
  Richard McKinley
  AbstractThis thesis introduces the notion of a classical doctrine: a semantics for proofs in first-order classical logic derived from the classical categories of Führmann and Pym, using Lawvere’s notion of hyperdoctrine. We introduce a hierarchy of classes of model, increasing in the strength of cut-reduction theory they model; the weakest captures cut reduction, and the strongest gives De Morgan duality between quantifiers as an isomorphism. Whereas classical categories admit the elimination of logical cuts as equalities, (and cuts against structural rules as inequalities), classical doctrines admit certain logical cuts as inequalities only. This is a result of the additive character of the quantifier introduction rules, as is illustrated by a concrete model based on families of sets and relations, using an abstract Geometry of Interaction construction.
  We establish that each class of models is sound and complete with respect to the relevant cut-reduction theory on proof nets based on those of Robinson for propositional classical logic. We show also that classical categories and classical doctrines are not only a class of models for the sequent calculus, but also for deep inference calculi due to Brünnler for classical logic. Of particular interest are the local systems for classical logic, which we show are modelled by categorical models with an additional axiom forcing monoidality of certain functors; these categorical models correspond to multiplicative presentations of the sequent calculus with additional additive features.
  Pdf30 March 2006
  PhD thesis, successfully defended on 17.3.2006

• The Three Dimensions of Proofs
  Yves Guiraud
  AbstractIn this document, we study a 3-polygraphic translation for the proofs of SKS, a formal system for classical propositional logic. We prove that the free 3-category generated by this 3-polygraph describes the proofs of classical propositional logic modulo structural bureaucracy. We give a 3-dimensional generalization of Penrose diagrams and use it to provide several pictures of a proof. We sketch how local transformations of proofs yield a non contrived example of 4-dimensional rewriting.

  • Conference version pdf14 April 2005
    Proceedings of Structures and Deduction '05, pp. 35–52
  • Full paper pdf18 November 2005
    Annals of Pure and Applied Logic, Vol. 141 (1-2) 2006, pp. 266–295

• On Two Forms of Bureaucracy in Derivations
  Kai Brünnler and Stéphane Lengrand
  AbstractWe call irrelevant information in derivations bureaucracy. An example of such irrelevant information is the order between two consecutive inference rules that trivially permute. Building on ideas by Guglielmi, we identify two forms of bureaucracy that occur in the calculus of structures (and, in fact, in every non-trivial term rewriting derivation). We develop term calculi that provide derivations that do not contain this bureaucracy. We also give a normalisation procedure that removes bureaucracy from derivations and find that in a certain sense the normalisation process is a process of cut elimination.
  Pdf10 June 2005
  Proceedings of Structures and Deduction '05, pp. 69–80
  This is further expanded in Ch. 11 of Stéphane Lengrand's PhD thesis Normalisation & Equivalence in Proof Theory & Type Theory (2007 Ackermann Award of the EACSL)

• From Deep Inference to Proof Nets
  Lutz Straßburger
  AbstractWe will see how derivations in (a variation of) SKS can be translated into proof nets. Since an SKS derivation contains more information about a proof than the corresponding proof net, we observe a loss of information which can be understood as "eliminating bureaucracy". Technically this is achieved by cut reduction on proof nets. As an intermediate step between the two extremes, SKS derivations and proof nets, we will see nets representing derivations in "Formalism A".
  Postscript13 April 2005
  Proceedings of Structures and Deduction '05, pp. 2–18

• Constructing Free Boolean Categories
  François Lamarche and Lutz Straßburger
  AbstractBy Boolean category we mean something which is to a Boolean algebra what a category is to a poset. We propose an axiomatic system for Boolean categories, which is different in several respects from the ones proposed recently. In particular everything is done from the start in a *-autonomous category and not in a weakly distributive one, which simplifies issues like the Mix rule. An important axiom, which is introduced later, is a "graphical" condition, which is closely related to denotational semantics and the Geometry of Interaction. Then we show that a previously constructed category of proof nets is the free "graphical" Boolean category in our sense. This validates our categorical axiomatization with respect to a real-life example. Another important aspect of this work is that we do not assume a-priori the existence of units in the *-autonomous categories we use. This has some retroactive interest for the semantics of linear logic, and is motivated by the properties of our example with respect to units.

