J. Otten, C. Kreitz. A Uniform Proof Procedure for Classical and Non-Classical Logics KI-96, LNAI 1137, pp. 307--319, Springer, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in order to provide global program extraction from the whole proof. Thus, the MJ proof has to be transformed back into an LJ proof which can be integrated as a proof plan for solving the original J goal. Since the matrix characterisation for J [16] as well as developed proof search procedures [12] are based on a multiple succedent sequent calculus LJ mc , proof reconstruction has to be done in two steps (T1 and T2 in Figure 1) For T 1 , reconstruction procedures for generating LJ mc proofs from matrix proofs have been developed in [14] For T 2 , the construction of LJ cut proofs from ....

J. Otten and C. Kreitz. A Uniform Proof Procedure for Classical and Non-classical Logics. In 20 th German Annual Conference on AI, LNAI 1137, pp. 307--319, 1996.

.... 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 sequent style proofs [19] could be developed. However, these approaches do not exactly give an answer to the question what a matrix characterization is. It is our aim to fill this gap by an abstract definition of matrix characterizations. We ....

....we extract the semantics from the matrix system. 5 Conclusion We have presented a formal definition of matrix systems as a framework for matrix characterizations, their relation to sequent calculi, and the semantics naturally induced by matrix systems. By this, we substantiate the results of [4, 20, 17, 11] concerning an exact definition of a matrix characterization. Additionally, we extend Wallen s approach [20] concerning multiplicities, connections, the integration of partial ordered prefix types, and polarities. It is noteworthy that the extension of polarities offers a natural way to handle ....

J. Otten, C. Kreitz. A Uniform Proof Procedure for Classical and Non-Classical Logics. KI96: Advances in Artificial Intelligence, LNAI 1137, pp. 307--319, Springer Verlag, 1996.

....simulated su#ciently well on a computer. Recently the characterizations of logical validity, which underly these systems, have been extended to intuitionistic logic and the modal logics K,K4,D,D4,T,S4, and S5 [17, 28, 29, 27] On this basis the existing proof methods have been extended accordingly [18, 19] in order to develop a coherent theorem prover that can deal with a variety of logics and many applications requiring mathematical reasoning. The use of automated theorem proving (ATP) in practical applications, however, causes another problem. As the e#ciency of proof search strongly depends on ....

....of proof tasks Uniform Proof Search Procedure Uniform Conversion Procedure Integration of solutions in application Fig. 1. Integrating Automated Theorem Proving into Application Systems Since currently only matrix based proof methods are able to handle di#erent logics in a uniform way [19] our starting point will be a proof according to Wallen s matrix characterizations [29] of logical validity. Here a formula F is valid if every path through a matrix representation of F contains at least one complementary pair of atomic formulae. In classical logic, complementarity means that the ....

[Article contains additional citation context not shown here]

J. Otten, C. Kreitz. A Uniform Proof Procedure for Classical and Non-Classical Logics KI-96, LNAI 1137, pp. 307--319, Springer, 1996.

....connection method for classical logic [2, 3, 5] avoid many kinds of redundancies contained in sequent calculi and yield a compact representation of the search space. They have been extended successfully to intuitionistic and modal logics [24] and serve as a basis for a uniform proof search method [20] and a method for translating matrix proofs back 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 ....

....can be found in [17] The Characterization. The characterization theorem proven in this 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. ....

[Article contains additional citation context not shown here]

J. Otten & C. Kreitz. A Uniform Proof Procedure for Classical and Non-classical Logics. KI-96: Advances in Artificial Intelligence, LNAI 1137, pp. 307--319. Springer, 1996.

....calculus and tableaux proof search methods. Originally developed as foundation of Bibel s connection method for classical logic [2,4] they have later been extended to nonclassical logics by Wallen [27] Wallen s formulation serves as a basis of a uniform proof method for a rich variety of logics [19,21] and also allows to transform matrix proofs into sequent style proofs by a uniform procedure [23,24] By Wallen s conjecture [27] matrix methods can be developed for any logic which has the same primary properties as classical logic. The linear connection method [3] has demonstrated that matrix ....

