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The usual measure based on the length of regular paths cannot be used, since there are proof-nets which can be normalized in polynomial time but whose regular paths have exponential length

Context semantics, linear logic, and computational complexity

logic in computer science, no. 4 (2009): ArticleNo.25-ArticleNo.25

Cited by: 63|Views132
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Abstract

We show that context semantics can be fruitfully applied to the quantitative analysis of proof normalization in linear logic. In particular, context semantics lets us define the weight of a proof-net as a measure of its inherent complexity: it is both an upper bound to normalization time (modulo a polynomial overhead, independently on the...More

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Introduction
  • Linear logic has always been claimed to be resource-conscious: structural rules are applicable only when the involved formulas are modal, i.e. in the form !A.
  • The weight WG of a proof-net G will be defined from the context semantics of G following two ideas: The cost of a given box inside G is the number of times it can possibly be copied during normalization; The weight of G is the sum of costs of boxes inside G, where boxes that are inside other boxes are possibly counted more than once.
  • In Section 3, context semantics is defined and some examples of proof-nets are presented, together with their interpretation.
Highlights
  • Linear logic has always been claimed to be resource-conscious: structural rules are applicable only when the involved formulas are modal, i.e. in the form !A
  • Completely forbidding duplication highlights strong relations between proofs and boolean circuits [20]. These results demonstrate the relevance of linear logic in implicit computational complexity, where the aim is obtaining machineindependent, logic-based characterization of complexity classes
  • Context semantics and the geometry of interaction have been used to prove the correctness of optimal reduction algorithms [14] and in the design of sequential and parallel interpreters for the lambda calculus [17, 18]
  • A class of proof-nets which are not just strongly normalizing but normalizable in elementary time can still be captured in the geometry of interaction framework, as suggested by Baillot and Pedicini [3]
  • The usual measure based on the length of regular paths cannot be used, since there are proof-nets which can be normalized in polynomial time but whose regular paths have exponential length
  • The weight WG of a proof-net G will be defined from the context semantics of G following two ideas: The cost of a given box inside G is the number of times it can possibly be copied during normalization; The weight of G is the sum of costs of boxes inside G, where boxes that are inside other boxes are possibly counted more than once
Results
  • The authors are ready to define the context semantics for a proof-net G.
  • Context semantics can be defined on sharing graphs as well, proof-nets have been considered here.
  • The authors define some proof-nets together with observations about how context-semantics reflects the complexity of normalization.
  • Consider a proof-net G, an edge e 2 BG such that @(e) = 0 and let H be the box whose conclusion is e.
  • Any proof-net G satisfying the conditions of Lemma 4 has strictly positive weights, WG = 0.
  • Lemma 6 gives them enough information to establish strong correspondences between TG, WG and the number of steps necessary to rewrite G to normal form: Proposition 1 (Positive Weights, Absense of Cycles and Monotonicity) Let G be a proof-net.
  • The following theorem holds: Theorem 2 Let G be a proof-net.
  • This has very interesting consequences: for example, a family G of proof-nets can be normalized in polynomial time iff there is a polynomial p such that WG p(jGj) for every G 2 G .
  • This observation can be slightly generalized into the following result: Proposition 2 (Subtree Property) Suppose t is a standard exponential signature.
  • The intuitive idea behind the subtree property is the following: whenever t is a copy of e under U and U is canonical for e, the exponenial signature t must be completely “consumed” along the canonical path leading from (e; U; t; +) to a final context C.
Conclusion
  • This restriction enforces the following property at the semantic level: Lemma 8 (Stratification) Let G be a ELL proof-net.
  • The second inequality is satisfied, because there are at most 2jGj+1 exponential signatures with length at most jGj. If @(e) > 0, the authors can observe that canonical sequences for @(e) are in the form V t, where V is canonical for G(e) and t is a copy for G(e) under V .
  • The authors define a context semantics for linear logic proof-nets, showing it gives precise quantitative information on the dynamics of normalization.
Tables
  • Table1: Rewrite Rules for Vertices R(, L(,R , L , R8, L8 e
  • Table2: Rewrite Rules for Vertices X, D, N , L! and R!
Download tables as Excel
Funding
  • The author is partially supported by PRIN project FOLLIA (2004) and ANR project NOCOST (2005)
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