A. Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Univ. Paul-Sabatier, Toulouse, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... calculi and known cut free Gentzen systems for these logics (see e.g. 11] An alternative approach is to translate propositional modal logics into classical firstorder logic since this allows us to use the wealth of knowledge in first order theorem proving to mechanize modal deduction (see e.g. [17, 20, 12, 5]) Let FO n be the fragment of classical first order logic using at most n individual variables and no function symbols. Any modal logic characterized by a first order definable class of modal frames can be translated into FO n where n 2 is the number of variables in the first order formula ....

A. Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Universit'e P. Sabatier, Toulouse, 1989.

....mention the goal of identifying the modal fragment of rstorder logic as a motivation for introducing the guarded fragment. A survey on translation methods for modal logics can be found in [ONdRG01] where more references are provided, for instance about the functional translation (see e.g. [Her89,Ohl93]) Apart from satisfying nice logical properties [ANB98] the guarded fragment GF has an EXPTIME complete satis ability problem, when the maximal arity of the predicate symbols is xed in advance [Gr a99b] Hence its worst case complexity is identical to some simple extensions of modal logic K, ....

A. Herzig. Raisonnement automatique en logique modale et algorithmes d'uni cation. PhD thesis, Universite Paul Sabatier, Toulouse, 1989.

....overview. Some important topics omitted in this overview include the following. Non standard translation approaches. Non standard translation methods include reductions derived from the functional semantics of normal modal logics with unparameterised modalities, namely the functional translation [1, 35, 59], the optimised functional translation [35, 63, 76] and the semi functional translation [56] Surveys of the di#erent translation methods of modal logics and other non classical logics are [60, 61, 62] These non standard translation approaches are all implemented in mspass [46] Experience shows ....

....include the following. Non standard translation approaches. Non standard translation methods include reductions derived from the functional semantics of normal modal logics with unparameterised modalities, namely the functional translation [1, 35, 59] the optimised functional translation [35, 63, 76] and the semi functional translation [56] Surveys of the di#erent translation methods of modal logics and other non classical logics are [60, 61, 62] These non standard translation approaches are all implemented in mspass [46] Experience shows that the performance of first order theorem provers ....

A. Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Univ. Paul-Sabatier, Toulouse, 1989.

....is lost during transformation to clausal form and subsequent deduction. By using a di erent translation method, for Resolution for Testing Modal Satis ability and Building Models 3 example, translation methods based on the functional translation where accessibility is encoded in terms of paths [2, 19, 36], this problem can be reduced and eliminated for certain logics [9, 12] There is another well known technique, called structural transformation or renaming, which enables the preservation of the structure of the original formula in the clausal form. This technique is used in connection with most ....

Herzig, A.: 1989, `Raisonnement automatique en logique modale et algorithmes d'unication.'. Ph.D. thesis, Univ. Paul-Sabatier, Toulouse.

....infinitely many non redundant resolvents are produced. We use a different translation, namely an optimised version of the functional translation for which infinite computation does not occur, as we will show. The functional translation method was proposed independently by Ohlbach (1988, 1991) Herzig (1989) and Auffray and Enjalbert (1992) for modal predicate logics and by Zamov (1989) for propositional modal logics. The optimised functional translation method applies only to propositional modal logics and is due to Herzig (1989) for non axioms (it is also implicit in Zamov (1989) Ohlbach and ....

....translation method was proposed independently by Ohlbach (1988, 1991) Herzig (1989) and Auffray and Enjalbert (1992) for modal predicate logics and by Zamov (1989) for propositional modal logics. The optimised functional translation method applies only to propositional modal logics and is due to Herzig (1989) for non axioms (it is also implicit in Zamov (1989) Ohlbach and Schmidt (1995) show the optimisation can also be applied to axioms. Important for this paper is, that the optimised functional translation eliminates in the clause forms all Skolem functions other than Skolem constants. The ....

Herzig, A. (1989), Raisonnement automatique en logique modale et algorithmes d'unification. , PhD thesis, Universit'e Paul-Sabatier, Toulouse.

....from lwffs, rwffs and lterms, i.e. Gamma; Delta; Theta L w:A, i) rwffs are derived from rwffs alone, and (ii) lterms are derived from rwffs and lterms but not from lwffs. In comparison, note that in approaches based on semantic embedding, also called semantics based translations, e.g. [1,13,17], a first order modal formula is translated into a formula in first order predicate logic and shown to be true (or false) in a first order theory formalizing the semantics of the modalities and quantification domains. However, with these translations all structure is lost as relations, predicates, ....

