## Least and greatest fixed points in linear logic Extended Version (2007)

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Citations: | 42 - 12 self |

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@MISC{Baelde07leastand,

author = {David Baelde and Dale Miller},

title = {Least and greatest fixed points in linear logic Extended Version},

year = {2007}

}

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### Abstract

david.baelde at ens-lyon.org dale.miller at inria.fr Abstract. The first-order theory of MALL (multiplicative, additive linear logic) over only equalities is an interesting but weak logic since it cannot capture unbounded (infinite) behavior. Instead of accounting for unbounded behavior via the addition of the exponentials (! and?), we add least and greatest fixed point operators. The resulting logic, which we call µMALL = , satisfies two fundamental proof theoretic properties. In particular, µMALL = satisfies cut-elimination, which implies consistency, and has a complete focused proof system. This second result about focused proofs provides a strong normal form for cut-free proof structures that can be used, for example, to help automate proof search. We then consider applying these two results about µMALL = to derive a focused proof system for an intuitionistic logic extended with induction and co-induction. The traditional approach to encoding intuitionistic logic into linear logic relies heavily on using the exponentials, which unfortunately weaken the focusing discipline. We get a better focused proof system by observing that certain fixed points satisfy the structural rules of weakening and contraction (without using exponentials). The resulting focused proof system for intuitionistic logic is closely related to the one implemented in Bedwyr, a recent model checker based on logic programming. We discuss how our proof theory might be used to build a computational system that can partially automate induction and co-induction. 1

### Citations

662 | Light linear logic
- Girard
- 1998
(Show Context)
Citation Context ...al inference rules for exponentials, the resulting proof system would yield µBt � ! µBt (and equivalently ? νBt � νBt) provided that B is fully synchronous. In the language of the Logic of Unity (LU) =-=[Gir93]-=-, fully asynchronous (resp. fully synchronous) would be negative (resp. positive) or right-permeable (resp. left-permeable) formulas. Mixing synchronous and asynchronous connectives would yield a neut... |

383 | uniform proofs as a foundation for logic programming
- Miller, Nadathur, et al.
- 1991
(Show Context)
Citation Context ...plementations can be justified by the completeness of SLDresolution [AvE82]; uniform proofs (goal-directed proofs) in intuitionistic and intuitionistic linear logics have been used to justify λProlog =-=[MNPS91]-=- and Lolli [HM94]; the classical linear logic programming languages LO [AP91] and Forum [Mil96] have used directly Andreoli’s general focusing result [And92] for linear logic. In this paper, we establ... |

343 | Logic programming with focusing proofs in linear logic
- Andreoli
- 1992
(Show Context)
Citation Context ...on) variables are used, which is the case in this paper where the rules for induction and co-induction use such higher-order variables. A second normal form theorem, usually related to focused proofs =-=[And92]-=- is also important to establish. Such “focusing” theorems provide normal forms that organize invertible and non-invertible inference rules into collections: such striping of the inference rules in a c... |

306 | Logic programming in a fragment of intuitionistic linear logic
- Hodas, Miller
- 1994
(Show Context)
Citation Context ...e justified by the completeness of SLDresolution [AvE82]; uniform proofs (goal-directed proofs) in intuitionistic and intuitionistic linear logics have been used to justify λProlog [MNPS91] and Lolli =-=[HM94]-=-; the classical linear logic programming languages LO [AP91] and Forum [Mil96] have used directly Andreoli’s general focusing result [And92] for linear logic. In this paper, we establish these two fou... |

272 |
Investigations into logical deduction
- Gentzen
- 1969
(Show Context)
Citation Context ...tic logic in which there are no atomic formulas but were there are (positive) equalities and the two fixed point constructors µ and ν. Let µLJ = be the proof system that extends Gentzen’s cut-free LJ =-=[Gen69]-=- with the following 16srules for equality and (co)inductive expressions. {(Γ ⊢ G)θ : θ ∈ csu(s . = t)} Γ, s = t ⊢ G BS x ⊢ S x Γ, S t ⊢ G Γ, µBt ⊢ G Γ, B(νB)t ⊢ G Γ, νBt ⊢ G νL µL = L Γ, νBt ⊢ νBt ν0 ... |

210 | Linear objects: logical processes with built-in inheritance
- Andreoli, Pareschi
- 1991
(Show Context)
Citation Context ...iform proofs (goal-directed proofs) in intuitionistic and intuitionistic linear logics have been used to justify λProlog [MNPS91] and Lolli [HM94]; the classical linear logic programming languages LO =-=[AP91]-=- and Forum [Mil96] have used directly Andreoli’s general focusing result [And92] for linear logic. In this paper, we establish these two foundational proof-theoretic properties for the following logic... |

191 |
Contributions to the theory of logic programming
- Apt, Emden
- 1982
(Show Context)
Citation Context ...mula property does not hold. Various computational systems have employed different focusing theorems: much of Prolog’s design and implementations can be justified by the completeness of SLDresolution =-=[AvE82]-=-; uniform proofs (goal-directed proofs) in intuitionistic and intuitionistic linear logics have been used to justify λProlog [MNPS91] and Lolli [HM94]; the classical linear logic programming languages... |

