Alessio Guglielmi's Research / Deep Inference and the Calculus of Structures / Language Design
Thanks to a self-dual non-commutative extension of linear logic one gets the first purely logical account of sequentiality in proof search. The new logical operators make possible a new approach to partial order planning and its relation to concurrency.
We establish a strict correspondence between the proof-search space of a logical formal system and computations in a simple process algebra. Sequential composition in the process algebra corresponds to a logical relation in the formal systemin this sense our approach is purely logical, no axioms or encodings are involved. The process algebra is a minimal restriction of CCS to parallel and sequential composition; the logical system is a minimal extension of multiplicative linear logic. This way we get the first purely logical account of sequentiality in proof search. Since we restrict attention to a small but meaningful fragment, which is then of very broad interest, our techniques should become a common basis for several possible extensions. In particular, we argue about this work being the first step in a two-step research for capturing most of CCS in a purely logical fashion.
We present a purely logical framework to planning where we bring the sequential and parallel composition in the plans to the same level, as in process algebras. The problem of expressing causality, which is very challenging for common logics and traditional deductive systems, is solved by resorting to a recently developed extension of multiplicative exponential linear logic with a self-dual, noncommutative operator. We present an encoding of the conjunctive planning problems in this logic, and provide a constructive soundness and completeness result. We argue that this work is the first, but crucial, step of a uniform deductive formalism that connects planning and concurrency inside a common language, and allow to transfer methods from concurrency to planning.
We present an approach to linear logic planning where an explicit correspondence between partial order plans and multiplicative exponential linear logic proofs is established. This is performed by extracting partial order plans from sound and complete encodings of planning problems in multiplicative exponential linear logic in a way that exhibits a non-interleaving behavioral concurrency semantics. Relying on this fact, we argue that this work is a crucial step for establishing a common language for concurrency and planning that will allow to carry techniques and methods between these two fields.
27 August 2007
Accepted at LPAR '07 (short paper)
We introduce the language CP, a compositional language for place transition petri nets for the purpose of modelling signalling pathways in complex biological systems. We give the operational semantics of the language CP by means of a proof theoretical deductive system which extends multiplicative exponential linear logic with a self-dual non-commutative logical operator. This allows to express parallel and sequential composition of processes at the same syntactic level as in process algebra, and perform logical reasoning on these processes. We demonstrate the use of the language on a model of a signaling pathway for Fc receptor-mediated phagocytosis.
This thesis studies the design of deep-inference deductive systems. In the systems with deep inference, in contrast to traditional proof-theoretic systems, inference rules can be applied at any depth inside logical expressions. Deep applicability of inference rules provides a rich combinatorial analysis of proofs. Deep inference also makes it possible to design deductive systems that are tailored for computer science applications and otherwise provably not expressible.
By applying the inference rules deeply, logical expressions can be manipulated starting from their sub-expressions. This way, we can simulate analytic proofs in traditional deductive formalisms. Furthermore, we can also construct much shorter analytic proofs than in these other formalisms. However, deep applicability of inference rules causes much greater nondeterminism in proof construction.
This thesis attacks the problem of dealing with nondeterminism in proof search while preserving the shorter proofs that are available thanks to deep inference. By redesigning the deep inference deductive systems, some redundant applications of the inference rules are prevented. By introducing a new technique which reduces nondeterminism, it becomes possible to obtain a more immediate access to shorter proofs, without breaking certain proof theoretical properties such as cut-elimination. Different implementations presented in this thesis allow to perform experiments on the techniques that we developed and observe the performance improvements. Within a computation-as-proof-search perspective, we use deep-inference deductive systems to develop a common proof-theoretic language to the two fields of planning and concurrency.
Pdf11 September 2006
PhD thesis, successfully defended on 21.12.2006