....logical connectives. Once a complementary connection has been identi ed all paths containing this connection are deleted. This is similar to Bibel s connection method for classical logic and formulas in clausal form [4] The theoretical basis of the following algorithm is described in detail in [21] where it is used for proof search in classical, intuitionistic and modal logics. Only a few modi cations were necessary to adapt it to MLL. De nition 7 ( related, related) Two positions u and v are related , denoted u v u v, i u6=v and the greatest common ancestor of u and v, wrt. ....

[Article contains additional citation context not shown here]

J. Otten, C. Kreitz. A uniform proof procedure for classical and non-classical logics. KI-96: Advances in Articial Intelligence, LNAI 1137, pp. 307-319, Springer Verlag, 1996.

....be simulated sufficiently well on a computer. Recently the characterizations of logical validity which underly these systems have been extended to intuitionistic logic and the modal logics K;K4;D;D4;T;S4, and S5 [16, 23, 24] On this basis the existing proof methods have been extended accordingly [17, 18] in order to develop a coherent theorem prover which can deal with a variety of logics and many applications which require mathematical reasoning. The use of automated theorem proving (ATP) in practical applications, however, causes another problem that needs to be solved. The efficiency of proof ....

....from application Uniform Proof Procedure Uniform Conversion Procedure Integration of solutions in application Fig. 1. Integrating Automated Theorem Proving into Application Systems Since currently only matrix based proof methods are able to handle different logics in a uniform way [18] the starting point of our procedure will be a proof according to Wallen s matrix characterizations [24] of logical validity. Here a formula F is valid if every path through a matrix representation of F contains at least one pair of atomic formulae which are complementary . In classical logic, ....

[Article contains additional citation context not shown here]

J. Otten, C. Kreitz. A Uniform Proof Procedure for Classical and Non-Classical Logics KI-96, LNAI 1137, pp. 307--319, Springer, 1996.

.... by the Adolf Messer Stiftung 2 Or via web http: aida.intellektik.informatik.th darmstadt.de jeotten ileantap 2 The Program We assume the reader to be familiar with free variable tableaux [4] and the leanTAP code (see [1, 2] as well as with some details of Wallen s approach (see [10] or [7, 5]) The following Prolog implementation is of course not as lean and compact as the original code of leanTAP . The logical connectives and quantifiers of intuitionistic logic need a separate treatment and we can not make use of any negation normal form. 3 We divide the description of the ....

.... A,C ) immediately yields a program implementing a theorem prover for the first order modal logic S4 (where [ and represent the corresponding modal operators) Modifying the algorithm for T string unification accordingly leads also to provers for the modal logics D, D4, S5, and T (see [6, 7] for details) Of course, there is still room for further research. The current implementation is not a decision procedure for the propositional intuitionistic logic (which is decidable) since we need multiplicities already in this fragment. For example it would be interesting to integrate some ....

J. Otten, C. Kreitz. A Uniform Proof Procedure for Classical and Non-Classical Logics. KI-96: Advances in Artificial Intelligence, LNAI, Springer Verlag, 1996.

....calculus and tableaux proof search methods. Originally developed as foundation of Bibel s connection method for classical logic [2, 3] they have later been extended to nonclassical logics by Wallen [26] Wallen s formulation serves as a basis of a uniform proof method for a rich variety of logics [18, 20] and also allows to transform matrix proofs into sequent style proofs by a uniform procedure [22, 23] By Wallen s conjecture matrix methods can be developed for any logic which has the same primary properties as classical logic [26] The desire for a matrix characterization of linear logic has ....

....logical connectives. Once a complementary connection has been identified all paths containing this connection are deleted. This is similar to Bibel s connection method for classical logic and formulas in clausal form [3] The theoretical basis of the following algorithm is described in detail in [20] where it is used for proof search in classical, intuitionistic and modal logics. Only a few modifications were necessary to adapt it to MLL. Definition7 (ff related, fi related) Two positions u and v are ff fi related , denoted u ff v u fi v, iff u6=v and the greatest common ancestor of u and ....