  • Conference version pdf8 April 2005
    Proceedings of LICS 2005, pp. 209–218
  • Full paper pdf18 February 2005

• Classical Categories and Deep Inference
  Richard McKinley
  AbstractDeep inference is a proof-theoretic notion in which proof rules apply arbitrarily deeply inside a formula. We show that the essence of deep inference is the bifunctoriality of the connectives. We demonstrate that, when given an inequational theory that models cut-reduction, a deep inference calculus for classical logic (SKSg) is a categorical model of the classical sequent calculus LK in the sense of Führmann and Pym. We observe that this gives a notion of cut-reduction for derivations in SKSg, for which the usual notion of cut in SKSg is a special case. Viewing SKSg as a model of the sequent calculus uncovers new insights into the Craig interpolation lemma and intuitionistic provability.
  Pdf31 March 2005
  Proceedings of Structures and Deduction '05, pp. 19–33

• Naming Proofs in Classical Propositional Logic
  François Lamarche and Lutz Straßburger
  AbstractWe present a theory of proof denotations in classical propositional logic. The abstract definition is in terms of a semiring of weights, and two concrete instances are explored. With the Boolean semiring we get a theory of classical proof nets, with a geometric correctness criterion, a sequentialization theorem, and a strongly normalizing cut-elimination procedure. With the semiring of natural numbers, we obtain a sound semantics for classical logic, in which fewer proofs are identified. Though a "real" sequentialization theorem is missing, these proof nets have a grip on complexity issues. In both cases the cut elimination procedure is closely related to its equivalent in the calculus of structures, and we get "Boolean" categories which are not posets.
  Pdf31 January 2005
  TLCA 2005, LNCS 3461, pp. 246–261

• Deep Inference Proof Theory Equals Categorical Proof Theory Minus Coherence
  Dominic Hughes
  AbstractThis paper links deep inference proof theory, as studied by Guglielmi et. al., to categorical proof theory in the sense of Lambek et. al. It observes how deep inference proof theory is categorical proof theory, minus the coherence diagrams/laws. Coherence yields a ready-made and well studied notion of equality on deep inference proofs. The paper notes a precise correspondence between the symmetric deep inference system for multiplicative linear logic (the linear fragment of SKSg and the presentation of *-autonomous categories as symmetric linearly distributive categories with negation. Contraction and weakening in SKSg corresponds precisely to the presence of (co)monoids.
  Pdf6 October 2004

• On Proof Nets for Multiplicative Linear Logic with Units
  Lutz Straßburger and François Lamarche
  AbstractIn this paper we present a theory of proof nets for full multiplicative linear logic, including the two units. It naturally extends the well-known theory of unit-free multiplicative proof nets. A linking is no longer a set of axiom links but a tree in which the axiom links are subtrees. These trees will be identified according to an equivalence relation based on a simple form of graph rewriting. We show the standard results of sequentialization and strong normalization of cut elimination. Furthermore, the identifications enforced on proofs are such that the proof nets, as they are presented here, form the arrows of the free (symmetric) *-autonomous category.
  Pdf30 June 2004
  CSL 2004, LNCS 3210, pp. 145–159

• Current research topics and open problems in formalism B, deductive nets and identity of derivations
• Current research topics and open problems in proof nets and semantics of proofs

3.8Language Design

Thanks to a self-dual non-commutative extension of linear logic one gets the first purely logical account of sequentiality in proof search. The new logical operators make possible a new approach to partial order planning and its relation to concurrency.

The following papers exist, in addition to Nondeterminism and Language Design in Deep Inference, mentioned above:

• On Linear Logic Planning and Concurrency
  Ozan Kahramanogullari
  AbstractWe present an approach to linear logic planning where an explicit correspondence between partial order plans and multiplicative exponential linear logic proofs is established. This is performed by extracting partial order plans from sound and complete encodings of planning problems in multiplicative exponential linear logic in a way that exhibits a non-interleaving behavioral concurrency semantics. Relying on this fact, we argue that this work is a crucial step for establishing a common language for concurrency and planning that will allow to carry techniques and methods between these two fields.
  Pdf2 February 2008
  Accepted at LATA '08, previously presented at LPAR '07 (short paper)

• A Deductive Compositional Approach to Petri Nets for Systems Biology
  Ozan Kahramanogullari
  AbstractWe introduce the language CP, a compositional language for place transition petri nets for the purpose of modelling signalling pathways in complex biological systems. We give the operational semantics of the language CP by means of a proof theoretical deductive system which extends multiplicative exponential linear logic with a self-dual non-commutative logical operator. This allows to express parallel and sequential composition of processes at the same syntactic level as in process algebra, and perform logical reasoning on these processes. We demonstrate the use of the language on a model of a signaling pathway for Fc receptor-mediated phagocytosis.
  Pdf18 September 2007
  Presented as poster at CMSB '07

• Labeled Event Structure Semantics of Linear Logic Planning
  Ozan Kahramanogullari
  Pdf17 January 2005
  Presented at 1st World Congress on Universal Logic

