4,823 research outputs found
Unifying type systems for mobile processes
We present a unifying framework for type systems for process calculi. The
core of the system provides an accurate correspondence between essentially
functional processes and linear logic proofs; fragments of this system
correspond to previously known connections between proofs and processes. We
show how the addition of extra logical axioms can widen the class of typeable
processes in exchange for the loss of some computational properties like
lock-freeness or termination, allowing us to see various well studied systems
(like i/o types, linearity, control) as instances of a general pattern. This
suggests unified methods for extending existing type systems with new features
while staying in a well structured environment and constitutes a step towards
the study of denotational semantics of processes using proof-theoretical
methods
Normalisation Control in Deep Inference via Atomic Flows
We 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
Linear Logic and Noncommutativity in the Calculus of Structures
In 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
A Non-Commutative Extension of MELL
We 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
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
A System of Interaction and Structure
This 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 submitted 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 allow the design of BV, yield a modular proof
of cut elimination.Comment: This is the authoritative version of the article, with readable
pictures, in colour, also available at
. (The published version contains
errors introduced by the editorial processing.) Web site for Deep Inference
and the Calculus of Structures at <http://alessio.guglielmi.name/res/cos
The Focused Calculus of Structures
The focusing theorem identifies a complete class of sequent proofs that have no inessential non-deterministic choices and restrict the essential choices to a particular normal form. Focused proofs are therefore well suited both for the search and for the representation of sequent proofs. The calculus of structures is a proof formalism that allows rules to be applied deep inside a formula. Through this freedom it can be used to give analytic proof systems for a wider variety of logics than the sequent calculus, but standard presentations of this calculus are too permissive, allowing too many proofs. In order to make it more amenable to proof search, we transplant the focusing theorem from the sequent calculus to the calculus of structures. The key technical contribution is an incremental treatment of focusing that avoids trivializing the calculus of structures. We give a direct inductive proof of the completeness of the focused calculus of structures with respect to a more standard unfocused form. We also show that any focused sequent proof can be represented in the focused calculus of structures, and, conversely, any proof in the focused calculus of structures corresponds to a focused sequent proof
Locality for Classical Logic
In 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. In addition, they enjoy a top-down symmetry and
some normal forms for derivations that are not available in the sequent
calculus. Identity axiom, cut, weakening and also contraction can be reduced to
atomic form. This leads to rules that are local: they do not require the
inspection of expressions of unbounded size
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