C. Kreitz, H. Mantel, J. Otten, and S. Schmitt. Connection-based proof construction in Linear Logic. In 14th Int. Conference on Automated Deduction, CADE-14, LNCS 1249, pages 207221, Townsville, North Queensland, Australia, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 ....

....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 natural. Since BI is conservative over MILL [11] we can de ne the rst connection based ....

C. Kreitz, H. Mantel, J. Otten, and S. Schmitt. Connection-based proof construction in linear logic. In 14th Int. Conference on Automated Deduction, pages 207221, Townsville, North Queensland, Australia, 1997.

....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 ....

C. Kreitz, H. Mantel, J. Otten, and S. Schmitt. Connection-based proof construction in Linear Logic. In 14th Int. Conference on Automated Deduction, CADE-14, LNCS 1249, pages 207221, Townsville, North Queensland, Australia, 1997.

....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 ....

C. Kreitz, H. Mantel, J. Otten, and S. Schmitt. Connection-based proof construction in linear logic. In 14th Int. Conference on Automated Deduction, pages 207221, Townsville, North Queensland, Australia, 1997.

....] F 4 [x 2 ] which is precisely what is generated by the techniques of [7] in this case. Development along these lines will help to form a bridge between sequent systems, which are good for analysis but generally make a poor basis for efficient implementations, and matrix or connection methods [12, 8], on which many efficient implementations are based. An implementation of the combined system using constraint logic programming techniques with Boolean constraint solver is underway. Acknowledgements We thank David Pym and Michael Winikoff for discussions related to this work. The author is ....

C. Kreitz, H. Mantel, J. Otten and S. Schmitt, Connection-based Proof Construction in Linear Logic, Proceedings of the International Conference on Automated Deduction (CADE-14) 207-221, W. McCune (ed.), Townsville, July, 1997. Published by Springer-Verlag as Lecture Notes in Computer Science 1249.

....example, 4, 56, 57, 141] For a more detailed example, there have recently been attempts to develop connection methods for linear logic. One possibility is to extend the previous works on intuitionistic logic and then to keep uniformity inside a global proof environment for constructive logics [92]. An alternative proposal comes from a direct analysis of proof nets and proposes, after de ning a connection based characterization, to use a proof net construction method in LL to derive a new connection method (see Galmiche s paper, in this volume) In this setting, the proofobject is not only ....

C. Kreitz, H. Mantel, J. Otten, and S. Schmitt. Connection-based proof construction in linear logic. In 14th Int. Conference on Automated Deduction, pages 207-221, Townsville, North Queensland, Australia, 1997.

....avoiding problems such as sequent reconstruction from the two following points of view: construction of a sequent proof from a proof net [6] or construction of a sequent from the connection characterization (i.e. the subset of connections leading to provability) 44] Independently, Kreitz and al. [35] have recently presented a characterization of logical validity in MLL and a connection based proof search procedure that is a straightforward extension of a method originally developed for classical and intuitionistic logics [42] Among its features, let us mention the generality and uniformity ....

.... 21, 22] With the derived connection method, we are able to avoid the sequent reconstruction from a proof net [6] or from the connection characterization (i.e. the subset of connections leading to provability) 44] A dual result is that a connection method for MLL, for instance like the one of [35], can be used to construct proof nets. Further works could analyse such an approach from complexity and algorithmic points of view. In fact, our approach is important for an analysis of computation (including failure) w.r.t. an initial logical speci cation. From these results in the MLL fragment ....

[Article contains additional citation context not shown here]

C. Kreitz, H. Mantel, J. Otten, and S. Schmitt. Connection-based proof construction in linear logic. In 14th Int. Conference on Automated Deduction, pages 207221, Townsville, North Queensland, Australia, 1997.

....example, 4, 56, 57, 141] For a more detailed example, there have recently been attempts to develop connection methods for linear logic. One possibility is to extend the previous works on intuitionistic logic and then to keep uniformity inside a global proof environment for constructive logics [92]. An alternative proposal comes from a direct analysis of proof nets and proposes, after defining connection based characterization, to use a proof net construction method in LL to derive a new connection method (see Galmiche s paper, in this volume) In this setting, the proofobject is not only a ....

C. Kreitz, H. Mantel, J. Otten, and S. Schmitt. Connection-based proof construction in linear logic. In 14th Int. Conference on Automated Deduction, pages 207--221, Townsville, North Queensland, Australia, 1997.

