M. Kracht. Tools and Techniques in Modal Logic, vol. 142 of Studies in Logic. Elsevier, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... to treat the general case) We are currently investigating if their map can be generalised by considering the map f # : FML 0, 1 FML inductively defined as f except that f # (2#, 1) 2(F(#) f # (#, 1) Another map in polynomial time from G into K4 is given in [Fit83, Chapter 5] Kracht [Kra99] notes that such maps exist for nearly all classical logics. 5 Pseudo Gentzenisable Logics Theorem 14 admits a natural counterpart when LF has a traditional Gentzen style calculus, see forthcoming Theorem 23. However, the DL framework appears to be much more flexible. Typically, #LF is sound ....

M. Kracht. Tools and Techniques in Modal Logic. Elsevier, 1999.

.... (m) to global satisfiability in K (m) or satisfiability in K (m) enhanced with the universal modality (which can be further reduced to local satisfiability in K (m) On a modal level the axiomatic translation is closely related to the reduction functions introduced independently by Kracht in [10] and developed further in [11] Kracht s reduction functions are defined di#erently, the most important di#erence being that they do not use new symbols for subformulae. Renaming would lead to an improvement of complexity of the reduction functions from polynomial to linear complexity. Kracht has ....

M. Kracht. Tools and Techniques in Modal Logic, vol. 142 of Studies in Logic. Elsevier, 1999.

....n be the square algebra (P (n n) E 1 , E 2 , D) Consider the family # = n## . From Lemma 4.1 it follows that # forms a # anti chain. For any subset # of #, let V # denote the variety generated by #. Using the standard splitting technique (see, e.g. That is V # = HSP(#) Kracht [6] for details) we can easily show that V # V # # whenever # # # . Therefore, there exist 2 # many subvarieties of RCA 2 . For n 1 let n denote the finite cylindric space obtained from the nn square by substituting a singleton non diagonal E 0 cluster by a two element E 0 cluster. ....

....# is a cover of V and V is finitely generated, then there are only finitely many subvarieties of V # , hence V # is finitely generated by Theorem 7.2. 2) The proof is analogous to the standard proof that a finitely generated variety of K4 algebras has only finitely many covers (see, e.g. Kracht [6] or Chagrov and Zakharyaschev [3] 7.1 Varieties of cylindric algebras of depth one In this subsection we give a complete charcterization of the lattice structure of the varieties of cylindric algebras of depth one. Let 2 denote the 2 element Df 2 algebra, where n i (a) 0 if a=0, 1 ....

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M. Kracht, Tools and Techniques in Modal Logic, North-Holland, 1999.

....term may have more than one G normal form. Proposition 9. Let G be a # base of E and t T (#, G) If G is recursive, there is an e#ective way of computing a term s(v) T (#, V ) and a sequence r of terms in G such that t = s(r) 2. 2 Constructors and Modal Logics For all normal modal logics [9], equivalence of formulae is a congruence relation on formulae that is closed under substitution [9] For example, consider the basic modal logic K. Here, the signature #K contains the Boolean operators (#, #, the Boolean constant (for truth) and the unary (modal) operator . Equivalence of ....

....If G is recursive, there is an e#ective way of computing a term s(v) T (#, V ) and a sequence r of terms in G such that t = s(r) 2. 2 Constructors and Modal Logics For all normal modal logics [9] equivalence of formulae is a congruence relation on formulae that is closed under substitution [9]. For example, consider the basic modal logic K. Here, the signature #K contains the Boolean operators (#, #, the Boolean constant (for truth) and the unary (modal) operator . Equivalence of formulae in K can be axiomatized [10] by the equational theory EK , which consists of the ....

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M. Kracht. Tools and Techniques in Modal Logic. Elsevier, Amsterdam, The Netherlands, 1999.