....of assumptions. 17 As a consequence, unlike our approach which leads to simple implementations, these systems cannot be directly formalized in standard logical frameworks such as Isabelle [18] or the Edinburgh LF [12] Our work is closely related to approaches based on semantic embeddings, e.g. [1,13,17]. In these approaches, a formula of quantified modal logic is translated into a formula of first order predicate logic, and shown to be true (or false) in a theory formalizing the semantics of the modalities and domains of quantification. For example, 2(A B) would be translated into a formula ....

A. Herzig. Raisonnement Automatique en Logique Modale et Algorithmes d'Unification. PhD thesis, Universit'e Paul-Sabatier, Toulouse, 1989.

....semantics, i.e. w:A iff w 0 :A for all w 0 2 W accessible from w. Then by formalizing the details of particular accessibility relations we can produce particular modal logics. This treatment has obvious similarities to traditional semantic embedding (i.e. translation into predicate logic [20, 24, 25]) but it offers substantial advantages: our formalization does not require all of first order logic and it yields structured ND systems where the strong separation between the base logic and the relational theory gives us better proof normalization results (cf. Theorem 30 and its commentary) In ....

....lwffs may depend, via rules for non local connectives, on derivations of rwffs, but not vice versa. III) Derivations normalize: the derivations of lwffs have a well structured normal form that satisfies the subformula property. Xi For comparison, consider the semantic embedding approach (e.g. [20, 24, 25]) in which a non classical logic is encoded as a suitable (e.g. intuitionistic or classical) first order theory by axiomatizing an appropriate definition of truth: i) a non classical logic constitutes a theory of full first order logic, as opposed to an extension of labelled propositional ....

A. Herzig. Raisonnement Automatique en Logique Modale et Algorithmes d'Unification. PhD thesis, Universit'e Paul-Sabatier, Toulouse, 1989.

....formulae into a set of rst order formulae, ii) transformation into clausal form, and (iii) saturation based resolution. In the current implementation of mspass the available translation methods include: the standard relational translation method, the functional translation method [1, 5, 14], the optimised functional translation method [15] including a variation de ned in terms of n ary predicates [7] as well as the semi functional translation method [12] The implementation of the relational translation method is most general and applies to the language described in ....

A. Herzig. Raisonnement automatique en logique modale et algorithmes d'unication. PhD thesis, Univ. Paul-Sabatier, Toulouse, 1989.

....with (15) Instead we use the derived equivalences (16) and (17) wherever possible first for rewriting quantifications as a whole. Applied to the modal logic case, our K transformation turns the relational translation of modal logic into predicate logic into the functional translation (c.f. [Wal87, Ohl88a, JR88, Her89, AE92, Ohl90, Gas92, Ohl93, Zam89]) tr f (2 ; w) 8fl:AF tr f ( fl(w) tr f (3 ; w) 9fl:AF tr f ( fl(w) The transformer (15) can be used in exactly the same way for optimizing the treatment of varying domains in the translation of quantified modal logics. The normal translation rules for the quantifiers in the ....

Andreas Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Universit'e Paul-Sabatier, Toulouse, 1989.

....We will show that in a functional language (which we ll use to replace the relational language) moving existential quantifiers outward over universal quantifiers is allowed in certain cases. The idea for this was first mentioned by Herzig and others (Fari nas del Cerro, Luis and Herzig 1988a, Herzig 1989). The functional language Every relation can be decomposed into a set of functions. The accessibility relation can therefore be decomposed into a set AF of accessibility functions mapping worlds to accessible worlds. Consider the relation R 1 : 3 There is a way of reconstructing quantifiers ....

....briefly recall how modal formulae can be translated into (first order or second order) predicate logic based on their relational semantics. We show how this relational translation can be transformed according to the functional translation (Ohlbach 1988a, Fari nas del Cerro, Luis and Herzig 1988b, Herzig 1989, Auffray and Enjalbert 1992, Zamov 1989) Then we prove a theorem that justifies the application of the rule for moving existential quantifiers outward over universal quantifiers. Finally, we show how the scan algorithm can be used to compute the frame properties in the functional language. 3 ....

Herzig, A. (1989), Raisonnement automatique en logique modale et algorithmes d'unification., PhD thesis, Universit'e Paul-Sabatier, Toulouse.