....and completeness of this semantics versus provability in the sequent calculus, i.e. a formula OE 2 F is provable iff OE is valid. It is noteworthy that exchanging DLE structures by boolean algebras leads to a semantic of firstorder modal logic S4 with cumulative domains [21] People familiar with [21, 16] may recognize that our semantics could be helpful to obtain a Wallen style matrix characterization of the DLE, whereby the accessibility relations reflect the conditions imposed on the substitutions. 3 Linear Proofs revisited We turn to an appropriate matrix characterization for regular ....

....disjunction and its dual. In this case we end up with Affine Logic, which might be a good candidate for representing planning tasks, but so far no reasonable proof methods have been developed for this logic. Our next step will be a matrix characterization for the full DLE based on the results of [21, 16]. We plan to extend these results to full Affine Logic in the future. ....

C. Kreitz, H. Mantel, J. Otten, S. Schmitt. Connection-based proof construction in Linear Logic. CADE--14, 1997.

....combining our transformation algorithm with a proof procedure for various non classical logics in order to guide the derivation of proofs in one of the existing generic tools for interactive proof development. We have already extended our procedure to the multiplicative fragment of linear logic [12, 26] and will investigate the integration of further fragments into our approach. Finally, we will consider the combination of induction techniques with logical reasoning (e.g. 22] for a uniform representation within a matrix based framework. This will eventually extend our proof reconstruction ....

C. Kreitz, H. Mantel, J. Otten, S. Schmitt. Connection-Based Proof Construction in Linear Logic. CADE--14, LNAI 1249, pp. 207--221, Springer, 1997.

....become leaves in a sequent proof, instead of the logical connectives of a proof goal. Although originally developed for classical logic, the connection method has been extended to a variety of non classical logics such as intuitionistic logic [22] modal logics [20] and fragments of linear logic [18]. In this section we will briefly summarize its essential concepts. A formula tree is the tree representation of a formula F . Each position u in the tree is marked with a unique name and a label that denotes the connective of the corresponding subformula or the subformula itself, if it is ....

C. Kreitz, H. Mantel, J. Otten & S. Schmitt. Connection-Based Proof Construction in Linear Logic. 14 Conference on Automated Deduction, LNAI 1249, pp. 207-- 221, 1997.

....into sequent proofs [21, 22] Resource management similar to multiplicative linear logic is addressed by the linear connection method [4] Fronhofer [8] gives a matrix characterization of that captures some aspects of weakening and contraction but does not appear to generalize any further. In [15] we have developed a matrix characterization for and extended the uniform proof search and translation procedures accordingly. In this paper we present a matrix characterization for the full multiplicative exponential fragment including the constants 1 and #. This characterization uses ....

....section is the foundation for matrix based proof search methods. It yields a compactified representation of the search space which can be exploited by proof search methods in the same way as for other logics [20] The method has been extended uniformly to multiplicative linear logic, as shown in [15]. Along the same lines an extension to is possible. Theorem 10 (Characterization Theorem) A formula # is valid in if and only if the corresponding matrix is complementary for some multiplicity. Proof. Correctness follows from theorems 8, 6, 4, 2, and the correctness of # # 1 . ....

[Article contains additional citation context not shown here]

C. Kreitz, H. Mantel, J. Otten, S. Schmitt. Connection-Based Proof Construction in Linear Logic. 14 Conference on Automated Deduction, LNCS 1249, pp. 207--221. Springer, 1997.

....proof search in linear logic is dicult to automate. Various calculi have been developed for linear logic. Beginning with the sequent calculus and proof nets by Girard [12] several optimizations have been proposed. More recently, the connection method has been extended to fragments of linear logic [8,9,15,17]. In this article, we propose a tableau calculus for MELL and for M LL which is the theoretical basis for our theorem prover linTAP. linTAP is implemented in a very compact way but uses sophisticated techniques to reduce the search space and thus follows the idea of lean theorem proving . It was ....

