On the Proof Theory of Regular Fixed Points

International audienceWe consider encoding finite automata as least fixed points in a proof theoretical framework equipped with a general induction scheme, and study automata inclusion in that setting. We provide a coinductive characterization of inclusion that yields a natural bridge to proof-theory. This leads us to generalize these observations to regular formulas, obtaining new insights about inductive theorem proving and cyclic proofs in particular

A proposal for broad spectrum proof certificates

International audienceRecent developments in the theory of focused proof systems provide flexible means for structuring proofs within the sequent calculus. This structuring is organized around the construction of ''macro'' level inference rules based on the ''micro'' inference rules which introduce single logical connectives. After presenting focused proof systems for first-order classical logics (one with and one without fixed points and equality) we illustrate several examples of proof certificates formats that are derived naturally from the structure of such focused proof systems. In principle, a proof certificate contains two parts: the first part describes how macro rules are defined in terms of micro rules and the second part describes a particular proof object using the macro rules. The first part, which is based on the vocabulary of focused proof systems, describes a collection of macro rules that can be used to directly present the structure of proof evidence captured by a particular class of computational logic systems. While such proof certificates can capture a wide variety of proof structures, a proof checker can remain simple since it must only understand the micro-rules and the discipline of focusing. Since proofs and proof certificates are often likely to be large, there must be some flexibility in allowing proof certificates to elide subproofs: as a result, proof checkers will necessarily be required to perform (bounded) proof search in order to reconstruct missing subproofs. Thus, proof checkers will need to do unification and restricted backtracking search

Completeness and Decidability Results for First-order Clauses with Indices

Session: Inference systems (www.cl.cam.ac.uk/~gp351/cade24)International audienceWe define a proof procedure that allows for a limited form of inductive reasoning. The first argument of a function symbol is allowed to belong to an inductive type. We will call such an argument an index. We enhance the standard superposition calculus with a loop detection rule, in order to encode a particular form of mathematical induction. The satisfiability problem is not semi-decidable, but some classes of clause sets are identified for which the proposed procedure is complete and/or terminating

Infinets: The parallel syntax for non-wellfounded proof-theory

Logics based on the µ-calculus are used to model induc-tive and coinductive reasoning and to verify reactive systems. A well-structured proof-theory is needed in order to apply such logics to the study of programming languages with (co)inductive data types and automated (co)inductive theorem proving. While traditional proof system suffers some defects, non-wellfounded (or infinitary) and circular proofs have been recognized as a valuable alternative, and significant progress have been made in this direction in recent years. Such proofs are non-wellfounded sequent derivations together with a global validity condition expressed in terms of progressing threads. The present paper investigates a discrepancy found in such proof systems , between the sequential nature of sequent proofs and the parallel structure of threads: various proof attempts may have the exact threading structure while differing in the order of inference rules applications. The paper introduces infinets, that are proof-nets for non-wellfounded proofs in the setting of multiplicative linear logic with least and greatest fixed-points (µMLL ∞) and study their correctness and sequentialization. Inductive and coinductive reasoning is pervasive in computer science to specify and reason about infinite data as well as reactive properties. Developing appropriate proof systems amenable to automated reasoning over (co)inductive statements is therefore important for designing programs as well as for analyzing computational systems. Various logical settings have been introduced to reason about such inductive and coinductive statements, both at the level of the logical languages modelling (co)induction (such as Martin Löf's inductive predicates or fixed-point logics, also known as µ-calculi) and at the level of the proof-theoretical framework considered (finite proofs with explicit (co)induction rulesà la Park [23] or infinite, non-wellfounded proofs with fixed-point unfold-ings) [6-8, 4, 1, 2]. Moreover, such proof systems have been considered over classical logic [6, 8], intuitionistic logic [9], linear-time or branching-time temporal logic [19, 18, 25, 26, 13-15] or linear logic [24, 16, 4, 3, 14]

Observed communication semantics for classical processes

Classical Linear Logic (CLL) has long inspired readings of its proofs as communicating processes. Wadler's CP calculus is one of these readings. Wadler gave CP an operational semantics by selecting a subset of the cut-elimination rules of CLL to use as reduction rules. This semantics has an appealing close connection to the logic, but does not resolve the status of the other cut-elimination rules, and does not admit an obvious notion of observational equivalence. We propose a new operational semantics for CP based on the idea of observing communication, and use this semantics to define an intuitively reasonable notion of observational equivalence. To reason about observational equivalence, we use the standard relational denotational semantics of CLL. We show that this denotational semantics is adequate for our operational semantics. This allows us to deduce that, for instance, all the cut-elimination rules of CLL are observational equivalences