[Article contains additional citation context not shown here]

J. Otten, C. Kreitz. A uniform proof procedure for classical and non-classical logics. KI-96: Advances in Artificial Intelligence, LNAI 1137, pp. 307--319, 1996.

....and goal oriented connection based provers. Unfortunately the extension to first order logic cannot be done so easily, since copying of appropriate subformulas is a difficult task. A non clausal prover can also be extended to some non classical logics, like intuitionistic, modal or linear logic [16,8]. Thus leanCoP can serve as a basis for lean connection based theorem provers for logics for which up to now only lean tableau based provers [17,12] have been realized. ....

J. Otten and C. Kreitz. A uniform proof procedure for classical and non-classical logics. KI-96: Advances in Artificial Intelligence, LNAI 1137, pp. 307--319, 1996.

....consists of an algorithm which checks the complementarity of all paths and uses an additional string unification procedure to unify the prefixes. Searching for a spanning set of connections is done by a general path checking algorithm which is driven by connections instead of logical connectives [28, 30]. Once a complementary connection has been identified all paths containing this connection are deleted. This is similar to Bibel s connection method for 3 The polarity of an atomic formula is either 0 or 1 and indicates whether it would occur negated (polarity 1) in the negational normal form or ....

....can only occur within 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 ....

[Article contains additional citation context not shown here]

J. Otten and C. Kreitz. A Uniform Proof Procedure for Classical and Non-classical Logics. In G. Gorz & S. Holldobler, eds., KI-96: Advances in Artificial Intelligence, LNAI 1137, pp. 307--

....connection method for classical logic [2,3,5] avoid many kinds of redundancies contained in sequent calculi and yield a compact representation of the search space. They have been extended successfully to intuitionistic and modal logics [24] and serve as a basis for a uniform proof search method [20] and a method for translating matrix proofs back 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 ....

....can be found in [17] The Characterization. The characterization theorem proven in this 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. ....

[Article contains additional citation context not shown here]

J. Otten & C. Kreitz. A Uniform Proof Procedure for Classical and Non-classical Logics. KI-96: Advances in Articial Intelligence, LNAI 1137, pp. 307-319. Springer, 1996.

....of atomic formulae that may 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 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 ....

....refiner or as mechanism 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 ....

J. Otten and C. Kreitz. A Uniform Proof Procedure for Classical and Non-classical Logics. In G. Gorz & S. Holldobler, eds., KI-96: Advances in Artificial Intelligence, LNAI 1137, pp. 307--319. Springer, 1996.

.... mainly for classical logic and formulas in clauseform, their theoretical 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 ....

....a very compact prover but compares favourable with other (larger) implementations. 7 Conclusion We have presented a uniform proof search procedure for classical and nonclassical logics that generalizes our previously developed proof procedures for intuitionistic [Otten and Kreitz, 1995] modal [Otten and Kreitz, 1996(b)] and multiplicative linear logic [Kreitz et al. 1997] It is based on a unified representation of matrix characterizations for logical validity, which enables us to abstract from the semantical differences between various logics and to focus on structural similarities during proof search. Our ....

J. Otten and C. Kreitz. A uniform proof procedure for classical and non-classical logics. KI-96: Advances in Artificial Intelligence, LNAI 1137, pp. 307--319, 1996.

.... of the extension procedure, a non normal form Davis Putnam prover for classical propositional logic as well as an inference machine which implements the non normal form version of the extension procedure thereby allowing for the use of various co processors for first order non classical logics [OK96b]. The output layer consists of various techniques which present the generated proofs within natural and logistic calculi for the respective input logic. Obviously their application depends on the way the input formula was processed. If, for instance, the formula was transformed by a logic morphism ....

....a readable output, i.e. to present the proof within sequent or natural deduction calculi. For this purpose we have developed a uniform framework for non normal form theorem proving. The resulting method basically consists of a two step algorithm, i.e. a uniform procedure finding the proofs [OK96b] and a uniform transformation procedure converting these proofs into sequent style systems [SK96] During the development of the algorithm we have put particular emphasis on the uniformity of our environment. First of all we have to unify all logics under consideration into one system of matrix ....