....logic (on which we will focus in the following section of this paper) we do not need to deal with characters of type E or E . Furthermore all pre xes to be uni ed have the form C 1 V 1 C 2 V 2 : C n Vn (where V i V and C i C) allowing us to drop rules R2, R4, R6, and R7 (see also [15]) De nition 12. Let V= M be a set of variables, C= M be a set of constants, and V 0 be a set of auxiliary variables (with V V 0 = The set of transformation rules for M LL is de ned in Table 5. R1. f = j g; fg; R3. fXs = jXtg; fs = jtg; R5. fV s = zj g; fs = ....

[Article contains additional citation context not shown here]

C. Kreitz, H. Mantel, J. Otten, S. Schmitt. Connection-Based Proof Construction in Linear Logic. 14 th Conference on Automated Deduction, LNCS 1249, pp. 207-221. Springer, 1997.

....a common substring at the beginning of the two prefixes. This enabled us to develop a much simpler algorithm computing a minimal set of most general unifiers. Our general proof procedure also allows a uniform treatment of other nonclassical logics like various modal logics [30] or linear logic [19]. We only have to change the notion of complementarity (i.e. the prefix unification) while leaving the path checking algorithm unchanged. Path checking can also be performed by using a semantic tableau [11] The prover ileanTAP [26] is based on free variable semantic tableaux extended by the ....

....at a fi position. This guarantees that no decisions on selecting proof relevant sub relations have to be made and hence, additional search wrt. these decisions will be avoided. Our approach for reconstructing LJmc proofs from MJ proofs has been uniformly extended to various non classical logics [35, 19] for which matrix characterizations exist. A uniform representation of different logics and proofs within logical calculi as well as abstracted descriptions for integrating special properties of these logics in an uniform way, e.g. the completion of reduction orderings , yields a general ....

C. Kreitz, H. Mantel, J. Otten, and S. Schmitt. Connection-Based Proof Construction in Linear Logic. In W. McCune, ed., 14 th Conference on Automated Deduction, LNAI 1249, pp. 207--221, Springer Verlag, 1997.

....into sequent proofs [21,22] Resource management similar to multiplicative linear logic is addressed by the linear connection method [4] Fronh ofer [8] gives a matrix characterization of MLL that captures some aspects of weakening and contraction but does not appear to generalize any further. In [15] we have developed a matrix characterization for MLL and extended the uniform proof search and translation procedures accordingly. In this paper we present a matrix characterization for the full multiplicative exponential fragment including the constants 1 and . This characterization uses ....

....section is the foundation for matrix based proof search methods. It yields a compacti ed representation of the search space which can be exploited by proof search methods in the same way as for other logics [20] The method has been extended uniformly to multiplicative linear logic, as shown in [15]. Along the same lines an extension to MELL is possible. Theorem 17 (Characterization Theorem) A formula is valid in MELL if and only if the corresponding matrix is complementary for some multiplicity. Proof. Correctness follows from theorems 14, 9, 5, 3, and the correctness of 0 1 . ....

[Article contains additional citation context not shown here]

C. Kreitz, H. Mantel, J. Otten, S. Schmitt. Connection-Based Proof Construction in Linear Logic. 14 th Conference on Automated Deduction, LNCS 1249, pp. 207-221. Springer, 1997.

....methods based on sequent style or tableau proofs. Starting with Bibel s [4, 6] connection method for classical logic matrix characterizations have later been extended to many non classical logics by Wallen [20] Recently, matrix characterizations for fragments of linear logic have been developed [11, 16, 15]. Different approaches have been undertaken to represent matrix characterizations in a uniform way [20, 7] They allow to share results among different characterizations, e.g. based on Wallen s style of formulation a uniform proof method [17] and a uniform procedure for transforming matrix into ....

....2 s for formulas in the context but none for deleted formulas. For the example in section 2, there is a one to one correspondence where 2 and 3 connectives are translated into 2m and 3m operators, respectively. For other logics, this is more complicated, e.g. intuitionistic logics [20] or MLL [15]. However, to construct an efficient matrix representation the number of such operators should be kept to minimum. Similarly, the multiplicity concept should be kept to a minimal size. For our S4 example the multiplicity concept DS4 = f W ; 3mg is sufficient. After M and lm have been determined ....

