# Volume 2, Issue 2

2006

### 1. Extending the Extensional Lambda Calculus with Surjective Pairing is Conservative

We answer Klop and de Vrijer's question whether adding surjective-pairing axioms to the extensional lambda calculus yields a conservative extension. The answer is positive. As a byproduct we obtain a "syntactic" proof that the extensional lambda calculus with surjective pairing is consistent.

### 2. The monadic second-order logic of graphs XVI : Canonical graph<br> decompositions

This article establishes that the split decomposition of graphs introduced by Cunnigham, is definable in Monadic Second-Order Logic.This result is actually an instance of a more general result covering canonical graph decompositions like the modular decomposition and the Tutte decomposition of 2-connected graphs into 3-connected components. As an application, we prove that the set of graphs having the same cycle matroid as a given 2-connected graph can be defined from this graph by Monadic Second-Order formulas.

### 3. On the Expressiveness of the Ambient Logic

The Ambient Logic (AL) has been proposed for expressing properties of process mobility in the calculus of Mobile Ambients (MA), and as a basis for query languages on semistructured data. In this paper, we study the expressiveness of AL. We define formulas for capabilities and for communication in MA. We also derive some formulas that capture finitess of a term, name occurrences and persistence. We study extensions of the calculus involving more complex forms of communications, and we define characteristic formulas for the equivalence induced by the logic on a subcalculus of MA. This subcalculus is defined by imposing an image-finiteness condition on the reducts of a MA process.

### 4. A System of Interaction and Structure II: The Need for Deep Inference

This 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.

### 5. Modal Logics of Topological Relations

Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity.

### 6. Context-Sensitive Languages, Rational Graphs and Determinism

We investigate families of infinite automata for context-sensitive languages. An infinite automaton is an infinite labeled graph with two sets of initial and final vertices. Its language is the set of all words labelling a path from an initial vertex to a final vertex. In 2001, Morvan and Stirling proved that rational graphs accept the context-sensitive languages between rational sets of initial and final vertices. This result was later extended to sub-families of rational graphs defined by more restricted classes of transducers. languages.<br><br> Our contribution is to provide syntactical and self-contained proofs of the above results, when earlier constructions relied on a non-trivial normal form of context-sensitive grammars defined by Penttonen in the 1970's. These new proof techniques enable us to summarize and refine these results by considering several sub-families defined by restrictions on the type of transducers, the degree of the graph or the size of the set of initial vertices.