....and the consequence relation are then defined as follows. A formula # is said to be true in a model M, abbreviated by M #, if # is true in every pointed model (M,w) based on M; and # follows globally from #, if for all models M with M # it holds that M #. As several authors have observed [5, 13, 22], a large number of logical properties, like completeness, canonicity, finite model property, and interpolation, come in two flavors: a local one and a global one. For instance, the notions of local and global consequence do not coincide: ## follows locally from #, but not globally, yet the ....

....and, second, it makes use of the notion of # saturated models which restricts its applicability to modal languages that lie inside first order logic. However, the result may serve as a good starting point for further work. In this context, Kracht and Wolter s work on transfer results [13, 14] deserves attention. They investigate the metalogical properties of modal logics, thereby considering both levels, the local and the global one. Though they take a 17 di#erent perspective on modal logic their aim is to explore the lattice of normal modal logic there are interesting ....

M. Kracht. Tools and Techniques in Modal Logic. Habilitationsschrift, Berlin 1996.

....without interaction axioms is the same as dovetailing. But difficulty arises with the extra interaction axiom. Then we need a more general concept like fibring. Our study starts with the assumption that the combination of two complete logics need not be complete when we add interaction axioms [8]. We want to identify conditions under which completeness can be preserved when we include interaction axioms like above. 2 BDI Multi Modal Logics The main advantage of using Multi Modal Logics in BDI is their ability to express complex modalities, that can capture the inter relationships ....

M. Kracht. Tools and Techniques for Modal Logics. Elsevier, 1999.

....instance, one can show (using some quite ad hoc method) that L 3 satis ability is EXPTIME hard with G = hf1g; f2; 3g; f1 2 2 3 3 1; 1 2g; 1i. G does not fall into any previous identi ed cases. Extensions of the EXPTIME lower bound to weakly transitive (poly)modal logics (see e.g. [Kra99]) are also expected. 42 8.3 Decidable criteria for classi cation Theorem 38, Theorem 40 and Theorem 43 provide sucient conditions to guarantee that a given regular grammar logic has an EXPTIME hard satis ability problem. Those results are partly satisfactory since no equivalence conditions ....

M. Kracht. Tools and Techniques in Modal Logic. Elsevier, 1999. 49

....b 2 D if ABox LI(A ; a; b) is de ned LI a : LI a u ABox LI(A ; a; b) GDC : GDC t LI a ; evaluate(GDC) T : T [ fD =GDCg; return T ; 2.2 Tableau Methods for Interpolation. The algorithms to calculate interpolants using logical tableaux presented here follow the lines of [7]. Interpolants for ALC concept subsumption can be constructed from a fully expanded closed tableau collecting contradicting literals on each branch using construction rules corresponding to traditional tableaux. For ABox interpolants the more complex interaction between role and concept ....

....Since concept interpolation is similar to interpolation in modal logic K and because of the close connection between ALC and K [11] we omit further details and assume that there is a procedure 7 concept LI(C 1 ; C 2 ) which calculates a concept interpolant for C 1 and C 2 . Please consult [12, 7] for further details. u) if (a : C 1 u C 2 ) 2 B, but not both (a : C 1 ) 2 B and (a : C 2 ) 2 B then B 0 : B [ f(a : C 1 ) a : C 2 )g. t) if (a : C 1 t C 2 ) 2 B, but neither (a : C 1 ) 2 B nor (a : C 2 ) 2 B. then B 0 : B [ fa : C 1 g and B 00 : B [ fa : C 2 g. 8) if (a : ....

M. Kracht. Tools and Techniques in Modal Logic. North Holland, 1999.

....destroy the bisimilarity and hence invalidate the counter examples, and perhaps restore interpolation and hence Beth de nability. The case is di erent for H( 3) and H(hR 1 i; 3) As we will now show, we can extend the constructive method for establishing arrow interpolation presented in (Kracht 1999, Section 3.8) to handle and 3. Again, we will use the normal form introduced in Proposition 4.3. Theorem 5.6 H( 3) and H(hR 1 i; 3) have arrow interpolation. Given that arrow interpolation implies global Beth de nability for these languages, implicit de nitions in H( 3) can be ....

