Galmiche, D.: Connection methods in linear logic and proof nets constructions. Theoretical Computer Science 232 (2000) 213--272  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... fragment of Mixed Linear Logic 1 (MMLL) extends the commutative multiplicative linear logic (MLL) and the non commutative (or cyclic) multiplicative linear logic (NCMLL) We naturally consider our previous works, based on the concept of proof net, dedicated to proof search in both fragments [6,7] but in MMLL an essential step consists in understanding and analyzing the interaction between commutative and non commutative connectives during the proof search process. We have recently studied two proof search methods in MMLL that were based on the de nition of labelled (with constraints) ....

....fragment. Moreover the de nition of dependencies (or constraints) attached to the calculi is a natural way to translate some proof theoretical results of MMLL into simple and tractable automated proof search procedures, that are based on the construction of semantical structures like proof nets [6,11]. After the study of the relationships between both systems (adequacy and sequentialization) we propose a based on proof nets algorithm for proof construction in MMLL that builds in parallel a proof structure (like in MLL) and a set of dependencies on which we have to verify some validity ....

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D. Galmiche. Connection Methods in Linear Logic and Proof nets Construction. Theoretical Computer Science, 232(1-2):231-272, 2000.

....e cient proof search procedures for such a mixed logic. Because of the cohabitation of linear and intuitionistic connectives in a same logic and even if BI includes IL and MILL as subsystems, we cannot directly extend proof search (sequent, tableau, or connection) calculi de ned for each of them [2,4,5,14] to BI. Moreover we aim to design a BI prover that builds proofs but also generates countermodels for non theorems. Such systems exist for IL [5] and MILL [14] and the tableau method directly provides countermodel generation for a wide range of logics, including IL, but not for substructural ....

....countermodels. 3 A Labelled Tableaux calculus for BI Because of the cohabitation of linear and intuitionistic connectives in BI and even if it includes IL and MILL as subsystems, we cannot directly extend to BI the proof search (sequent, tableau, or connection) calculi de ned for both [2,4,5,14]. Here, we apply the methodology of Labelled Deductive Systems (LDS) 3] to the tableau method in order to propose a labelled tableau calculus TBI for propositional BI , the use of labels making it possible to generate countermodels. 3.1 A labelling algebra We de ne a set of labels and ....

D. Galmiche. Connection Methods in Linear Logic and Proof nets Construction. Theoretical Computer Science, 232(1-2):231272, 2000.

....X, A # B, Y [#] # X, A # B, Y # X, A, B [ # X, A B # t [t] # X [f] # X, f # X, # [#] # X, A [ # X, A # X, A [ # X, A # X [K ] # X, A # X, A, A [WI ] # X, A 2.9.2 Proof Nets [ This section must be added. Relevant citations will be from among [26, 43, 57, 66, 107, 117, 119]] 2.10 Curry Howard Some logicians have found that it is possible to analyse proofs more closely by giving them names. After all, if proofs are first class entities, we will be better off if we can distinguish different proofs. I can illustrate this by looking at an example from intuitionistic ....

D. GALMICHE. "Connection methods in Linear Logic and Proof Nets Construction". Theoretical Computer Science, 232:231--272, 2000.

....branches of proofs and, in particular, that we do not address the issue of which rules are used in which order in order to find proofs. It follows that our work is essentially orthogonal to studies such as Andreoli s [1] on focusing, and somewhat more related (though indirectly) to Galmiche s [3] proof nets construction mechanism. 1 The most closely related work of which we are aware is that of Galmiche, Mery and Pym [4, 5, 6] on semantic tableaux for BI, in which the resource distribution problem is handled via calculations using labels drawn from an algebra of Kripke worlds. Our ....

D. Galmiche, Connection methods in linear logic and proof nets construction, Theoretical Computer Science 232 (2000) 231-272.

....linear substitution oe L which makes every path through some F complementary. It terminates with failure if F is not valid. Attempts for obtaining matrix characterizations in fragments of linear logic have been made on the basis of acyclic connection graphs [Fronhofer, 1996, Galmiche, 1996, Galmiche, 1999]. This acyclicity condition is very close to proof nets and therefore these attempts will very likely have similar limitations. In contrast to that our approach is based on prefixes and unifies the advantages of several approaches to proof search in linear logic without sharing their problems. ....

D. Galmiche. Connection methods in linear logic and proof nets construction. Theoretical Computer Science, 1999 (to appear).

.... DEFINITIONS A proof net is a particular graph, the nodes of which are formulae of linear logic [23] It is naturally defined for the multiplicative linear logic MLL (without constants) as recalled in this subsection and can be seen as an efficient tool for automated deduction in such a fragment [15]. There are also proof nets definitions for other CLL fragments, like MALL (Multiplicative and Additive Linear Logic) 23] but they are more complicated to handle. Definition 19 A proof structure is a graph, the vertices of which are formulae, inductively built from the following substructures ....

.... generally based on criteria that characterize the proof structures that are proof nets [21] like for instance the Danos Regnier characterization [12] The construction of a proof net can be seen as the search of the connections (axiom links) that characterize the provability of a given sequent [15]. 18 LABELLED DEDUCTION To simplify the different representations, we often omit the formulae labelling the nodes and only represents the connectives. Here we recall the inductive definition of proof net, introduced by Bellin [5] Definition 20 A MLL proof net and its set of conclusions are ....

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D. Galmiche. Connection methods in Linear Logic and Proof nets Construction. Theoretical Computer Science, 1999. Accepted for publication.

