## Focusing the inverse method for linear logic (2005)

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Venue: | Proceedings of CSL 2005 |

Citations: | 37 - 11 self |

### BibTeX

@TECHREPORT{Chaudhuri05focusingthe,

author = {Kaustuv Chaudhuri and Stephen Brookes},

title = {Focusing the inverse method for linear logic},

institution = {Proceedings of CSL 2005},

year = {2005}

}

### Years of Citing Articles

### OpenURL

### Abstract

1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10

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Citation Context ...ely absent in the forward direction. (For details, see section 3.1.) The particular forward search strategy we use is the inverse method [73, 35]. The inverse method is a generalisation of resolution =-=[102, 84, 83]-=- that applies to a wide variety of logics, unlike resolution which only works for classical logic. Indeed, the inverse method can be seen to be logic-independent because it has very minimal requiremen... |

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331 | Logic programming with focusing proofs in linear logic
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281 | Computational interpretations of linear logic
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Symbolic Logic and Mechanical Theorem Proving
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Citation Context ... has a contraction-free sequent calculus that only needs to duplicate negative atoms and implications [13]. 7.2 Generalising hyperresolution Hyperresolution is a complete strategy for classical logic =-=[9, 19]-=- that in practice gives an efficient search procedure for Horn and near-Horn fragments [20]. We concentrate on the following intuitionistic (nonlinear) Horn-fragment: (goals) G ::= p | G1 ∧ G2 | t (cl... |

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Citation Context ...wo methods are commonly used to combat the state explosion problem – compositional reasoning [31, 85], which attempts to break up a big problem into manageable components, and partial order reduction =-=[45, 69, 90, 114]-=-. Combinations of such methods can nowadays handle model-checking problems of the 16sorder of 10 100 states. Efficient model checking back-ends are, nevertheless, ultimately unsatisfactory as descript... |

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215 | Logical frameworks - Pfenning - 2001 |

207 | Linear Objects: logical processes with built-in inheritance
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Citation Context ...ounded state systems [14]. Instead of traditional approaches based on finite abstraction, using human input or more automated methods, Bozzano translates the verification problem to a the language LO =-=[9]-=- parametrised over constraint domains such as Herbrand universes. Bozzano extended LO with universal quantification, and gave a top down saturation-based inference mechanism that was shown to be a dec... |

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Citation Context ...r normalising proof terms. The algorithm presented for normalisation will be a straightforward constructive proof of the malleability of linear logic. We follow the general schema outlined by Prawitz =-=[97]-=- by describing normal proofs as those that consist of two halves. One half reasons by deconstructing hypotheses into their component judgements using elimination rules; in this half, information flows... |

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Citation Context ...derivations, which, as demonstrated in the previous section, subsumes hyperresolution on Horn theories. The interaction of focusing and cut-elimination has been studied by Danos, Joinet and Schellinx =-=[33, 32]-=-. Although none of their translations are explicitly focusing aware, their calculi, particularly the constraints in the LK η p system bear unmistakable similarities to focusing. A more recent work by ... |

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Citation Context ...ombined procedure. This can either be done at the level of the logic by having separate intuitionistic connectives that operate in the unrestricted context (similar to the approach taken in LNL logic =-=[12]-=-), or the inverse method can be specialised for the image of one of the above translations to use more efficient algorithms that ignore the linear aspects of the sequents. The feasibility of any such ... |

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Citation Context ...e and 1GB of main memory; “—” denotes unsuccessful proof within≈ten hours. Table 1: some experimental results. problems, and translations of various quantified Boolean formulas using the algorithm in =-=[16]-=-. Focusing was faster in every case, with an average speedup of about three orders of magnitude over the non-focusing version. 6 Embedding non-linear logics 6.1 Intuitionistic logic When we move from ... |

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Citation Context ...rs considerably because of the different nature of forward search. Linear logic programming has also been examined from the perspective of specification languages, first by Miller in the system Forum =-=[78]-=- and Andreoli in the system LinLog [7]. The latter work, in fact, introduced the dyadic notation for resources used extensively in the present work. More recently, Cervesato and Pfenning [24] have att... |

