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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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Cited by 8 (0 self)
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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.
HigherOrder Functional Reactive Programming in Bounded Space
"... Functional reactive programming (FRP) is an elegant and successful approach to programming reactive systems declaratively. The high levels of abstraction and expressivity that make FRP attractive as a programming model do, however, often lead to programs whose resource usage is excessive and hard to ..."
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Cited by 4 (1 self)
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Functional reactive programming (FRP) is an elegant and successful approach to programming reactive systems declaratively. The high levels of abstraction and expressivity that make FRP attractive as a programming model do, however, often lead to programs whose resource usage is excessive and hard to predict. In this paper, we address the problem of space leaks in discretetime functional reactive programs. We present a functional reactive programming language that statically bounds the size of the dataflow graph a reactive program creates, while still permitting use of higherorder functions and highertype streams such as streams of streams. We achieve this with a novel linear type theory that both controls allocation and ensures that all recursive definitions are wellfounded. We also give a denotational semantics for our language by combining recent work on metric spaces for the interpretation of higherorder causal functions with lengthspace models of spacebounded computation. The resulting category is doubly closed and hence forms a model of the logic of bunched implications.
The Weak Lambda Calculus as a Reasonable Machine
, 2006
"... We define a new cost model for the callbyvalue lambdacalculus satisfying the invariance thesis. That is, under the proposed cost model, Turing machines and the callbyvalue lambdacalculus can simulate each other within a polynomial time overhead. The model only relies on combinatorial propertie ..."
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Cited by 4 (2 self)
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We define a new cost model for the callbyvalue lambdacalculus satisfying the invariance thesis. That is, under the proposed cost model, Turing machines and the callbyvalue lambdacalculus can simulate each other within a polynomial time overhead. The model only relies on combinatorial properties of usual betareduction, without any reference to a specific machine or evaluator. In particular, the cost of a single beta reduction is proportional to the difference between the size of the redex and the size of the reduct. In this way, the total cost of normalizing a lambda term will take into account the size of all intermediate results (as well as the number of steps to normal form).
A Semantic Proof of Polytime Soundness of Light Affine Logic
"... Abstract. We define a denotational semantics for Light Affine Logic (LAL) which has the property that denotations of functions are polynomial time computable by construction of the model. This gives a new proof of polytimesoundness of LAL which is considerably simpler than the standard proof based ..."
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Abstract. We define a denotational semantics for Light Affine Logic (LAL) which has the property that denotations of functions are polynomial time computable by construction of the model. This gives a new proof of polytimesoundness of LAL which is considerably simpler than the standard proof based on proof nets and also is entirely semantical in nature. The model construction uses a new instance of a resource monoid; a general method for interpreting variations of linear logic with complexity restrictions introduced earlier by the authors. 1
An Invariant Cost Model for the Lambda Calculus
, 2005
"... We define a new cost model for the callbyvalue lambdacalculus satisfying the invariance thesis. That is, under the proposed cost model, Turing machines and the callbyvalue lambdacalculus can simulate each other within a polynomial time overhead. The model only relies on combinatorial properties ..."
Abstract
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We define a new cost model for the callbyvalue lambdacalculus satisfying the invariance thesis. That is, under the proposed cost model, Turing machines and the callbyvalue lambdacalculus can simulate each other within a polynomial time overhead. The model only relies on combinatorial properties of usual betareduction, without any reference to a specific machine or evaluator. In particular, the cost of a single beta reduction is proportional to the difference between the size of the redex and the size of the reduct. In this way, the total cost of normalizing a lambda term will take into account the size of all intermediate results (as well as the number of steps to normal form). 1
Stratified Bounded Affine Logic for Logarithmic Space (Draft)
, 2007
"... 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), ..."
Abstract
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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. 1
Spaceefficient Computation by Interaction A Type System for Logarithmic Space
"... Abstract. We introduce a typed functional programming language for logarithmic space. Its type system is an annotated subsystem of Hofmann’s polytime LFPL. To guide the design of the programming language and to enable the proof of LOGSPACEsoundness, we introduce a realisability model over a variant ..."
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Abstract. We introduce a typed functional programming language for logarithmic space. Its type system is an annotated subsystem of Hofmann’s polytime LFPL. To guide the design of the programming language and to enable the proof of LOGSPACEsoundness, we introduce a realisability model over a variant of the Geometry of Interaction. This realisability model, which takes inspiration from MøllerNeergaard and Mairson’s work on BC − ε, provides a general framework for modelling spacerestricted computation. 1
Abstract
, 2004
"... We investigate on the extensional expressive power of Light Affine Logic, analysing the class of uniformly representable functions for various fragments of the logic. We give evidence on the incompleteness (for polynomial time) of the propositional fragment. Following previous work, we show that sec ..."
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We investigate on the extensional expressive power of Light Affine Logic, analysing the class of uniformly representable functions for various fragments of the logic. We give evidence on the incompleteness (for polynomial time) of the propositional fragment. Following previous work, we show that second order leads to polytime unsoundness. We then introduce simple constraints on second order quantification and least fixpoints, proving the obtained fragment to be polytime sound and complete. 1
Proofs as Efficient Programs
"... There may, indeed, be other uses of the system than its use as a logic. ..."