Proceedings of the international conference on Mathematical foundations of programming semantics
Algebraic approaches to program semantics
Algebraic approaches to program semantics
A mathematical modeling of pure, recursive algorithms
Logic at Botik'89 Symposium on logical foundations of computer science
The geometry of optimal lambda reduction
POPL '92 Proceedings of the 19th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
New foundations for the geometry of interaction
Information and Computation
Geometry of interaction III: accommodating the additives
Proceedings of the workshop on Advances in linear logic
The Pattern-of-Calls Expansion Is the Canonical Fixpoint for Recursive Definitions
Journal of the ACM (JACM)
Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculi
TLCA '97 Proceedings of the Third International Conference on Typed Lambda Calculi and Applications
Geometry of interaction 2: deadlock-free algorithms
COLOG '88 Proceedings of the International Conference on Computer Logic
Retracting Some Paths in Process Algebra
CONCUR '96 Proceedings of the 7th International Conference on Concurrency Theory
From Hilbert Spaces to Dilbert Spaces: Context Semantics Made Simple
FST TCS '02 Proceedings of the 22nd Conference Kanpur on Foundations of Software Technology and Theoretical Computer Science
Glueing and orthogonality for models of linear logic
Theoretical Computer Science - Category theory and computer science
Geometry of Interaction and linear combinatory algebras
Mathematical Structures in Computer Science
The Geometry of Linear Higher-Order Recursion
LICS '05 Proceedings of the 20th Annual IEEE Symposium on Logic in Computer Science
A categorical model for the geometry of interaction
Theoretical Computer Science - Automata, languages and programming: Logic and semantics (ICALP-B 2004)
Context Semantics, Linear Logic and Computational Complexity
LICS '06 Proceedings of the 21st Annual IEEE Symposium on Logic in Computer Science
A token machine for full geometry of interaction
TLCA'01 Proceedings of the 5th international conference on Typed lambda calculi and applications
Towards a typed geometry of interaction
CSL'05 Proceedings of the 19th international conference on Computer Science Logic
| Hi-index | 5.23 |
Any mathematical theory of algorithms striving to offer a foundation for programming needs to provide a rigorous definition for an abstract algorithm. The works reported by Girard (1988) in [10] and by Moschovakis (1989, 1995) in [29-31] are among the best examples of such attempts. They both try to offer a mathematically precise and rigorous formulation of an abstract algorithm, intend to keep the algorithmic flavour present and take the notion of recursion as primary and central. The present work is motivated by Girard's GoI 2 paper (Girard (1988) [10], which offers a treatment of recursion in terms of fixed points of linear functions. It is situated in the context of the geometry of interaction (GoI) program and is carried out in the concrete setting of the space of bounded linear maps on a Hilbert space. In this paper, we extend the work in Girard (1988) [10] to the context of traced unique decomposition categories, once again emphasizing the role of abstract trace in the theory of computing. We show that traced unique decomposition categories enriched over partially additive monoids or their variations suffice to axiomatize and hence extend the work in Girard's GoI 2 paper. The theory developed here allows us to formulate an abstract notion of an algorithm as a pair of morphisms in a traced unique decomposition category, an abstract notion of computation as the execution formula (defined using the trace operator) applied to an algorithm, and finally a notion of deadlock-freeness for algorithms. In addition, we can treat recursive definitions, fixed points and fixed point operators in a uniform way in terms of traced unique decomposition categories.