arXiv subjects

Luca Roversi

Publications and source records attributed to Luca Roversi.

24 records · Page 2Linked to original sources

arXiv2011

Linear lambda Calculus with Explicit Substitutions as Proof-Search in Deep Inference

SBV is a deep inference system that extends the set of logical operators of multiplicative linear logic with the non commutative operator Seq. We introduce the logical system SBVr which extends SBV by adding a self-dual atom-renaming operator to it. We prove that the cut elimination holds on SBVr. SBVr and its cut free subsystem BVr are complete and sound with respect to linear lambda calculus with explicit substitutions. Under any strategy, a sequence of evaluation steps of any linear lambda-term M becomes a process of proof-search in SBVr (BVr) once M is mapped into a formula of SBVr. Completeness and soundness follow from simulating linear beta-reduction with explicit substitutions as processes. The role of the new renaming operator of SBVr is to rename channel-names on-demand. This simulates the substitution that occurs in a beta-reduction. Despite SBVr is a minimal extension of SBV its proof-search can compute all boolean functions, as linear lambda-calculus with explicit substitutions can compute all boolean functions as well. So, proof search of SBVr and BVr is at least ptime-complete.

cs.LO↗

arXiv2010

Safe Recursion on Notation into a Light Logic by Levels

We embed Safe Recursion on Notation (SRN) into Light Affine Logic by Levels (LALL), derived from the logic L4. LALL is an intuitionistic deductive system, with a polynomial time cut elimination strategy. The embedding allows to represent every term t of SRN as a family of proof nets |t|^l in LALL. Every proof net |t|^l in the family simulates t on arguments whose bit length is bounded by the integer l. The embedding is based on two crucial features. One is the recursive type in LALL that encodes Scott binary numerals, i.e. Scott words, as proof nets. Scott words represent the arguments of t in place of the more standard Church binary numerals. Also, the embedding exploits the "fuzzy" borders of paragraph boxes that LALL inherits from L4 to "freely" duplicate the arguments, especially the safe ones, of t. Finally, the type of |t|^l depends on the number of composition and recursion schemes used to define t, namely the structural complexity of t. Moreover, the size of |t|^l is a polynomial in l, whose degree depends on the structural complexity of t. So, this work makes closer both the predicative recursive theoretic principles SRN relies on, and the proof theoretic one, called /stratification/, at the base of Light Linear Logic.

cs.LO↗

arXiv2008

Taming Modal Impredicativity: Superlazy Reduction

Pure, or type-free, Linear Logic proof nets are Turing complete once cut-elimination is considered as computation. We introduce modal impredicativity as a new form of impredicativity causing the complexity of cut-elimination to be problematic from a complexity point of view. Modal impredicativity occurs when, during reduction, the conclusion of a residual of a box b interacts with a node that belongs to the proof net inside another residual of b. Technically speaking, superlazy reduction is a new notion of reduction that allows to control modal impredicativity. More specifically, superlazy reduction replicates a box only when all its copies are opened. This makes the overall cost of reducing a proof net finite and predictable. Specifically, superlazy reduction applied to any pure proof nets takes primitive recursive time. Moreover, any primitive recursive function can be computed by a pure proof net via superlazy reduction.

cs.LO↗

arXiv2008

Weak Affine Light Typing is complete with respect to Safe Recursion on Notation

Weak affine light typing (WALT) assigns light affine linear formulae as types to a subset of lambda-terms of System F. WALT is poly-time sound: if a lambda-term M has type in WALT, M can be evaluated with a polynomial cost in the dimension of the derivation that gives it a type. The evaluation proceeds under any strategy of a rewriting relation which is a mix of both call-by-name and call-by-value beta-reductions. WALT weakens, namely generalizes, the notion of "stratification of deductions", common to some Light Systems -- those logical systems, derived from Linear logic, to characterize the set of Polynomial functions -- . A weaker stratification allows to define a compositional embedding of Safe recursion on notation (SRN) into WALT. It turns out that the expressivity of WALT is strictly stronger than the one of the known Light Systems. The embedding passes through the representation of a subsystem of SRN. It is obtained by restricting the composition scheme of SRN to one that can only use its safe variables linearly. On one side, this suggests that SRN, in fact, can be redefined in terms of more primitive constructs. On the other, the embedding of SRN into WALT enjoys the two following remarkable aspects. Every datatype, required by the embedding, is represented from scratch, showing the strong structural proof-theoretical roots of WALT. Moreover, the embedding highlights a stratification structure of the normal and safe arguments, normally hidden inside the world of SRN-normal/safe variables: the less an argument is "polyomially impredicative", the deeper, in a formal, proof-theoretical sense, it is represented inside WALT. Finally, since WALT is SRN-complete it is also polynomial-time complete since SRN is.

cs.LO↗

arXiv2008

Weak Affine Light Typing: Polytime intensional expressivity, soundness and completeness

Weak affine light typing (WALT) assigns light affine linear formulae as types to a subset of lambda-terms in System F. WALT is poly-time sound: if a lambda-term M has type in WALT, M can be evaluated with a polynomial cost in the dimension of the derivation that gives it a type. In particular, the evaluation can proceed under any strategy of a rewriting relation, obtained as a mix of both call-by-name/call-by-value beta-reductions. WALT is poly-time complete since it can represent any poly-time Turing machine. WALT weakens, namely generalizes, the notion of stratification of deductions common to some Light Systems -- we call as such those logical systems, derived from Linear logic, to characterize FP, the set of Polynomial functions -- . A weaker stratification allows to define a compositional embedding of the Quasi-linear fragment QlSRN of Safe recursion on notation (SRN) into WALT. QlSRN is SRN, which is a recursive-theoretical system characterizing FP, where only the composition scheme is restricted to linear safe variables. So, the expressivity of WALT is stronger, as compared to the known Light Systems. In particular, using the types, the embedding puts in evidence the stratification of normal and safe arguments hidden in QlSRN: the less an argument is impredicative, the deeper, in a formal, proof-theoretical sense, gets its representation in WALT.

cs.LO↗

arXiv2000

Light Affine Logic (Proof Nets, Programming Notation, P-Time Correctness and Completeness)

This paper is a structured introduction to Light Affine Logic, and to its intuitionistic fragment. Light Affine Logic has a polynomially costing cut elimination (P-Time correctness), and encodes all P-Time Turing machines (P-Time completeness). P-Time correctness is proved by introducing the Proof nets for Intuitionistic Light Affine Logic. P-Time completeness is demonstrated in full details thanks to a very compact program notation. On one side, the proof of P-Time correctness describes how the complexity of cut elimination is controlled, thanks to a suitable cut elimination strategy that exploits structural properties of the Proof nets. This allows to have a good catch on the meaning of the ``paragraph'' modality, which is a peculiarity of light logics. On the other side, the proof of P-Time completeness, together with a lot of programming examples, gives a flavor of the non trivial task of programming with resource limitations, using Intuitionistic Light Affine Logic derivations as programs.

cs.LO↗