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Stratified Bounded Affine Logic for Logarithmic Space
"... A number of complexity classes, most notably PTIME, have been characterised by subsystems of linear logic. In this paper we show that the functions computable in logarithmic space can also be characterised by a restricted version of linear logic. We introduce Stratified Bounded Affine Logic (SBAL), ..."
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A number of complexity classes, most notably PTIME, have been characterised by subsystems of linear logic. In this paper we show that the functions computable in logarithmic space can also be characterised by a restricted version of linear logic. We introduce Stratified Bounded Affine Logic (SBAL), a restricted version of Bounded Linear Logic, in which not only the modality! but also the universal quantifier is bounded by a resource polynomial. We show that the proofs of certain sequents in SBAL represent exactly the functions computable logarithmic space. The proof that SBALproofs can be compiled to LOGSPACE functions rests on modelling computation by interaction dialogues in the style of game semantics. We formulate the compilation of SBALproofs to spaceefficient programs as an interpretation in a realisability model, in which realisers are taken from a Geometry of Interaction situation.
Bounded Linear Logic, Revisited
, 904
"... We present QBAL, an extension of Girard, Scedrov and Scott’s bounded linear logic. The main novelty of the system is the possibility of quantifying over resource variables. This generalization makes bounded linear logic considerably more flexible, while preserving soundness and completeness for poly ..."
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We present QBAL, an extension of Girard, Scedrov and Scott’s bounded linear logic. The main novelty of the system is the possibility of quantifying over resource variables. This generalization makes bounded linear logic considerably more flexible, while preserving soundness and completeness for polynomial time. In particular, we provide compositional embeddings of Leivant’s RRW and Hofmann’s LFPL into QBAL. 1
Quantitative Models and Implicit Complexity
, 2005
"... We give new proofs of soundness (all representable functions on base types lies in certain complexity classes) for Elementary Affine Logic, LFPL (a language for polytime computation close to realistic functional programming introduced by one of us), Light Affine Logic and Soft Affine Logic. The proo ..."
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We give new proofs of soundness (all representable functions on base types lies in certain complexity classes) for Elementary Affine Logic, LFPL (a language for polytime computation close to realistic functional programming introduced by one of us), Light Affine Logic and Soft Affine Logic. The proofs are based on a common semantical framework which is merely instantiated in four different ways. The framework consists of an innovative modification of realizability which allows us to use resourcebounded computations as realisers as opposed to including all Turing computable functions as is usually the case in realizability constructions. For example, all realisers in the model for LFPL are polynomially bounded computations whence soundness holds by construction of the model. The work then lies in being able to interpret all the required constructs in the model. While being the first entirely semantical proof of polytime soundness for light logics, our proof also provides a notable simplification of the original already semantical proof of polytime soundness for LFPL. A new result made possible by the semantic framework is the addition of polymorphism and a modality to LFPL thus allowing for an internal definition of inductive datatypes. 1