[Article contains additional citation context not shown here]

C. Kreitz, H. Mantel, J. Otten, S. Schmitt. Connection-Based Proof Construction in Linear Logic. CADE--14, Springer Verlag, 1997. to appear

....in a sequent proof, instead of the logical connectives of a proof goal. Although originally developed for classical logic, the connection method has recently been extended to a variety of non classical logics such as intuitionistic logic [18] modal logics [20] and fragments of linear logic [14, 17]. Furthermore, algorithms for converting matrix proofs into sequent proofs have been developed [23, 24] which makes it possible to view matrix proofs as plans for predicate logic proofs that can be executed within a proof assistant [6, 15] Viewing matrix proofs as proof plans also suggests the ....

....which generates a proof plan that will later be executed by the proof assistant. Obviously, this concept is not restricted to first order or inductive theorem proving. In a similar way we can also integrate proof procedures for other important logics, such as modal logics [20] or linear logic [14], or higher level strategies for program synthesis [12, 13] In many of these cases we can rely on already known successful techniques that were originally implemented independently and view their results as plans for the actual derivation. By executing this plan within a proof assistant like ....

C. Kreitz, H. Mantel, J. Otten, and S. Schmitt. Connection-Based Proof Construction in Linear Logic. In W. McCune, ed., 14 th Conference on Automated Deduction, LNAI 1249, pp. 207--221. Springer, 1997.

.... foundations could be extended to intuitionistic and various modal logics [Wallen, 1990] On this basis we have extended the connection method to non clausal form, intuitionistic logic [Otten and Kreitz, 1995] modal logics [Otten and Kreitz, 1996(b) and also to fragments of linear logic [Kreitz et al. 1997, Mantel and Kreitz, 1998] Isabelle NuPRL Program Synthesis Mathematica Linear formula Modal formula Intuitionistic formula Classical formula Matrix proof (non clausal form) NK=LK proof NJ =LJ proof NM=LM proof LL proof Isabelle ....

.... a 13 a 17 ) 0 ff a 18 T 1 a 13 a 19 R 0 a 13 a 20 0 fi a 21 : 0 ff a 22 : 1 ff a 23 P 0 a a 24 0 fi a 25 S 0 a a 26 S 0 b a 27 Figure 2: Formula tree for F 1 with (a2) 2 The matrix characterizations of logical validity [Bibel, 1981, Bibel, 1987] Wallen, 1990, Kreitz et al. 1997] depend on the concepts of paths, connections, and complementarity. A path through a formula F is a maximal set of mutually ff related atomic positions of its formula tree. It can be visualized as a maximal horizontal line through the matrix representation of F . A connection is a pair of atomic ....

[Article contains additional citation context not shown here]

C. Kreitz, H. Mantel, J. Otten, S. Schmitt. Connection-based proof construction in linear logic. 14 th Conference on Automated Deduction, LNAI 1249, pp. 207--221, 1997.

....search in linear logic is difficult to automate. Various calculi have been developed for linear logic. Beginning with the sequent calculus and proof nets by Girard [12] several optimizations have been proposed. More recently, the connection method has been extended to fragments of linear logic [8,9,15,17]. In this article, we propose a tableau calculus for MELL and for M LL which is the theoretical basis for our theorem prover linTAP. linTAP is implemented in a very compact way but uses sophisticated techniques to reduce the search space and thus follows the idea of lean theorem proving . It was ....

....(on which we will focus in the following section of this paper) we do not need to deal with characters of type OE E or E . Furthermore all prefixes to be unified have the form C 1 V 1 C 2 V 2 : C nVn (where V i ffl V and C i ffl C) allowing us to drop rules R2, R4, R6, and R7 (see also [15]) Definition 12. Let V= Phi M be a set of variables, C= Psi M be a set of constants, and V 0 be a set of auxiliary variables (with V V 0 = The set of transformation rules for M LL is defined in Table 5. R1. f = j g; oe fg; oe R3. fXs = jXtg; oe fs = jtg; oe R5. fV s = zj g; ....

[Article contains additional citation context not shown here]

C. Kreitz, H. Mantel, J. Otten, S. Schmitt. Connection-Based Proof Construction in Linear Logic. 14 th Conference on Automated Deduction, LNCS 1249, pp. 207--221. Springer, 1997.

....into sequent proofs [21,22] Resource management similar to multiplicative linear logic is addressed by the linear connection method [4] Fronhofer [8] gives a matrix characterization of MLL that captures some aspects of weakening and contraction but does not appear to generalize any further. In [15] we have developed a matrix characterization for MLL and extended the uniform proof search and translation procedures accordingly. In this paper we present a matrix characterization for the full multiplicative exponential fragment including the constants 1 and . This characterization uses ....

