As described in Section  , there has been a decade-long effort to design a logical framework for reasoning about logic specifications. Finally in 2008 and 2009 team members have reached what appears to be a natural culmination of this development. In particular, David Baelde's PhD and Andrew Gacek's PhD thesis provided rich analysis of how the -quantification can be related to fixed point definitions and their associated induction and co-induction inference rules. Baelde has concentrated on proving focusing-style results that are critical for proof automation and on a minimal genericinterpretation of the -quantifier. Gacek has concentrated on a nominal genericinterpretation of the -quantifier. We now understand the difference between these logics: the nominal approach resembles much more closely the approach developed by Pitts .

Full proofs of the important meta-theory results of the logic in Gacek's thesis have been submitted for publication . Gacek has also provided an implementation of his logic within the Abella prover that he has worked on as part of his PhD thesis.

We have developed extensive examples of our this new logic: significant examples taken from the -calculus have been published in and the Abella distribution contains a large number of examples.

Relational descriptions have been widely used in formalizing diverse computational notions, including, for example, operational semantics, typing, acceptance by non-deterministic machines, etc. Such relational specifications can be faithfully captured by a (restricted) logical theory over relations. Such a specification logiccan be picked so that it explicitly treats binding in object languages. Once such a logic is fixed, a natural next question is what devices should be used to prove theorems about specifications written in it. Within the team, we have a second logic, called the reasoning logic, to reason about provability in the first logic. To be adequate for this purpose, the reasoning logic should be able to completely encode the specification logic, including notions of binding, such as quantifiers within formulas, for eigenvariables within sequents, and for abstractions within terms. To provide a natural treatment of these aspects, the reasoning logic must encode binding structures as well as their associated notions of scope, free/bound variables, and capture-avoiding substitution. Furthermore, the reasoning logic should possess strong mechanisms for constructing proofs by induction and co-induction.

Within the context of the Gacek's PhD thesis and the submitted paper by Gacek, Miller, and Nadathur, the logic was present: this logic represents relations over -terms via definitions of atomic judgments, contains inference rules for induction and co-induction, and includes a special quantifier called and a related generalization of equality over -terms called nominal abstraction. The interactive theorem prover Abella implements and supports this two-level logic approach to reasoning about computation. Gacek and others have now contributed a large number of interesting examples showing the utility of using this two-level approach to reasoning: see the Abella web site for many examples. In particular, the POPLMark challenge problems 1a and 2a have nice, declarative solutions within Abella.

New insights into -quantification

The team has been actively extending the scope of effectiveness -quantification. As Tiu and Miller have shown in , the quantifier (developed in previous years within the team) provides a completely satisfactory treatment of binding structures in the finite -calculus. Moving this quantifier to treat infinite behaviors via induction and co-induction, required new advances in the underlying proof theory of -quantification.

The team has explored two different approaches to this problem. David Baelde , has developed a minimalist generalization of previous work by Miller and Tiu: he has found what seems to be the simplest extension to that earlier work that allows to interact properly with fixed points and their inference rules (namely, induction and co-induction). His logical approach allows for a rather careful and rigid understanding of scope in the treatment of the meta-theory of logics and computational specifications.

Another angle has been developed as a result of our close international collaborations. Alwen Tiu, now at the Australian National University, has developed a logic, called which extends the earlier, “minimal” approach by introducing the structural rules of strengthening and exchange into the context of generic variables. As a result, the behavior of bindings becomes much more like the behavior of names more generally, while still maintaining much of the status as being binders. In combination with our close colleagues at the University of Minnesota, we have extended this work to include a new definitional principle, called nabla-in-the-head, that strengthens our ability to declaratively describe the structure of contexts and proof invariants. This new definitional principle was first presented in and examples of it were presented in . Our colleague, Andrew Gacek (a PhD student at the University of Minnesota and former intern with Parsifal) has also built the Abella proof editor that allows for the direct implementation of this new definitional principle. His system is in distribution and has been used by a number of people to develop examples in this logic.

Since focusing proof systems seem to be behind much of our computational logic framework, the team has spent some energies developing further some foundational aspects of this approach to proof systems.

Chuck Liang and Miller have recently finished the paper in which a comprehensive approach to focusing in intuitionistic and classical logic was developed. In particular, they present a compact sequent calculus LKU for classical logic organized around the concept of polarization.Focused sequent calculi for classical, intuitionistic, and multiplicative-additive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structural rules to polarity information. They identify a general set of criteria under which cut elimination holds in such fragments. From cut elimination leads to a unified proof of the completeness of focusing. Furthermore, each sublogic can interact with other fragments through cut. Under certain circumstances, for example, it is possible for a classical lemma to be used in an intuitionistic proof while preserving intuitionistic provability. They also examine the possibility of defining some classical-linear hybrid logics.

