Fixed Points In Quantitative Semantics

We describe an interpretation of recursive computation in a symmetric monoidal category with infinite biproducts and cofree commutative comonoids (for instance, the category of free modules over a complete semiring). Such categories play a significant role in "quantitative" models of computation: they bear a canonical complete monoid enrichment, but may not be cpo-enriched, making standard techniques for reasoning about fixed points unavailable. By constructing a bifree algebra for the cofree exponential, we obtain fixed points for morphisms in its co-Kleisli category without requiring any order-theoretic structure. These fixed points corresponding to infinite sums of finitary approximants indexed over the nested finite multisets, each representing a unique call-pattern for computation of the fixed point. We illustrate this construction by using it to give a denotational semantics for PCF with non-deterministic choice and scalar weights from a complete semiring, proving that this is computationally adequate with respect to an operational semantics which evaluates a term by taking a weighted sum of the residues of its terminating reduction paths.

[1]  Nicolas Tabareau,et al.  An explicit formula for the free exponential modality of linear logic , 2009, Mathematical Structures in Computer Science.

[2]  Paul-André Melliès.  CATEGORICAL SEMANTICS OF LINEAR LOGIC , 2009 .

[3]  Thomas Ehrhard.  A Finiteness Structure on Resource Terms , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[4]  Laurent Regnier,et al.  The differential lambda-calculus , 2003, Theor. Comput. Sci..

[5]  François Lamarche,et al.  Quantitative Domains and Infinitary Algebras , 1992, Theor. Comput. Sci..

[6]  Antonio Bahamonde.  Tensor product of partially-additive monoids , 1985 .

[7]  Paolo Tranquilli,et al.  Nets between determinism and nondeterminism , 2009 .

[8]  Jean-Yves Girard,et al.  Normal functors, power series and λ-calculus , 1988, Ann. Pure Appl. Log..

[9]  Giulio Manzonetto,et al.  Constructing differential categories and deconstructing categories of games , 2013, Inf. Comput..

[10]  P. Freyd.  Algebraically complete categories , 1991 .

[11]  Michael Barr,et al.  Algebraically compact functors , 1992 .

[12]  Gérard Boudol,et al.  The Lambda-Calculus with Multiplicities (Abstract) , 1993, CONCUR.

[13]  Dominic R. Verity,et al.  ∞-Categories for the Working Mathematician , 2018 .

[14]  Thomas Ehrhard,et al.  Differential Interaction Nets , 2005, WoLLIC.

[15]  Dominic R. Verity,et al.  Traced monoidal categories , 1996, Mathematical Proceedings of the Cambridge Philosophical Society.

[16]  J. Robin B. Cockett,et al.  Differential categories , 2006, Mathematical Structures in Computer Science.

[17]  Alex K. Simpson,et al.  Computational adequacy for recursive types in models of intuitionistic set theory , 2002, Proceedings 17th Annual IEEE Symposium on Logic in Computer Science.

[18]  Gordon D. Plotkin,et al.  Complete axioms for categorical fixed-point operators , 2000, Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332).

[19]  Masahito Hasegawa.  Linearly Used Effects: Monadic and CPS Transformations into the Linear Lambda Calculus , 2001, APLAS.

[20]  Giulio Manzonetto,et al.  Weighted Relational Models of Typed Lambda-Calculi , 2013, 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science.

[21]  Zohar Manna,et al.  Proving termination with multiset orderings , 1979, CACM.