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Citation Context ...the present work. More recently, Cervesato and Pfenning [24] have attempted to provide a sound type-theoretic foundation to linear logic programming in terms of extensions to the LF logical framework =-=[51]-=-. Watkins et al have further extended this line of work by giving a manifestly decidable equational theory for an extension of LF with a monad for concurrency [117, 25]. 14s1.5.2 General theorem provi... |

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Citation Context ...A promising recent approach is to capture the refinement relations, to be turned into proof-obligations for a model-checking back-end, into behavioral types that are exposed in the abstraction itself =-=[63, 26]-=-. While an interesting use of type-theory, this approach must still be regarded as a tour de force of existing model checking frameworks such as SLAM. It is important to stress that theorem proving an... |

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Fundamentals of a Theory of Asynchronous Information Flow
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Citation Context ...t systems. In the domain of concurrent processes, particularly, many abstractions have been proposed to model concurrency and communication — CSP [52, 19], CCS [81], the π-calculus [82] or Petri-nets =-=[91]-=-. (These abstractions all fall naturally into fragments of linear logic [25].) Combining model checking approaches with such abstractions is a non-trivial task. A promising recent approach is to captu... |

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Citation Context ...istic linear logic [14, 3, 8] with an additional lax modality [22]. This choice is motivated by two considerations. First, it includes the propositional core of the Concurrent Logical Framework (CLF) =-=[21]-=-, so our theorem prover, and its first-order extension, can reason with specifications written in CLF; many such specifications, including Petri nets, the π-calculus and Concurrent ML, are described i... |

74 |
Classicaly and intuitionistically provable recursive functions
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- 1978
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Citation Context ...pressiveness of the possibility judgement. There are several well known translations from classical to intuitionistic logics, the most well known of them perhaps being the double-negation translation =-=[38]-=-. We use this idea in our case, using the definition of linear falsehood, ? 0, from the previous section. First, we briefly sketch the classical sequent calculus, which we present in a two-sided dyadi... |

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Citation Context ... for (classical) linear logic. This idea has seen considerable use since then: Hodas and Miller in the setting of logic programming in the uniform fragment [56], Benton et al. for linear term calculi =-=[13]-=-; more recently by Barber and Plotkin for the system DILL [11], Polakow and Pfenning for ordered logic [96, 95], and Howe in the setting of focused (backward) proof search for linear logic [57]. (The ... |

69 | A Uniform Proof-Theoretic Investigation of Linear Logic Programming
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Citation Context ...long history given the age of linear and substructural logics, with the earliest identification of this issue in uniform proof search dating back to the work of Harland and Pym in 1991 [48] (see also =-=[98]-=-). Their approach centred on rewriting the hereditary Harrop fragment of linear logic in a clausal form suitable for resolution search. Although they stopped short of identifying the resource manageme... |

65 |
The Collected Papers of Gerhard Gentzen
- Gentzen
- 1969
(Show Context)
Citation Context ...us, leaving a discussion of the natural deduction formulation for purely proof-presentation purposes (sec. 2.2). We import this judgemental view into a Gentzen-style sequent calculus for linear logic =-=[41, 40]-=-. In this calculus, we discard the general judgement of truth in favour of more basic notions of resources and goals, that is, with the judgements “A is a resource” and “A is a goal”. These correspond... |

62 | A technique of a state space search based on unfolding
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- 1995
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Citation Context ...wo methods are commonly used to combat the state explosion problem – compositional reasoning [31, 85], which attempts to break up a big problem into manageable components, and partial order reduction =-=[45, 69, 90, 114]-=-. Combinations of such methods can nowadays handle model-checking problems of the 16sorder of 10 100 states. Efficient model checking back-ends are, nevertheless, ultimately unsatisfactory as descript... |

60 |
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Citation Context ...eir use in proving completeness—are sufficiently interesting in and of themselves as they substantiate the logical basis of focusing. Similar notions of cut and cut-admissibility also exist in Ludics =-=[6]-=-, though our calculus and Ludics are philosophically dissimilar enough that we cannot simply import the cut-admissibility argument from Ludics. Rather, we choose to view our proof of cutadmissibility ... |

54 | Efficient resource management for linear logic proof search
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Citation Context ... search strategies postpone this split either by an input/output interpretation, where proving a subgoal consumes some of the resources from the input and passes the remaining resources on as outputs =-=[12]-=-, or via Boolean constraints on the occurrences of linear hypotheses [13]. Interestingly, multiplicative non-determinism is entirely absent in a forward reading of multiplicative rules: the linear con... |