• Towards Planning as Concurrency
  Ozan Kahramanogullari
  AbstractWe present a purely logical framework to planning where we bring the sequential and parallel composition in the plans to the same level, as in process algebras. The problem of expressing causality, which is very challenging for common logics and traditional deductive systems, is solved by resorting to a recently developed extension of multiplicative exponential linear logic with a self-dual, noncommutative operator. We present an encoding of the conjunctive planning problems in this logic, and provide a constructive soundness and completeness result. We argue that this work is the first, but crucial, step of a uniform deductive formalism that connects planning and concurrency inside a common language, and allow to transfer methods from concurrency to planning.
  Pdf19 October 2004
  Artificial Intelligence and Applications 2005, ACTA Press, pp. 197–202

• A Purely Logical Account of Sequentiality in Proof Search
  Paola Bruscoli
  AbstractWe establish a strict correspondence between the proof-search space of a logical formal system and computations in a simple process algebra. Sequential composition in the process algebra corresponds to a logical relation in the formal system—in this sense our approach is purely logical, no axioms or encodings are involved. The process algebra is a minimal restriction of CCS to parallel and sequential composition; the logical system is a minimal extension of multiplicative linear logic. This way we get the first purely logical account of sequentiality in proof search. Since we restrict attention to a small but meaningful fragment, which is then of very broad interest, our techniques should become a common basis for several possible extensions. In particular, we argue about this work being the first step in a two-step research for capturing most of CCS in a purely logical fashion.
  Pdf12 August 2002
  ICLP 2002, LNCS 2401, pp. 302–316

• Current research topics and open problems in language design

3.9Implementations

Ozan Kahramanogullari, Pierre-Etienne Moreau and Antoine Reilles are implementing calculus-of-structures proof systems in Maude and in Tom. Ozan managed to achieve efficiency without sacrificing proof theoretic cleanliness, and he is obtaining results of independent theoretical interest. There are two slides presentations:

• Implementing Deep InferenceOzan KahramanogullariTalk24 November 2004

• Interaction and Depth Against Nondeterminism in Proof SearchOzan KahramanogullariTalk15 December 2006

Max Schäfer has built a graphical proof editor in Java, called GraPE, for the Maude modules written by Ozan Kahramanogullari; this means that one can interactively build and find proofs in several deep-inference systems.

The following papers exist, in addition to Nondeterminism and Language Design in Deep Inference, mentioned above:

• Interaction and Depth Against Nondeterminism in Deep Inference Proof Search
  Ozan Kahramanogullari
  AbstractDeep inference is a proof theoretical methodology that generalises the traditional notion of inference of the sequent calculus. Deep inference provides more freedom in design of deductive systems for different logics and a rich combinatoric analysis of proofs. In particular, construction of exponentially shorter analytic proofs becomes possible, but with the cost of a greater nondeterminism than in the sequent calculus. In this paper, we extend our previous work on proof search with deep inference deductive systems. We argue that, by exploiting an interaction and depth scheme in the logical expressions, the nondeterminism in proof search can be reduced without losing the shorter proofs and without sacrificing from proof theoretical cleanliness.
  Pdf25 June 2008

• Ingredients of a Deep Inference Theorem Prover
  Ozan Kahramanogullari
  AbstractDeep inference deductive systems for classical logic provide exponentially shorter proofs than the sequent calculus systems, however with the cost of higher nondeterminism and larger search space in proof search. We report on our ongoing work on proof search with deep inference deductive systems. We present systems for classical logic where nondeterminism in proof search is reduced by constraining the context management rule of these systems. We argue that a deep inference system for classical logic can outperform sequent calculus deductive systems in proof search when nondeterminism and the application of the contraction rule are controlled by means of invertible rules.
  Pdf24 June 2008
  Short paper at CL&C'08

• Maude as a Platform for Designing and Implementing Deep Inference Systems
  Ozan Kahramanogullari
  AbstractDeep inference is a proof theoretical methodology that generalizes the traditional notion of inference in the sequent calculus: In contrast to the sequent calculus, the deductive systems with deep inference do not rely on the notion of main connective, and permit the application of the inference rules at any depth inside logical expressions, in a way which resembles the application of term rewriting rules. Deep inference provides a richer combinatoric analysis of proofs for different logics. In particular, construction of exponentially shorter proofs becomes possible. In this paper, aiming at the development of computation as proof search tools, we propose the Maude language as a means for designing and implementing different deep inference deductive systems and proof strategies that work on these systems. We demonstrate these ideas on classical logic and argue that these ideas can be analogously carried to other deductive systems for other logics.
  Pdf16 September 2007
  RULE '07, Electronic Notes in Theoretical Computer Science 219, 2008, pp. 35–50

• Canonical Abstract Syntax Trees
  Antoine Reilles
  AbstractThis paper presents GOM, a language for describing abstract syntax trees and generating a Java implementation for those trees. GOM includes features allowing to specify and modify the interface of the data structure. These features provide in particular the capability to maintain the internal representation of data in canonical form with respect to a