A Lightweight Formalization of the Metatheory of Bisimulation-Up-To

International audienceBisimilarity of two processes is formally established by producing a bisimulation relation that contains those two processes and obeys certain closure properties. In many situations, particularly when the under-lying labeled transition system is unbounded, these bisimulation relations can be large and even infinite. The bisimulation-up-to technique has been developed to reduce the size of the relations being computed while retaining soundness, that is, the guarantee of the existence of a bisimulation. Such techniques are increasingly becoming a critical ingredient in the automated checking of bisimilarity. This paper is devoted to the formalization of the meta theory of several major bisimulation-up-to techniques for the process calculi CCS and the π-calculus (with replication). Our formalization is based on recent work on the proof theory of least and greatest fixpoints, particularly the use of relations defined (co-)inductively, and of co-inductive proofs about such relations, as implemented in the Abella theorem prover. An important feature of our formalization is that our definitions of the bisimulation-up-to relations are, in most cases, straightforward translations of published informal definitions, and our proofs clarify several technical details of the informal descriptions. Since the logic behind Abella also supports λ-tree syntax and generic reasoning using the ∇-quantifier, our treatment of the π-calculus is both direct and natural

An infinitary model of linear logic

In this paper, we construct an infinitary variant of the relational model of linear logic, where the exponential modality is interpreted as the set of finite or countable multisets. We explain how to interpret in this model the fixpoint operator Y as a Conway operator alternatively defined in an inductive or a coinductive way. We then extend the relational semantics with a notion of color or priority in the sense of parity games. This extension enables us to define a new fixpoint operator Y combining both inductive and coinductive policies. We conclude the paper by sketching the connection between the resulting model of lambda-calculus with recursion and higher-order model-checking.Comment: Accepted at Fossacs 201

Towards standardized criteria for diagnosing chronic intervillositis of unknown etiology: A systematic review

Research into fetal development and medicin

(Mathematical) Logic for Systems Biology (Invited Paper)

International audienceWe advocates here the use of (mathematical) logic for systems biology, as a unified framework well suited for both modeling the dynamic behaviour of biological systems, expressing properties of them, and verifying these properties. The potential candidate logics should have a traditional proof theoretic pedigree (including a sequent calculus presentation enjoying cut-elimination and focusing), and should come with (certified) proof tools. Beyond providing a reliable framework, this allows the adequate encodings of our biological systems. We present two candidate logics (two modal extensions of linear logic, called HyLL and SELL), along with biological examples. The examples we have considered so far are very simple ones-coming with completely formal (interactive) proofs in Coq. Future works includes using automatic provers, which would extend existing automatic provers for linear logic. This should enable us to specify and study more realistic examples in systems biology, biomedicine (diagnosis and prognosis), and eventually neuroscience

Mutational analysis of a heterogeneous nuclear ribonucleoprotein A2 response element for RNA trafficking

Cytoplasmic transport and localization of mRNA has been reported for a range of oocytes and somatic cells. The heterogeneous nuclear ribonucleoprotein (hnRNP) A2 response element (A2RE) is a 21-nucleotide segment of the myelin basic protein mRNA that is necessary and sufficient for cytoplasmic transport of this message in oligodendrocytes. The predominant A2RE-binding protein in rat brain has previously been identified as hnRNP A2. Here we report that an 11-nucleotide subsegment of the A2RE (A2RE11) was as effective as the full-length A2RE in binding hnRNP A2 and mediating transport of heterologous RNA in oligodendrocytes. Point mutations of the A2RE11 that eliminated binding to hnRNP A2 also markedly reduced the ability of these oligoribonucleotides to support RNA transport. Oligodendrocytes treated with antisense oligonucleotides directed against the translation start site of hnRNP A2 had reduced levels of this protein and disrupted transport of microinjected myelin basic protein RNA. Several A2RE-like sequences from localized neuronal RNAs also bound hnRNP A2 and promoted RNA transport in oligodendrocytes. These data demonstrate the specificity of A2RE recognition by hnRNP A2, provide direct evidence for the involvement of hnRNP A2 in cytoplasmic RNA transport, and suggest that this protein may interact with a wide variety of localized messages that possess A2RE-like sequences