50 |
Avoiding the state explosion problem in temporal logic model checking
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- 1987
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Citation Context ... surprisingly subtle because of the exponential blowup in the state space in the presence of concurrency. Two methods are commonly used to combat the state explosion problem – compositional reasoning =-=[31, 85]-=-, which attempts to break up a big problem into manageable components, and partial order reduction [45, 69, 90, 114]. Combinations of such methods can nowadays handle model-checking problems of the 16... |

50 |
Logic Programming in Intuitionistic Linear Logic: Theory, Design and Implementation
- Hodas
- 1994
(Show Context)
Citation Context ...the premisses to construct the conclusion. In the domain of top-down linear logic programming—refining goals by applying inference rules in the backward direction until they become initial (eg. Lolli =-=[55]-=- or Lygon [118])—approaches to combating this kind of non-determinism fall into two broad kinds. The first kind commit to an input-output interpretation of hypotheses. For the ⊗R rule for example, pro... |

50 |
Automatic deduction with hyper-resolution
- Robinson
- 1965
(Show Context)
Citation Context ...formulas [80] and its generalisation to uniform proof [79] that describes how to treat compound implications as “procedures” in a programming interpretation. Another famous example is hyperresolution =-=[103]-=- in the domain of automated theorem proving for classical logic, where the input theory is “cooked” into a clausal form that allows large inferences. Both uniform proofs and hyperresolution are logica... |

49 | A judgmental analysis of linear logic
- Chang, Chaudhuri, et al.
- 2003
(Show Context)
Citation Context ...nd some of the experiments in Sec. 5.2 use them. Propositions are written using uppercase letters A, B, C, with p standing for atomic propositions. The sequent calculus is a standard fragment of JILL =-=[6]-=-, containing dyadic two-sided sequents of the formΓ;∆=⇒ C: the zoneΓ 2scontains the unrestricted hypotheses and∆contains the linear hypotheses. Both contexts are unordered. For the rules of this calcu... |

48 |
Games Semantics for Linear Logic
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- 1991
(Show Context)
Citation Context ...state; for example, planning (finite and infinite state) [71], protocol verification [14, 17, 16], concurrent computation [5], process calculi [77], Petri-nets [20], security protocols [25] and games =-=[62, 6]-=-. It is remarkable that linear logic serves as a uniform language for such diverse systems. Automated deduction for linear logic therefore has wide-ranging appeal. Besides allowing a direct analysis o... |

46 | A concurrent logical framework ii: Examples and applications
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(Show Context)
Citation Context ...so our theorem prover, and its first-order extension, can reason with specifications written in CLF; many such specifications, including Petri nets, the π-calculus and Concurrent ML, are described in =-=[7]-=-. For many of these applications, the intuitionistic nature of the framework is essential. Second, it is almost a worst-case scenario, combining the difficulties of modal logic, intuitionistic logic, ... |

46 | The uniform proof-theoretic foundation of linear logic programming (extended abstract
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- 1991
(Show Context)
Citation Context ...s a relatively long history given the age of linear and substructural logics, with the earliest identification of this issue in uniform proof search dating back to the work of Harland and Pym in 1991 =-=[48]-=- (see also [98]). Their approach centred on rewriting the hereditary Harrop fragment of linear logic in a clausal form suitable for resolution search. Although they stopped short of identifying the re... |

43 | Experiments with Discrimination-Tree Indexing and Path Indexing for Term Retrieval
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- 1992
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42 | A resolution theorem prover for intuitionistic logic
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- 1996
(Show Context)
Citation Context ...e possible to improve the efficiency of the inverse method search procedure by incorporating hyper-resolution strategies; see, for example, Tammet’s treatment of classical logic in the Gandalf prover =-=[108]-=-. Such specialised strategies have practical benefits, but they are not very satisfactory from both design and engineering. Furthermore, it is a denouncement of the versatility of the inverse method i... |

40 |
Term indexing
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Citation Context ...to construct the database in such a fashion that we can efficiently ask for more general forms of a given sequent. For our implementation of the database, we use a global forest of substitution trees =-=[46, 99]-=-. Substitution trees are what is known as a perfect filter: the results of querying a substitution tree gives exactly the answers that satisfy the query. This differs from imperfect filters such as di... |