On Formal Methods for Collective Adaptive System Engineering. {Scalable Approximated, Spatial} Analysis Techniques. Extended Abstract

In this extended abstract a view on the role of Formal Methods in System Engineering is briefly presented. Then two examples of useful analysis techniques based on solid mathematical theories are discussed as well as the software tools which have been built for supporting such techniques. The first technique is Scalable Approximated Population DTMC Model-checking. The second one is Spatial Model-checking for Closure Spaces. Both techniques have been developed in the context of the EU funded project QUANTICOL.Comment: In Proceedings FORECAST 2016, arXiv:1607.0200

Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents

We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants

Two-variable Logic with Counting and a Linear Order

We study the finite satisfiability problem for the two-variable fragment of first-order logic extended with counting quantifiers (C2) and interpreted over linearly ordered structures. We show that the problem is undecidable in the case of two linear orders (in the presence of two other binary symbols). In the case of one linear order it is NEXPTIME-complete, even in the presence of the successor relation. Surprisingly, the complexity of the problem explodes when we add one binary symbol more: C2 with one linear order and in the presence of other binary predicate symbols is equivalent, under elementary reductions, to the emptiness problem for multicounter automata

Recursion Schemes and the WMSO+U Logic

We study the weak MSO logic extended by the unbounding quantifier (WMSO+U), expressing the fact that there exist arbitrarily large finite sets satisfying a given property. We prove that it is decidable whether the tree generated by a given higher-order recursion scheme satisfies a given sentence of WMSO+U

Cost Automata, Safe Schemes, and Downward Closures

Higher-order recursion schemes are an expressive formalism used to define languages of possibly infinite ranked trees. They extend regular and context-free grammars, and are equivalent to simply typed ?Y-calculus and collapsible pushdown automata. In this work we prove, under a syntactical constraint called safety, decidability of the model-checking problem for recursion schemes against properties defined by alternating B-automata, an extension of alternating parity automata for infinite trees with a boundedness acceptance condition. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes

An Infinitary Proof Theory of Linear Logic Ensuring Fair Termination in the Linear ?-Calculus

Fair termination is the property of programs that may diverge "in principle" but that terminate "in practice", i.e. under suitable fairness assumptions concerning the resolution of non-deterministic choices. We study a conservative extension of ?MALL^?, the infinitary proof system of the multiplicative additive fragment of linear logic with least and greatest fixed points, such that cut elimination corresponds to fair termination. Proof terms are processes of ?LIN, a variant of the linear ?-calculus with (co)recursive types into which binary and (some) multiparty sessions can be encoded. As a result we obtain a behavioral type system for ?LIN (and indirectly for session calculi through their encoding into ?LIN) that ensures fair termination: although well-typed processes may engage in arbitrarily long interactions, they are fairly guaranteed to eventually perform all pending actions

Phase Semantics for Linear Logic with Least and Greatest Fixed Points

The truth semantics of linear logic (i.e. phase semantics) is often overlooked despite having a wide range of applications and deep connections with several denotational semantics. In phase semantics, one is concerned about the provability of formulas rather than the contents of their proofs (or refutations). Linear logic equipped with the least and greatest fixpoint operators (?MALL) has been an active field of research for the past one and a half decades. Various proof systems are known viz. finitary and non-wellfounded, based on explicit and implicit (co)induction respectively. In this paper, we extend the phase semantics of multiplicative additive linear logic (a.k.a. MALL) to ?MALL with explicit (co)induction (i.e. ?MALL^{ind}). We introduce a Tait-style system for ?MALL called ?MALL_? where proofs are wellfounded but potentially infinitely branching. We study its phase semantics and prove that it does not have the finite model property