Given the team's ambitious to automate logics that require induction and co-induction, we have also looked in detail at the proof theory of fixed points. In particular, David Baelde's recent PhD thesis contains a number of important, foundational theorems regarding focusing and fixed points. In particular, he has examined the logic MALL (multiplicative and additive linear logic). To strengthen this decidable logic into a more general logic, Girard added the exponentials, which allowed for modeling unbounded (“infinite”) behavior. Baelde considers, however, the addition of fixed points instead and he has developed the proof theory of the resulting logic. We see this logic as being behind much of the work that the team will be doing in the coming few years.

Alexis Saurin's recent PhD also contains a wealth of new material concerning focused proof system. In particular, he provides a new and modular approach to proving the completeness of focused proof systems as well as develops the theme of multifocusing.

A particular outcome of our work on focused proof search is the use of maximally multifocused proofsto help provide sequent calculus proofs a canonicity. In particular, Chaudhuri, Miller, and Saurin have shown in that it is possible to show that maximally multifocused sequent proofs can be placed in one-to-one correspondence with more traditional proof net structures for subsets of MALL.

A couple of years ago, Miller and Saurin proposed a neutral approach to proof and refutation. The goal was to describe an entirely neutral setting where a step in a “proof search” could be seen as a step in either building a proof of a formula or a proof of its negation. The early work was limited to essentially a simple generalization to additive logic. Delande was able to generalize that work to capture multiplicative connectives as well. His thesis contains two game semantics for multiplicative additive linear logic (MALL): the first is sequential and the second is concurrent. The concurrent game was used to capture full completeness results between MALL (focused) sequent calculus proofs and winning strategies.

The paper provides a detailed description of the sequent approach to both additive and multiplicative linear logic. In that paper, the search for a proof of Bor a refutation of B( i.e., a proof of ¬ B) can be carried out simultaneously: in contrast, the usual approach in automated deduction views proving Bor proving ¬ Bas two, possibly unrelated, activities. Their approach to proof and refutation is described as a two-player game in which each player follows the same rules. A winning strategy translates to a proof of the formula and a counter-winning strategy translates to a refutation of the formula. The game is described for multiplicative and additive linear logic (MALL). A game theoretic treatment of the multiplicative connectives is intricate and involves two important ingredients. First, labeled graph structures are used to represent positions in a game and, second, the game playing must deal with the failure of a given player and with an appropriate resumption of play. This latter ingredient accounts for the fact that neither player might win (that is, neither Bnor ¬ Bmight be provable).

The inference rules of a logic define a logical connective in a canonical fashion if the following test is passed: assume that there are two copies of a logical connective, say a red and blue copy, and assume that both of these connectives have the same introduction rules. If it is possible to prove that the red and blue versions are equivalent within the extended proof system, then we say that the connective is defined canonically. In linear logic, all connectives are canonical in this sense except for the exponentials (!, ?). That is, it is possible to have many exponentials and they do not need to all support weakening and contraction but some subsets of these structural rules. Since they do not need to provide all structural rules, we have called these not exponentials but subexponentials .

Proof theory does not provide canonical solutions for many things in computational logic: for example, the domain of first-order quantification is seldom addressed by proof theory as well as the exact nature of worlds within, say, Kripke models. These non-canonical aspects of logic provide, however, important opportunities for computer scientists to attach structures that they need to logic. Since the exponentials are not canonical, maybe there are possible exploitations of such non-canonical exponentials in computer science. Nigam and Miller have provided a partial answer to this question. In particular, they have shown that rich forms of multiset computation can be supported using subexponentials . In particular, it is possible to specify various multisets with different locations (identified with different exponentials) and to test them for emptiness. In this way, linear logic with an array of subexponential can then be used to declaratively and faithfully specify a wide range of deterministic and non-deterministic algorithms.

In earlier work by Pimentel and Miller , it was clear that linear logic could be used to encode provability in classical and intuitionistic logics using simple and elegant linear logic theories. Recently, Nigam and Miller have extended that work to show that by using polarity and focusing within linear logic, it is possible to account for a range of proof systems, such as, for example, sequent calculus, natural deduction, tableaux, free deduction, etc. The initial work in this area by Nigam, Pimentel, and Miller only captured relative completenesswhereas the most recent our these papers are able to capture a much more refined notion of “adequate encoding”, namely, inference rules in one system are captured exactly as (focused) inference rules in the linear logical framework. In particular, Nigam and Miller argue that linear logic can be used as a meta-logic to specify a range of object-level proof systems. In particular, they showed that by providing different polarizationswithin a focused proof systemfor linear logic, one can account for natural deduction (normal and non-normal), sequent proofs (with and without cut), and tableaux proofs. Armed with just a few, simple variations to the linear logic encodings, more proof systems can be accommodated, including proof system using generalized elimination and generalized introduction rules. In general, most of these proof systems are developed for both classical and intuitionistic logics. By using simple results about linear logic, they could also give simple and modular proofs of the soundness and relative completeness of all the proof systems considered.

