Results 1  10
of
13
Soft lambdacalculus: a language for polynomial time computation
 In Proc. FoSSaCS, Springer LNCS 2987
, 2004
"... Abstract. Soft linear logic ([Lafont02]) is a subsystem of linear logic characterizing the class PTIME. We introduce Soft lambdacalculus as a calculus typable in the intuitionistic and affine variant of this logic. We prove that the (untyped) terms of this calculus are reducible in polynomial time. ..."
Abstract

Cited by 31 (4 self)
 Add to MetaCart
(Show Context)
Abstract. Soft linear logic ([Lafont02]) is a subsystem of linear logic characterizing the class PTIME. We introduce Soft lambdacalculus as a calculus typable in the intuitionistic and affine variant of this logic. We prove that the (untyped) terms of this calculus are reducible in polynomial time. We then extend the type system of Soft logic with recursive types. This allows us to consider nonstandard types for representing lists. Using these datatypes we examine the concrete expressiveness of Soft lambdacalculus with the example of the insertion sort algorithm. 1
A soft type assignment system for λcalculus
, 2007
"... Abstract. Soft Linear Logic (SLL) is a subsystem of secondorder linear logic with restricted rules for exponentials, which is correct and complete for PTIME. We design a type assignment system for the λcalculus (STA), which assigns to λterms as types (a proper subset of) SLL formulas, in such a ..."
Abstract

Cited by 11 (8 self)
 Add to MetaCart
(Show Context)
Abstract. Soft Linear Logic (SLL) is a subsystem of secondorder linear logic with restricted rules for exponentials, which is correct and complete for PTIME. We design a type assignment system for the λcalculus (STA), which assigns to λterms as types (a proper subset of) SLL formulas, in such a way that typable terms inherit the good complexity properties of the logical system. Namely STA enjoys subject reduction and normalization, and it is correct and complete for PTIME and FPTIME.
Linear Logic by Levels and Bounded Time Complexity
, 2009
"... This work deals with the characterization of elementary and deterministic polynomial time computation in linear logic through the proofsasprograms correspondence. Girard’s seminal results, concerning elementary and light linear logic, use a principle called stratification to ensure the complexity b ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
This work deals with the characterization of elementary and deterministic polynomial time computation in linear logic through the proofsasprograms correspondence. Girard’s seminal results, concerning elementary and light linear logic, use a principle called stratification to ensure the complexity bound on the cutelimination procedure. Here, we propose a more flexible control principle, that of indexing, which allows us to extend Girard’s systems while keeping the same complexity properties. A consequence of the higher flexibility of indexing with respect to stratification is the absence of boxes for handling the § modality. We finally propose a variant of our polytime system in which the § modality is only allowed on atoms, and which may thus serve as a basis for developing λcalculus type assignment systems with more efficient typing algorithms than existing ones.
Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity
 LOGIC JOURNAL OF THE IGPL
, 2003
"... Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessi ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural twosorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result
Proofs as executions
, 2012
"... Abstract. This paper proposes a new interpretation of the logical contents of programs in the context of concurrent interaction, wherein proofs correspond to valid executions of a processes. A type system based on linear logic is used, in which a given process has many different types, each typing c ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
Abstract. This paper proposes a new interpretation of the logical contents of programs in the context of concurrent interaction, wherein proofs correspond to valid executions of a processes. A type system based on linear logic is used, in which a given process has many different types, each typing corresponding to a particular way of interacting with its environment and cut elimination corresponds to executing the process in a given interaction scenario. A completeness result is established, stating that every lockavoiding execution of a process in some environment corresponds to a particular typing. Besides traces, types contain precise information about the flow of control between a process and its environment, and proofs are interpreted as composable schedulings of processes. In this interpretation, logic appears as a way of making explicit the flow of causality between interacting processes. 1
Uniform circuits, & Boolean proof nets
"... The relationship between Boolean proof nets of multiplicative linear logic (APN) and Boolean circuits has been studied [Ter04] in a nonuniform setting. We refine the results taking care of uniformity: the relationship can be expressed in term of the (Turing) polynomial hierarchy. We give a proofsa ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
The relationship between Boolean proof nets of multiplicative linear logic (APN) and Boolean circuits has been studied [Ter04] in a nonuniform setting. We refine the results taking care of uniformity: the relationship can be expressed in term of the (Turing) polynomial hierarchy. We give a proofsasprograms correspondence between proof nets and deterministic as well as nondeterministic Boolean circuits with a uniform depthpreserving simulation of each other. The Boolean proof nets class m&BN(poly) is built on multiplicative and additive linear logic with a polynomial amount of additive connectives as the nondeterministic circuit class NNC(poly) is with nondeterministic variables. We obtain uniformAPN = NC and m&BN(poly) = NNC(poly) = NP.