[Article contains additional citation context not shown here]

J. Otten, C. Kreitz. A Uniform Proof Procedure for Classical and Non-Classical Logics. To appear in KI--96, Springer, 1996.

....and other such systems is limited by its low degree of automation. In order to overcome this drawback, we suggest to incorporate and combine logical and inductive theorem proving techniques. Logical proof search methods have been successfully used for proving formulas in first order logic. Otten [16] and Schmitt [17] show we can use efficient proof search methods like matrix methods in the framework of NuPRL. A major drawback of these methods is that they cannot deal with induction and inductive proof obligations. Inductive theorem proving techniques, on the other hand, have been very ....

....sequent prover for the logical decomposition of a given formula. For (constructive) first order logic, however, there are well known proof search methods [15, 19] which are much more efficient than sequent based proof search. Among those, matrix based proof techniques such as the connection method [4, 16] can be understood as a very compact representation of sequent proof techniques. They avoid the usual redundancies contained in the sequent calculus and are driven by complementary connections , i.e. pairs of atomic formulae that may become leaves in a sequent proof, instead of the logical ....

[Article contains additional citation context not shown here]

Otten, J. and Kreitz, C.: A Uniform Proof Procedure for Classical and Non-classical Logics. In G. Gorz & S. Holldobler, eds., KI-96: Advances in Artificial Intelligence, LNAI 1137, Springer, 1996, 307--319.

....connection method for classical logic [2,3,5] avoid many kinds of redundancies contained in sequent calculi and yield a compact representation of the search space. They have been extended successfully to intuitionistic and modal logics [24] and serve as a basis for a uniform proof search method [20] and a method for translating matrix proofs back 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 ....

....can be found in [17] The Characterization. The characterization theorem proven in this 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. ....

[Article contains additional citation context not shown here]

J. Otten & C. Kreitz. A Uniform Proof Procedure for Classical and Non-classical Logics. KI-96: Advances in Artificial Intelligence, LNAI 1137, pp. 307--319. Springer, 1996.

....simulated sufficiently well on a computer. Recently the characterizations of logical validity which underly these systems have been extended to intuitionistic logic and the modal logics K;K4;D;D4;T ; S4, and S5 [17, 24, 25] On this basis the existing proof methods have been extended accordingly [18, 19] in order to develop a coherent theorem prover which can deal with a variety of logics and many applications which require mathematical reasoning. The use of automated theorem proving (ATP) in practical applications, however, causes another problem that needs to be solved. The efficiency of proof ....

....of proof tasks from application Uniform Proof Procedure Uniform Conversion Procedure Integration of solutions in application Fig. 1. Integrating Automated Theorem Proving into Application Systems Since currently only matrix based proof methods are able to handle different logics in a uniform way [19] the starting point of our procedure will be a proof according to Wallen s matrix characterizations [25] of logical validity. Here a formula F is valid if every path through a matrix representation of F contains at least one pair of atomic formulae which are complementary . In classical logic, ....

[Article contains additional citation context not shown here]

J. Otten, C. Kreitz. A Uniform Proof Procedure for Classical and Non-Classical Logics KI-96, LNAI 1137, pp. 307--319, Springer, 1996.

....is either 0 or 1 and indicates whether it would occur negated (polarity 1) in the negational normal form or not (polarity 0) Searching for a spanning set of connections. Proof search is done by a general path checking algorithm which is driven by connections instead of logical connectives [30, 32]. Once a complementary connection has been identified all paths containing this connection are deleted. This is similar to Bibel s connection method for classical logic but without necessity for transforming the given formula to normal form. Dealing with arbitrary formulas is necessary since there ....

....can only occur within 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 ....