There are modal logics like S4 or K, for which it is rather straightforward to provide a cut-free sequent system, and there are others, like S5 for which this is difficult or impossible. We (in a joint work with Kai Brünnler, Univ. Bern) used “nested sequents”  (a generalization of hypersequents  ) to give a completely modular account to the whole modal cube below S5. That is to say, we have cut-free sequent systems for the basic normal modal logics formed by any combination of the axioms d, t, b, 4, 5, such that each axiom has a corresponding rule and each combination of these rules is complete for the corresponding frame conditions. This result are published in  .

We were able (in a joint work with Agata Ciabattoni, TU Wien, and Kazushige Terui, Kyoto University) to make further progress in the development of a systematic and algebraic proof theory for nonclassical logics. Continuing the work of we defined a hierarchy on Hilbert axioms in the language of classical linear logic without exponentials, and gave a systematic procedure for transforming axioms up to the level P₃^'of the hierarchy into inference rules in multiple-conclusion (hyper)sequent calculi, which enjoy cut-elimination under a certain condition. This allows a systematic treatment of logics which could not be dealt with in previous approaches. Our method also works as a heuristic principle for finding appropriate rules for axioms located at levels higher than P₃^'. The work is published in .

Kleene's theorem on the coincidence of the rational and the recognizable languages in a free monoid is a fundamental result of theoretical computer science. In we present a generalization of Kleene's theorem to forest languages, which are a generalization of tree languages. However, our result is not a generalization of the result by Thatcher and Wright on tree languages . We proposed an alternative approach to the standard notion of rational (or regular) expression for tree languages. The main difference is that in our new notion we have only one concatenation operation and only one star-operation, instead of many different ones. This is achieved by considering forests instead of trees over a ranked alphabet, or, algebraicly speaking, by considering cartesian categories instead of term-algebras. The main result is that in the free cartesian category the rational languages and the recognizable languages coincide.

Meta-variables are central in proof search mechanisms to represent incomplete proofs and incomplete objects. They are used in almost all implementations of proof-related software, yet their meta-theory remains less explored than that of complete proofs and objects such as in the -calculus.

In 2009, Stéphane Lengrand and Jamie Murdoch Gabbay have published in  a first proposal for a computational model taking these features into account.

This proposal, extending the calculus with a particular kind of meta-variables originating from nominal logic, is more sparing than previous approaches like Higher-Order Abstract Syntax, which explicitly represents all potentialdependencies between incomplete objects (this leads to computational inefficiencies as potential dependencies that are not effectively used still incur a computational cost).

Lengrand and Gabbay's proposal is only a first step, as it does not have a neat theory of normal forms (i.e. output values). A more complete version of such a -calculus, with incomplete objects and arbitrary binding dependencies but also with better normalization properties, has been in development since.

Joëlle Despeyroux and Kaustuv Chaudhuri have given an encoding of the synchronous stochastic -calculus in a hybrid extension of intuitionistic linear logic (called HyLL). Precisely, they have shown that focused partial sequent derivations in the encoding are in bijection with stochastic traces. The modal worlds are used to represent the rates of stochastic interactions, and the connectives of hybrid logic are used to represent the constraints in the stochastic transition rules. These results have been submitted to a journal and an extended report is available from HAL  .

One of the most successful applications of the stochastic -calculus has been in representing signal transduction networks in cellular biology. An interesting application of this work would therefore be the direct representations of biological processes in HyLL, the original motivation for this line of investigation. Furthermore, other stochastic systems can, at least in principle, be similarly encoded in HyLL, giving us the linguistic ability to compare and combine systems represented using different stochastic formalisms.

A central research topic of the team are the connections between proof theory and computation. A proof-theoretic method which is often used for modeling computational systems is cut-elimination. The computational aspects of this procedure are rather well understood for intuitionistic and linear logic while the situation in classical logic is less satisfactory. shows that in the general case the number of possible computational interpretations of a classical proof increases as strongly as its computational power and thus provides a lower bound on the set computations encoded by proof. on the other hand considers proof in classical first-order logic and Peano arithmetic and derives an upper bound which is characterized by a regular tree grammar. This grammar can be used as alternative algorithm for computing cut-free proofs and is therefore a contribution to the reduction of syntax in proof theory.