....of the linear sequent calculus, i.e. a counterpart of natural deduction in LL. We have defined an algorithm for automated proof net construction for the Multiplicative Linear Logic (MLL) and shown that it could naturally provide a connection (proof search) method for this logical fragment [7]. In fact, automated theorem proving or verification can strongly benefit from such investigations on proof nets. In this paper, we consider the multiplicative fragment of Non Commutative Linear logic (NCMLL) 1,2] and we design an algorithm for automatic construction of non commutative proof nets ....

.... calculus is reflected by planarity conditions on proof nets, justified by a graph theoretic characterization of non commutativity [18] A main point is to keep the construction principles used in the MLL case that lead to a direct construction of proof nets without an a posteriori verification [7]. Moreover because of the relationships between Lambek calculus and noncommutative linear logic [3] another interesting result is that we can derive, from the previous results, an alternative algorithm for construction of Lambek proof nets [17,20] that is not based on the explicit use of labels ....

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D. Galmiche. Connection methods in Linear Logic and Proof nets Construction. Theoretical Computer Science, 1998/99. Accepted for publication. 19 D. Galmiche and B. Martin

....valid in MLL and returns an admissible linear substitution oe L which makes every path through some F complementary. It terminates with failure if F is not valid. Attempts for obtaining matrix characterizations in fragments of linear logic have been made on the basis of acyclic connection graphs [14, 15, 16]. This acyclicity condition is very close to proof nets and therefore these attempts will very likely have similar limitations. In contrast to that our approach is based on prefixes and unifies the advantages of several approaches to proof search in linear logic without sharing their problems. ....

D. Galmiche. Connection methods in linear logic and proof nets construction. Theoretical Computer Science, 1999 (to appear).

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D. Galmiche. Connection Methods in Linear Logic and Proof nets Construction. Theoretical Computer Science, 232(1-2):231272, 2000.

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D. Galmiche. Connection Methods in Linear Logic and Proof nets Construction. Theoretical Computer Science, 232(1-2):231-272, 2000.

....analyzed but the interest of constraint based proof calculi [15] for substructural logics is clearly con rmed. 5 A new connection based characterization for MILL There exists connection based characterizations and related connection methods for multiplicative (commutative) linear logic (MLL) [5,8] but not for its intuitionistic fragment, namely MILL. The matrix characterization proposed in [8] is based on particular pre xes and substitutions dedicated to MLL. In order to extend or adapt it to MILL, it would be necessary to add intuitionistic pre xes but it seems to be di cult and not ....

....with minor changes on the constraints to satisfy and thus we can show that this formula is provable. We have previously de ned a connection based characterization of provability in MLL and studied the relationships between connection methods and proof nets construction in linear logic fragments [5]. We can show, like for MLL, that a connection based method for MILL provides an algorithm for the construction of MILL proof nets [2] Conversely, we have de ned a connection method for MLL that is based on the automatic construction of proof nets (and of sequent proofs in parallel) This method ....

D. Galmiche. Connection Methods in Linear Logic and Proof nets Construction. Theoretical Computer Science, 232(1-2):231272, 2000.

.... Tableaux Calculus for BI Because of the cohabitation of multiplicative (linear) and additive (intuitionistic) connectives in BI and even if it includes IL and MILL as subsystems, we cannot directly extend proof search (sequent, tableau, or connection) calculi de ned for both sublogics to BI [3, 11, 13, 36]. We introduce labels and labelled formulas in a speci c way compared to the general LDS approach of [10] in order to de ne a labelled tableau calculus TBI , for propositional BI , with generation of countermodels. Let us start de ning this particular set of labels and constraints. 3.1 ....

....the restriction of the initial TBI system to the multiplicative connectives, i.e. the expansion rules of gure 16, directly provides a tableau method for MILL. There exist proof search methods for (propositional) multiplicative linear logic (MLL) based on di erent notions like connections [11, 22], tableaux [25, 26] with pre xes or canonical proofs [1, 18] They cannot be easily extended to MILL and the design of a tableau method with labels from the Urquhart s semantics appears as an appropriate solution. The restriction of TBI to the multiplicatives leads to a suitable tableau system for ....

[Article contains additional citation context not shown here]

D. Galmiche. Connection Methods in Linear Logic and Proof nets Construction. Theoretical Computer Science, 232(1-2):231272, 2000.

.... calculus for BI Because of the cohabitation of multiplicative (linear) and additive (intuitionistic) connectives in BI and even if it includes IL and MILL as subsystems, we cannot directly extend to BI the proof search (sequent, tableau, or connection) calculi de ned for both sublogics [3, 10, 12, 35]. Here, we apply the methodology of Labelled Deductive Systems (LDS) 6] to the tableau method but in a speci c way compared to the general approach of [9] in order to de ne a labelled tableau calculus TBI for propositional BI , the use of labels making it possible to generate ....

....the redundancy schema given in gure 17. 8.2 Proof and countermodel construction In this section, we illustrate how to build proofs and countermodels in TBI MILL. There are some proof search methods for (propositional) multiplicative linear logic (MLL) based on di erent notions like connections [10, 19], tableaux [23, 24] or canonical proofs [1, 14] Some of them can or could be adapted to MILL, but our approach based on labelled semantic tableau allows to directly build countermodels in case of non provability. Let us illustrate these points with the following examples. Figure 18 gives a ....

D. Galmiche. Connection Methods in Linear Logic and Proof nets Construction. Theoretical Computer Science, 232(1-2):231272, 2000.

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Galmiche, D.: Connection methods in linear logic and proof nets constructions. Theoretical Computer Science 232 (2000) 213--272

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Galmiche, D.: Connection methods in linear logic and proof nets constructions. Theoretical Computer Science 232 (2000) 213--272

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