Kracht, M. 1999. Tools and Techniques in Modal Logic. Amsterdam: NorthHolland Publishing Co.

....which attack its supporting sub claims, OE i . 17 14 They are called step warrants in Verheij s legal argumentation system [75] 15 Note that our arguments in this Section could be formalised with use of valuation functions and models, as these terms are used within mathematical logic [34, 55], and we are attempting this in [43] 16 However, even pure mathematicians may have variable belief in an assertion depending upon the means used to prove it. For example, constructivist mathematicians (e.g. 6, 70] do not accept inference based on proof techniques which purport to demonstrate ....

M. Kracht. Tools and Techniques in Modal Logic. North Holland, Amsterdam, The Netherlands, 1999. 19

....provable for each natural number n. Indeed, if Proof (n; F ) holds, then F is provable. If Proof (n; F ) does not hold then its negation 4 also known under the names G, GL, K4.W, PRL. 5 There are many adequate non provability models for S4 known: algebraic, topological, Kripke, etc. cf. 30] [55]) EXPLICIT PROVABILITY 7 :Proof (n; F ) is provable, since Proof (x; y) is a decidable relation. In both cases Proof (n; F ) F is provable. This consideration suggests the idea of developing an explicit provability logic by switching from the formal provability 9xProof (x; F ) to Proof(t,F) and ....

M. Kracht, Tools and techniques in modal logic, Elsevier, 1999.

....but not needed in sequel. For example, if the branches have well order type , the tree is countable, but if the well order type is at least 1 and countable, then jT j = 2 0 . The present paper assumes a fair amount of knowledge in modal logic. For background in modal logic we refer to [13], in which all notions relevant to this paper are explained. We assume that the reader knows the systems S5 and G and has some understanding of tense logic. We will consider not only modal logics of a single operator, but in fact logics with arbitrarily many operators; we only require that the set ....

.... ; i in the language of tense logic for both orders, where hR; i is the real line and hR; i a well order. Then has no countable models. In particular, Kz( 2 0 . The resulting logic is a 4 modal logic. To get a monomodal logic with these properties we invoke the simulation theorem from [13]. This theorem states that for every nite number k there is an isomorphism 7 s from the lattice of k modal logics onto an interval in the lattice of monomodal logics such that the property of completeness is left invariant. It is easy to see that Kz( s ) k Kz( k 1. Theorem ....

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Marcus Kracht. Tools and Techniques in Modal Logic. Elsevier, 1999.

....may have more than one G normal form. Proposition 9. Let G be a # base of E and t # T (#, G) If G is recursive, there is an e#ective way of computing a term s(v) # T (#, V ) and a sequence r of terms in G such that t = s(r) 2. 2 Constructors and Modal Logics For all normal modal logics [9], equivalence of formulae is a congruence relation on formulae that is closed under substitution [9] For example, consider the basic modal logic K. Here, the signature #K contains the Boolean operators (#, #, the Boolean constant # (for truth) and the unary (modal) operator 2. ....

....If G is recursive, there is an e#ective way of computing a term s(v) # T (#, V ) and a sequence r of terms in G such that t = s(r) 2. 2 Constructors and Modal Logics For all normal modal logics [9] equivalence of formulae is a congruence relation on formulae that is closed under substitution [9]. For example, consider the basic modal logic K. Here, the signature #K contains the Boolean operators (#, #, the Boolean constant # (for truth) and the unary (modal) operator 2. Equivalence of formulae in K can be axiomatized [10] by the equational theory EK , which consists of the ....