.... translational approach consists in translating modal logics into logics for which theorem provers already exist, typically classical first order logic (FOL) The relational translation into FOL (see e.g. Mor76, Sch99, GHM98] is the most common such translation although not the only one (see e.g. [Mor76, Ohl88, Her89, dMP95, Ohl98]) These two approaches cannot always be applied with equal success (see e.g. HS97] For instance, for the provability logic G which is characterized by a second order class of modal frames (see e.g. Boo93] the relational translation is not possible unless FOL is augmented with fixedpoint ....

A. Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Universit'e P. Sabatier, Toulouse, 1989.

....very often infinitely many non redundant resolvents are produced. We use a different translation, namely an optimised version of the functional translation for which infinite computation does not occur, as we will show. The functional translation method was proposed independently by Ohlbach 1988, Herzig 1989 and Auffray and Enjalbert 1992 for modal predicate logics and by Zamov 1989 for propositional modal logics. The optimised functional translation method applies only to propositional modal logics and is due to Ohlbach and Herzig (and implicitly also to Zamov) Ohlbach and Schmidt 1995 deal with ....

Herzig, A. 1989. Raisonnement automatique en logique modale et algorithmes d'unification. Doctoral dissertation, Univ. Paul-Sabatier, Toulouse.

....between tableaux calculi and known cut free Gentzen systems for these logics. An alternative approach is to translate propositional modal logics into classical first order logic since this allows us to use the wealth of knowledge in firstorder theorem proving to mechanize modal deduction (see e.g. [Mor76, Ohl88, Her89, dMP95, Non96, Ohl98]) Let FO n be the fragment of classical first order logic using at most n individual variables and no function symbols. Any modal logic characterized by a first order definable class of modal frames can be translated into FO n for some fixed n 2. The decidable modal logic K4, for example, is ....

A. Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Universit'e P. Sabatier, Toulouse, 1989.

....(9ffi z = ffi(x) 9 z = y) 8x; y; z 8fl; ffi 9 y = fl(x) z = ffi(x) z = y) 8x; y; z 8fl; ffi 9 ffi(x) fl(x) Applied to the modal logic case, our K transformation turns the relational translation of modal logic into predicate logic into the functional translation (c.f. [Wal87a, Ohl88a, JR88, Her89, AE92, Ohl90, Gas92, Ohl93, Zam89]) serial: tr f (2 ; w) 8fl:AF tr f ( fl(w) tr f (3 ; w) 9fl:AF tr f ( fl(w) non serial: tr f (2 ; w) End(w) 8fl:AF tr f ( fl(w) tr f (3 ; w) End(w) 9fl:AF tr f ( fl(w) The transformer (29) can be used in exactly the same way for optimizing the treatment of ....

Andreas Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Universit'e Paul-Sabatier, Toulouse, 1989.

....from lwffs, rwffs and lterms, i.e. Gamma; Delta; Theta L w:A, i) rwffs are derived from rwffs alone, and (ii) lterms are derived from rwffs and lterms but not from lwffs. In comparison, note that in approaches based on semantic embedding, also called semantics based translations, e.g. [1, 13, 17], a first order modal formula is translated into a formula in first order predicate logic and shown to be true (or false) in a first order theory formalizing the semantics of the modalities and quantification domains. However, with these translations all structure is lost as relations, predicates, ....

....of assumptions. 17 As a consequence, unlike our approach which leads to simple implementations, these systems cannot be directly formalized in standard logical frameworks such as Isabelle [18] or the Edinburgh LF [12] Our work is closely related to approaches based on semantic embeddings, e.g. [1, 13, 17]. In these approaches, a formula of quantified modal logic is translated into a formula of first order predicate logic, and shown to be true (or false) in a theory formalizing the semantics of the modalities and domains of quantification. For example, 2(A B) would be translated into a formula ....

A. Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Universit'e Paul-Sabatier, Toulouse, 1989.

....are produced. We use a different translation, namely an optimised version of the functional translation for which infinite computation does not occur, as we will show. The functional translation method was proposed independently by Ohlbach (1988a, 1988b) Fari nas Del Cerro and Herzig (1988) Herzig (1989) and Auffray and Enjalbert (1992) for modal predicate logics. Zamov (1989) claims to have a decision procedure for the logic S4 using lock resolution. The optimised functional translation method applies only to propositional modal logics and is due to Ohlbach and Herzig (and it is also implicit in ....

Herzig, A. (1989), Raisonnement automatique en logique modale et algorithmes d'unification., PhD thesis, Univ. Paul-Sabatier, Toulouse.