....section is the foundation for matrix based proof search methods. It yields a compactified representation of the search space which can be exploited by proof search methods in the same way as for other logics [20] The method has been extended uniformly to multiplicative linear logic, as shown in [15]. Along the same lines an extension to MELL is possible. Theorem 17 (Characterization Theorem) A formula is valid in MELL if and only if the corresponding matrix is complementary for some multiplicity. Proof. Correctness follows from theorems 14, 9, 5, 3, and the correctness of Sigma 0 1 . ....

[Article contains additional citation context not shown here]

C. Kreitz, H. Mantel, J. Otten, S. Schmitt. Connection-Based Proof Construction in Linear Logic. 14 th Conference on Automated Deduction, LNCS 1249, pp. 207--221. Springer, 1997.

....a common substring at the beginning of the two prefixes. This enabled us to develop a much simpler algorithm computing a minimal set of most general unifiers. Our general proof procedure also allows a uniform treatment of other nonclassical logics like various modal logics [32] or linear logic [21]. We only have to change the notion of complementarity (i.e. the prefix unification) while leaving the path checking algorithm unchanged. Path checking can also be performed by using a semantic tableau [13] The prover ileanTAP [28] is based on free variable semantic tableaux extended by the ....

....at a fi position. This guarantees that no decisions on selecting proof relevant subrelations have to be made and hence, additional search wrt. these decisions will be avoided. Our approach for reconstructing LJmc proofs from MJ proofs has been uniformly extended to various non classical logics [37, 21] for which matrix characterizations exist. A uniform representation of different logics and proofs within logical calculi as well as abstract descriptions for integrating special properties of these logics in a uniform way, e.g. the completion of reduction orderings , yields a general proof ....

C. Kreitz, H. Mantel, J. Otten, S. Schmitt. Connection-Based Proof Construction in Linear Logic. In W. McCune, ed., 14 th Conference on Automated Deduction, LNAI 1249, pp. 207-- 221, Springer Verlag, 1997.

....in a sequent proof, instead of the logical connectives of a proof goal. Although originally developed for classical logic, the connection method has recently been extended to a variety of non classical logics such as intuitionistic logic [19] modal logics [20] and fragments of linear logic [16]. In this section we will briefly summarize the theoretical foundations of this method A formula tree is the tree representation of a formula F . Each position u in the tree is marked with a unique name and a label that denotes the connective of the corresponding subformula or the subformula ....

C. Kreitz, H. Mantel, J. Otten, and S. Schmitt. Connection-Based Proof Construction in Linear Logic. In W. McCune, ed., 14 th Conference on Automated Deduction, LNAI 1249, pp. 207--221. Springer, 1997.

.... logic and formulas in clause form, their theoretical foundations could be extended to intuitionistic and various modal logics [45] On this basis we have extended the connection method to non clausal form, intuitionistic logic [33] modal logics [35] and also to fragments of linear logic [20, 28]. Since the actual proofs generated by automated proof search procedures tend to have a very technical look, we have also developed a uniform algorithm for converting matrix proofs in these logics into sequent proofs [41, 42, 23] This allows us to view matrix proofs as plans for predicate logic ....

....the greatest common ancestor of u and v w.r.t. the tree ordering is of principal type ff (or fi) u is ff related (fi related) to a set S of positions if u ff v (u fi v) for all v 2S. The matrix representation of F 1 is given in Figure 3. The matrix characterizations of logical validity [5, 6, 45, 20] depend on the concepts of paths, connections, and complementarity. A path through a formula F is a maximal set of mutually ff related atomic positions of its formula tree. It can be visualized as a maximal horizontal line through the matrix representation of F . A connection is a pair of atomic ....

[Article contains additional citation context not shown here]

C. Kreitz, H. Mantel, J. Otten, S. Schmitt. Connection-based proof construction in linear logic. 14 th Conference on Automated Deduction, LNAI 1249, pp. 207--221, 1997.

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C. Kreitz, H. Mantel, J. Otten, and S. Schmitt. Connection-based proof construction in Linear Logic. In 14th Int. Conference on Automated Deduction, CADE-14, LNCS 1249, pages 207221, Townsville, North Queensland, Australia, 1997.

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