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### From Proof-Nets to Linear Logic Type Systems for Polynomial Time Computing [chapter]

Patrick Baillot
2007 Lecture Notes in Computer Science
We stress the importance of proof-nets from Light linear logic for the design of this type system and for a result establishing that typable lambda-terms can be evaluated efficiently with optimal reduction  ...  We also discuss the issue of DLAL type inference, which can be performed in polynomial time for system F terms.  ...  This resolution procedure is correct and complete and it can be performed in polynomial time w.r.t. the size of the original system F term.  ...

### Soft linear logic and polynomial time

Y LAFONT
2004 Theoretical Computer Science
We present a subsystem of second-order linear logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms, and vice versa.  ...  It is quite natural to look for an intermediate system corresponding to polynomial time computation.  ...  Soft linear logic can be seen as a subsystem of bounded linear logic which is powerful enough to encode polynomial time. It is simpler than bounded linear logic and light linear logic.  ...

### Soft linear logic and polynomial time

Yves Lafont
2004 Theoretical Computer Science
We present a subsystem of second-order linear logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms, and vice versa.  ...  It is quite natural to look for an intermediate system corresponding to polynomial time computation.  ...  Soft linear logic can be seen as a subsystem of bounded linear logic which is powerful enough to encode polynomial time. It is simpler than bounded linear logic and light linear logic.  ...

### Elementary Linear Logic Revisited for Polynomial Time and an Exponential Time Hierarchy [chapter]

Patrick Baillot
2011 Lecture Notes in Computer Science
Elementary linear logic is a simple variant of linear logic, introduced by Girard and which characterizes in the proofs-as-programs approach the class of elementary functions (computable in time bounded  ...  Other systems, like light linear logic have then been defined to capture in a similar way polynomial time functions, but at the price of either a more complicated syntax or of more involved encodings.  ...  Thanks also to Christophe Raffalli, for useful discussions about the typing of Scott integers in system F. This work was partially supported by project ANR-08-BLANC-0211-01 "COM-PLICE".  ...

### Bounded linear logic: a modular approach to polynomial-time computability

Jean-Yves Girard, Andre Scedrov, Philip J. Scott
1992 Theoretical Computer Science
Theorem 5. 3 . 3 Any function from dyadic lists to dyadic lists represented by a proof of t-Nf;-+N&, in bounded linear logic is computable in polynomial time.  ...  Our aim in this paper is not just to express polynomial time computability as provability of formulas in a logical system, but to provide a notion of polynomial time computation intrinsically within a  ...  In the iterative step for S,, first consider the BLL proof of N~@N~@N~EN~+, defined "by Cases" from Axiom N&N;+, , from the BLL proof s, of N~EN~+, g iven in Fig. 17 (with z for x), and from the BLL  ...

### Page 911 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews
He then proves that, for proof-nets of a fixed depth, the time required for cut-elimination is bounded by a polynomial in the size.  ...  Second, every polynomial time computable function can be encoded as a proof in the system in such a way that applica- tion of the function to an argument can be regarded as combining the proof with (a  ...

### Paths-based criteria and application to linear logic subsystems characterizing polynomial time [article]

Matthieu Perrinel
2017 arXiv   pre-print
Several previous subsystems of linear logic characterizing polynomial time (LLL, mL^4, maximal system of MS) are embedded in SDNLL, proving its Ptime completeness.  ...  Light linear logic (LLL) ensures a polynomial bound on reduction time, and characterizes in this way polynomial time (Ptime).  ...  Bounded linear logic: a modular approach to polynomial-time computability. Theoretical computer science, 97(1), 1992.  G. Gonthier, M. Abadi, and J.J. Lévy.  ...

### Some Complexity and Expressiveness Results on Multimodal and Stratified Proof Nets [chapter]

Luca Roversi, Luca Vercelli
2009 Lecture Notes in Computer Science
Every subsystem generates stratified proof nets whose normalization preserves stratification, a structural property of nodes and edges, like in Light Linear/Affine logical systems.  ...  P q (m 1 , . . . , m k ) can be applied to every proof net Π : A 1 , . . . , A k ⊢ B of MS, with assumptions of type A 1 , . . . , A k and conclusion of type B.  ...  We thank Ugo Dal Lago and Marco Gaboardi for the useful discussions leading to this work and the three referees for their high quality comments and suggestions.  ...

### Page 6100 of Mathematical Reviews Vol. , Issue 98J [page]

1998 Mathematical Reviews
Appro- priate proof nets and sequential systems are introduced and co- herence results for bilinear categories with respect to these nets are proven.  ...  of the progressions be polynomial-time computable.  ...

### Page 5967 of Mathematical Reviews Vol. , Issue 2004h [page]

2004 Mathematical Reviews
Girard [Inform. and Comput. 143 (1998), no. 2, 175-204; MR 2000b:03214], is a weakening of linear logic designed to satisfy polynomial time computability in various ways.  ...  Logic 120 (2003), no. 1-3, 65-102. Summary: “We introduce the new notion of additive ‘multibox’ for linear logic proof-nets.  ...

### Intuitionistic Light Affine Logic

Andrea Asperti, Luca Roversi
2002 ACM Transactions on Computational Logic
The bound on the normalization cost is proved by introducing the proof-nets for ILAL. The bound follows from a suitable normalization strategy that exploits structural properties of the proof-nets.  ...  This article is a structured introduction to Intuitionistic Light Affine Logic (ILAL).  ...  ACKNOWLEDGMENT We warmly thanks the anonymous referee and Simone Martini who greatly helped us by suggesting how to correct and clarify this paper.  ...

### Page 3554 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews
Summary: “It is a known approach to translate propositional for- mulas into systems of polynomial inequalities and consider proof systems for the latter.  ...  The well-studied proof systems of this type are the cutting plane proof system (CP), utilizing linear inequal- ities, and the Lovasz-Schrijver calculi (LS), utilizing quadratic inequalities.  ...

### Page 5861 of Mathematical Reviews Vol. , Issue 98I [page]

1998 Mathematical Reviews
Jager, Some proof theory of first order logic programming (201-228); Piotr Kosiuczenko and Martin Wirsing, Timed rewriting logic for the specification of time-sensitive systems (229-264); Dale Miller  ...  Trakhtenbrot, From finite automata toward hybrid systems (extended abstract) (411-422); Zenon Sadowski, On an optimal quantified propositional proof system and a complete language for NPM co-NP (423-428  ...

### Soft Linear Logic and Polynomial Complexity Classes

Marco Gaboardi, Jean-Yves Marion, Simona Ronchi Della Rocca
2008 Electronical Notes in Theoretical Computer Science
Useful tools for this aim are the so called light logics, derived from Girard's Linear Logic [Gir87] , where cut elimination is bounded in time by the size of the proof.  ...  So, assuming the proof-nets as computational model, and the cut-elimination as computational rule, SLL is correct for polynomial time computations, once the degree is fixed.  ...  . • The size |Π| of a proof Π is the number of rules in Π. • The rank of a rule (m), as defined in Table 2 , Let r be the the maximum rank of a rule (m) in Π.  ...

### Page 6001 of Mathematical Reviews Vol. , Issue 94j [page]

1994 Mathematical Reviews
Our technique is used to prove that every language in the polynomial-time hierarchy has an interactive proof system. This technique played'a pivotal role in the recent proofs that IP = PSPACE [J.  ...  Summary: “In this paper, we consider the multiplicative fragment of linear logic (MLL) from an automated deduction point of view.  ...
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