[Article contains additional citation context not shown here]

J. Otten & C. Kreitz. A Uniform Proof Procedure for Classical and Non-classical Logics. In G. Gorz & S. Holldobler, eds., KI-96: Advances in Artificial Intelligence, LNAI 1137, pp. 307--

....analysis and to solve synthesis problems like the above automatically. For (constructive) first order logic, however, there are well known proof methods [19, 25] which are much more efficient than sequent based proof search. Among those, matrixbased proof techniques such as the connection method [4, 20] can be understood as a very compact representation of sequent proof techniques. They avoid the usual redundancies contained in the sequent calculus and are driven by complementary connections , i.e. possible leaves in a sequent proof, instead of the logical connectives of a proof goal. On the ....

....of atomic formulae that may 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 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 ....

[Article contains additional citation context not shown here]

J. Otten and C. Kreitz. A Uniform Proof Procedure for Classical and Non-classical Logics. In G. Gorz & S. Holldobler, eds., KI-96: Advances in Artificial Intelligence, LNAI 1137, pp. 307--319. Springer, 1996.

....proof goal. Although developed mainly for classical 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 ....

....number of most general unifiers is finite but may grow exponentially with the length of the prefixes. 7 Conclusion We have presented a uniform proof search procedure for classical and nonclassical logics that generalizes our previously developed proof procedures for intuitionistic [33] modal [35], and multiplicative linear logic [20] It is based on a unified representation of matrix characterizations for logical validity, which enables us to abstract from the semantical differences between various logics and to focus on structural similarities during proof search. Our procedure consists ....

J. Otten and C. Kreitz. A uniform proof procedure for classical and non-classical logics. KI-96: Advances in Artificial Intelligence, LNAI 1137, pp. 307--319, 1996.

....calculus and tableaux proof search methods. Originally developed as foundation of Bibel s connection method for classical logic [2, 4] they have later been extended to nonclassical logics by Wallen [27] Wallen s formulation serves as a basis of a uniform proof method for a rich variety of logics [19, 21] and also allows to transform matrix proofs into sequent style proofs by a uniform procedure [23, 24] By Wallen s conjecture [27] matrix methods can be developed for any logic which has the same primary properties as classical logic. The linear connection method [3] has demonstrated that matrix ....

....logical connectives. Once a complementary connection has been identified all paths containing this connection are deleted. This is similar to Bibel s connection method for classical logic and formulas in clausal form [4] The theoretical basis of the following algorithm is described in detail in [21] where it is used for proof search in classical, intuitionistic and modal logics. Only a few modifications were necessary to adapt it to MLL. Definition 7 (ff related, fi related) Two positions u and v are ff fi related , denoted u ff v u fi v, iff u6=v and the greatest common ancestor of u ....

[Article contains additional citation context not shown here]

J. Otten, C. Kreitz. A uniform proof procedure for classical and non-classical logics. KI-96: Advances in Artificial Intelligence, LNAI 1137, pp. 307--319, Springer Verlag, 1996.

....into characterizations of validity for intuitionistic logic (J) and the modal logics K;K4;D;D4;T;S4, and S5. Since then attempts have been undertaken to extend the existing proof methods accordingly in order to create efficient proof procedures for intuitionistic logic [12] and modal logics [13]. The efficiency of automated proof methods strongly depends on a compact representation of a proof. The characterizations of logical validity, on which these methods are based, avoid the notational redundancies contained in mathematical languages or sequent calculi. As a result, an automatically ....

.... domain variants concerning the Kripke semantics of these logics [6, 7, 18] These characterizations have been adopted from [18] which again can be viewed as an extension of Bibel s matrix method [4] Within this paper we focus on the basic ideas and syntactical concepts and refer to [18] or [13] for details. Position trees, Types, and Prefixes The basic structure for representing matrix proofs is a tree ordering which will be constructed from a formula tree. We classify a formula A and its sub formulae according to the tableaux scheme in table 1. We use the concept of signed formulae ....

[Article contains additional citation context not shown here]

J. Otten, C. Kreitz. A Uniform Proof Procedure for Classical and Non-Classical Logics Technical Report, TH Darmstadt, 1996.

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