[Article contains additional citation context not shown here]

M. Kracht. Tools and Techniques in Modal Logic. Elsevier, Amsterdam, The Netherlands, 1999.

....less obvious, class of examples for which our combination approach could provide fresh insights and results. 3. 2 Constructors and Modal Logics For all normal modal logics, equivalence of formulae is a congruence relation on formulae that is closed under substitution (see [Gol76] or Chapter 4 in [Kra99]) For example, consider the basic modal logic K [Fit93] Here, the signature Sigma K contains the Boolean operators ( the Boolean constant (for truth) and the unary (modal) operator 2. 7 Equivalence of formulae in K can be axiomatized [Lem66] by the equational theory E K , which ....

Marcus Kracht. Tools and Techniques in Modal Logic. Elsevier, Amsterdam, The Netherlands, 1999. 51

....us however note that it makes a difference whether Theta is given by means of an axiomatization or by means of a finite frame. For in general it is undecidable given a finite frame F and a finite set X of axioms whether or not K Phi X is the theory of F. This has been shown by A. Chagrov. See [6] for a proof. Now, if a tabular logic contains K4 this question becomes in fact decidable. For a tabular logic has a representation the form K4=N , where N is a finite set of finite frames. We will with the exception of x8 assume that our logics contain K4. If that is assumed, we have reduced the ....

....are the open sets of a topological space together with the operations of intersection and infinitary union. A locale is continuous if it also satisfies the dual law x t i2I y i = i2I x t y i There is a representation theory for locales. The background can be found in [5] and, for modal logic, in [6]. Let I(L) be the set of all meet irreducible elements. An element x is meet irreducible if from x = y u z follows x = y or x = z. Now let x y : fy 2 I(L) y xg. Then the following holds: 1. x u y) y = x y y y 2. i2I y i ) y = S i2I y y i Put Spec(L) hI(L) fx y : x ....

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Marcus Kracht. Tools and Techniques in Modal Logic. Habilitationsschrift, 1997.

....of problems raised in [14] and [13] which cannot be attacked this way because they require completeness properties to be preserved as well. We show here that this is in fact the case. We will apply this to solve some open problems in modal logic. Moreover, using these techniques it is proved in [18] that there exist logics which have finite model property locally but are globally incomplete. Other problems, such as the decidability of finite model property or of decidability itself have a straightforward solution in bimodal logics (using word problems) By appealing to the simulation method, ....

....of p in P is in a subformula of the form Q R or j Q, i n, if that occurrence is in the scope of some Sigma k , k n. This describes a wider class of formulae. However, for every Sahlqvist van Benthem formula P there exists a Sahlqvist formula Q such that K Phi P = K Phi Q. See [18]. Definition 42 An n modal logic has local interpolation if whenever P Q there exists a formula R such that var(R) var(P ) var(Q) and P R Q. has global interpolation if whenever P fl Q there exists a R such that var (R) var (P ) var(Q) and P fl R fl Q. Theorem 43 The ....

Marcus Kracht. Tools and Techniques in Modal Logic. Habilitationsschrift, Department of Mathematics, FU Berlin, 1996.

.... Features Marcus Kracht II. Mathematisches Institut Freie Universitat Berlin Arnimallee 3 D 14195 Berlin kracht math.fu berlin.de Abstract. If one converts surface filters into context free rules, one has to introduce new features. These features are strictly nonlexical, and their distribution is predictable from ....

Marcus Kracht. Tools and Techniques in Modal Logic. Habilitationsschrift, Department of Mathematics, FU Berlin, 1996.

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M. Kracht. Tools and Techniques in Modal Logic, vol. 142 of Studies in Logic. Elsevier, 1999.

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M. Kracht, Tools and techniques in modal logic, Elsevier, 1999.

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M. Kracht, Tools and Techniques in Modal Logic, North-Holland, 1999.

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Kracht, Marcus. 1999. Tools and Techniques in Modal Logic. Studies in Logic and the Foundations of Mathematics, Vol. 142, Elsevier Science Ltd., Amsterdam.

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