.... calculi and known cut free Gentzen systems for these logics (see e.g. 11] An alternative approach is to translate propositional modal logics into classical firstorder logic since this allows us to use the wealth of knowledge in first order theorem proving to mechanize modal deduction (see e.g. [17, 20, 12, 5]) Let FO n be the fragment of classical first order logic using at most n individual variables and no function symbols. Any modal logic characterized by a first order definable class of modal frames can be translated into FO n where n 2 is the number of variables in the first order formula ....

A. Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Universit'e P. Sabatier, Toulouse, 1989.

....facilitated by theory resolution via the socalled functional translation or its variation for propositional modal logics, the optimised functional translation approach. The functional translation method was proposed independently in the late eighties by a number of groups. Fari nas del Cerro and Herzig (1989, 1995) describe a transformation of quantified modal logics into so called deterministic logics and use a modal resolution calculus. Ohlbach (1988, 1991) and Auffray and Enjalbert (1992) embed quantified modal logics into fragments of first order logic and employ first order resolution theorem ....

....Enjalbert (1992) embed quantified modal logics into fragments of first order logic and employ first order resolution theorem proving. Zamov (1989) describes a lock decision procedure for the translation of S4. All procedures involve theory unification. The optimised functional translation method (Herzig 1989, Ohlbach and Schmidt 1997) applies to propositional normal modal logics and gives rise to a class of path logics, which this paper considers. Very much like modal logics, path logics form a lattice with the weakest being the basic path logic associated with the basic modal logic K and also KD. ....

Herzig, A. (1989), Raisonnement automatique en logique modale et algorithmes d'unification., PhD thesis, Univ. Paul-Sabatier, Toulouse.

....i.e. w:2A iff w 0 :A for all w 0 2 W accessible from w. Then by formalizing the details of particular accessibility relations we can produce particular modal logics. This treatment has obvious similarities to traditional semantic embeddings (i.e. translations into predicate logic [27, 35, 36]) but it offers substantial advantages: our formalization does not require all of first order logic and it yields structured ND systems where the separation between the base logic and the relational theory gives us better proof normalization results (see Theorem 29 and its commentary) In [5] we ....

....of lwffs may depend, via rules for non local connectives, on derivations of rwffs, but not vice versa. iii) Derivations normalize: the derivations of lwffs have a well structured normal form that satisfies the subformula property. For comparison, consider the semantic embedding approach, e.g. [27, 35, 36], in which a non classical logic is encoded as a suitable (e.g. intuitionistic or classical) first order theory by axiomatizing an appropriate definition of truth: i) a non classical logic with Horn axiomatizable properties of the relations constitutes a theory of full first order logic, as ....

A. Herzig. Raisonnement Automatique en Logique Modale et Algorithmes d'Unification. PhD thesis, Universit'e Paul-Sabatier, Toulouse, 1989.

....of is a theorem in the new logic. This translation is only an intermediary step in a translation to predicate logic. In the second step, we translate the multi modal logic into a predicate logic using the functional translation of (Ohlbach 1988, Ohlbach 1991, Fari nas del Cerro and Herzig 1988, Herzig 1989, Auffray and Enjalbert 1992, Zamov 1989) The reason for using the functional translation instead of the usual relation translation is this: The multi modal logic of graded modalities can have frame properties that are not first order definable in terms of R n relations. However, the frame ....

....standard relational translation we use the functional translation as proposed in Ohlbach and Schmidt (1995) for non first order axioms like McKinsey s axiom. The functional translation method was proposed by various authors, for example Ohlbach (1988, 1991) Fari nas del Cerro and Herzig (1988) Herzig (1989), Auffray and Enjalbert (1992) and Zamov (1989) It exploits the fact that every binary relation can be decomposed into a set AFR of functions, called accessibility functions. Any (non empty) relation R is defined by: R(x; y) 9fl 2 AFR y = fl(x) In the functional translation we quantify over ....

Herzig, A. (1989), Raisonnement automatique en logique modale et algorithmes d'unification., PhD thesis, Univ. Paul-Sabatier, Toulouse.

No context found.

A. Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Univ. Paul-Sabatier, Toulouse, 1989.

No context found.

A. Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Universite Paul-Sabatier, Toulouse, 1989.

No context found.

A. Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Univ. Paul-Sabatier, Toulouse, 1989.

No context found.

A. Herzig. Raisonnement automatique en logique modale et algorithmes d'unification. PhD thesis, Univ. Paul-Sabatier, Toulouse, 1989.

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