86 | Forum: A multiple-conclusion specification logic
- Miller
- 1996
(Show Context)
Citation Context ...l-directed proofs) in intuitionistic and intuitionistic linear logics have been used to justify λProlog [MNPS91] and Lolli [HM94]; the classical linear logic programming languages LO [AP91] and Forum =-=[Mil96]-=- have used directly Andreoli’s general focusing result [And92] for linear logic. In this paper, we establish these two foundational proof-theoretic properties for the following logic. We first extend ... |

65 | Cut-elimination for a logic with definitions and induction
- McDowell, Miller
(Show Context)
Citation Context ...d inference rules that could only unfold fixed point descriptions: as a consequence, such logics could not discriminate between a least and greatest fixed point. Stronger systems that allow induction =-=[MM00]-=- as well as co-induction [Tiu04,MT03] include inference rules using a higher-order variable that ranges over prefixed or postfixed points (invariants). Of course, approaches that use (co)induction are... |

59 | Rules of definitional reflection - Schroeder-Heister - 1993 |

54 | Soft linear logic and polynomial time
- Lafont
(Show Context)
Citation Context ...es for the promotion and structural rules for exponentials and some of these choices lead to different versions of logic (for example, elementary and light linear logics [Gir98] and soft linear logic =-=[Laf04]-=-). Even if we fix the inference rules for the exponentials, as in standard linear logic, the rules do not describe unique exponentials. If one gives a red tensor and a blue tensor the same inference r... |

42 | A fixpoint theorem in linear logic, an email posting to the mailing list linear@cs.stanford.edu - Girard - 1992 |

36 | A Logical Framework for Reasoning about Logical Specifications
- Tiu
- 2004
(Show Context)
Citation Context ...h two of the derivations (for the totality of half and that the sum of natural numbers is a natural number) do not satisfy the restriction on co-invariants. The logic µLJ = is closely related to LINC =-=[Tiu04]-=-. The main difference is the absence of the ∇ quantifier in our system: we suspect that ∇ can be added to µMALL = in the same relatively orthogonal fashion that LINC added it to LJ. The resulting exte... |

33 | Focusing and polarization in intuitionistic logic
- Liang, Miller
- 2007
(Show Context)
Citation Context ..., database updates, Petri nets, etc). Instead, we find the proof theory of linear logic an appropriate and powerful setting for exploring the structure of proofs in various intuitionistic logics (see =-=[LM07]-=- for another such use of linear logic). In the next section, we define µMALL = and prove some of the most basic aspects of its proof theory, including the cut-elimination theorem. Section 3 presents a... |

23 | Induction and co-induction in sequent calculus - Momigliano, Tiu - 2003 |

18 | The structure of exponentials: Uncovering the dynamics of linear logic proofs
- Danos, Joinet, et al.
- 1993
(Show Context)
Citation Context ...All of linear logic connectives except the exponentials yield similar theorems. It is certainly possible to consider a (partially ordered) collection of exponentials on top of MALL (see, for example, =-=[DJS93]-=-). An alternative to strengthen MALL with exponentials is to extend it with fixed points. Early approaches to adding fixed points [Gir92,SH93] involved inference rules that could only unfold fixed poi... |

17 | From proofs to focused proofs: a modular proof of focalization in linear logic
- Miller, Saurin
- 2007
(Show Context)
Citation Context ... ′ x, S ′ x ⊥ ). The transformation is straightforward and relies on monotonicity, and obviously does not increase the µ-height. ⊓⊔ We now present some interesting notions introduced by Alexis Saurin =-=[MS07]-=-, which make the focalization proof clear and simple. Definition 6 (Descendant). We distinguish several occurences of a formula in a sequent, and in a derivation. We define the notion of immediate des... |

15 | Model checking for π-calculus using proof search - Tiu |

14 | Mixing finite success and finite failure in an automated prover
- Tiu, Nadathur, et al.
- 2005
(Show Context)
Citation Context ... from linear logic, does not share such restrictions, neither difference is relevant when we restrict our attention to formulas in G. Interestingly, the fragment G has already been identified in LINC =-=[TNM05]-=-, and the Bedwyr system [BGM + 07] implements a proof-search strategy for it that is complete under the assumption that all fixed points are noetherian (and hence that least and greatest fixed points ... |

13 | Gopalan Nadathur, and Alwen Tiu. The Bedwyr system for model checking over syntactic expressions - Baelde, Gacek, et al. - 2007 |

13 | On the specification of sequent systems
- Pimentel, Miller
- 2005
(Show Context)
Citation Context ...e connection to the focused proof systems of [LM07] should be explored). Such an extension to µMALL = could also be used to generalize the uses of induction in the linear logic programming setting of =-=[PM05]-=-. At the system designing and implementation level, our focused proof system should help in designing a logic engine that attempts to prove formulas involving induction and co-induction. Our hope is t... |

10 |
A game semantics for proof search: Preliminary results
- Miller, Saurin
(Show Context)
Citation Context ...e applications in proofsearch since it reduces the proof-search space by limiting the situations when backtracking is necessary. Focused proofs are also useful for justifying game theoretic semantics =-=[MS05]-=- and have been central to the design of Ludics [Gir01]. 9sA good focused proof system for µMALL = is not a simple consequence of the translation of fixed points into LL2 that is used in the proof of T... |

4 | A User Guide to Bedwyr - Baelde, Gacek, et al. - 2006 |

2 | Games for bisimulation and model checking. Notes for Mathfit Workshop on Finite Model Theory - Stirling - 1996 |