Measurements in Proof Nets as HigherOrder Quantum Circuits
"... Abstract. We build on the series of work by Dal Lago and coauthors and identify proof nets (of linear logic) as higherorder quantum circuits. By accommodating quantum measurement using additive slices, we obtain a comprehensive framework for programming and interpreting quantum computation. Specif ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We build on the series of work by Dal Lago and coauthors and identify proof nets (of linear logic) as higherorder quantum circuits. By accommodating quantum measurement using additive slices, we obtain a comprehensive framework for programming and interpreting quantum computation. Specifically, we introduce a quantum lambda calculus MLLqm and define its geometry of interaction (GoI) semantics—in the style of token machines—via the translation of terms into proof nets. Its soundness, i.e. invariance under reduction of proof nets, is established. The calculus MLLqm attains a pleasant balance between expressivity (it is higherorder and accommodates all quantum operations) and concreteness of models (given as token machines, i.e. in the form of automata). 1
Obsessional cliques: a semantic characterization of bounded time complexity
 In Proceedings of LICS’06
, 2006
"... We give a semantic characterization of bounded complexity proofs. We introduce the notion of obsessional clique in the relational model of linear logic and show that restricting the morphisms of the category REL to obsessional cliques yields models of ELL and SLL. Conversely, we prove that these m ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
We give a semantic characterization of bounded complexity proofs. We introduce the notion of obsessional clique in the relational model of linear logic and show that restricting the morphisms of the category REL to obsessional cliques yields models of ELL and SLL. Conversely, we prove that these models are relatively complete: an LL proof whose interpretation is an obsessional clique is always an ELL/SLL proof. These results are achieved by introducing a system of ELL/SLL untyped proofnets, which is both correct and complete with respect to elementary/polynomial time complexity. 1.
unknown title
"... We study the relationship between proof nets for mutiplicative linear logic (with unbounded fanin logical connectives) and Boolean circuits. We give simulations of each other in the style of the proofsasprograms correspondence; proof nets correspond to Boolean circuits and cutelimination correspo ..."
Abstract
 Add to MetaCart
(Show Context)
We study the relationship between proof nets for mutiplicative linear logic (with unbounded fanin logical connectives) and Boolean circuits. We give simulations of each other in the style of the proofsasprograms correspondence; proof nets correspond to Boolean circuits and cutelimination corresponds to evaluation. The depth of a proof net is defined to be the maximum logical depth of cut formulas in it, and it is shown that every unbounded fanin Boolean circuit of depth n, possibly with stCONN2 gates, is polynomially simulated by a proof net of depth O(n) and vice versa. Here, stCONN2 stands for stconnectivity gates for undirected graphs of degree 2. Let APN i be the class of languages for which there is a polynomial size, log idepth family of proof nets. We then have APN i = AC i (st CONN2). 1.
Under consideration for publication in Math. Struct. in Comp. Science On Light Logics, Uniform Encodings and Polynomial Time
, 2006
"... Light Affine Logic is a variant of Linear Logic with a polynomial cutelimination procedure. We study the extensional expressive power of Light Affine Logic with respect to a general notion of encoding of functions, in the setting of the CurryHoward correspondence. We consider Light Affine Logic wi ..."
Abstract
 Add to MetaCart
(Show Context)
Light Affine Logic is a variant of Linear Logic with a polynomial cutelimination procedure. We study the extensional expressive power of Light Affine Logic with respect to a general notion of encoding of functions, in the setting of the CurryHoward correspondence. We consider Light Affine Logic with both fixpoints of formulae and secondorder quantifiers and analyze the properties of polytime soundness and polytime completeness for various fragments of this system. We show in particular that the implicative propositional fragment is not polytime complete, if we add some reasonable conditions on the encodings. Following previous work, we show that second order leads to polytime unsoundness. We then introduce simple constraints on second order quantification and fixpoints, proving the obtained fragments to be polytime sound and complete. 1.