The predicate modal logic of provability, Notre Dame Journal of Formal Logic, 25, pp. 179--189.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that simplest and least useful part of the subject, the logic of absolute terms, which, De Morgan1864b] Volume 10, pp. 331 358. De Morgan s second and third papers are [De Morgan1856] and [De Morgan1864a] De Morgan1966] p. 208. Peirce1870] See, for example, Brink1978] Brink1979] Brunning1980] Martin1976] Martin1978] and [Merrill1984] See [Brink1981] and [Brink1988] when he wrote, was the only formal logic known. The object of this paper is to show that an affirmative answer can be given to this question. I think there can be no doubt that a calculus, or art ....

, The algebra of relatives, Notre Dame Journal of Formal Logic 20 (1979), 900--908.

....lower bound, we use a reduction from the global satis ability problem for the standard modal logic K. A formula is globally satis able if there exists a model 10 such that is satis ed in every state of the model. The global satis ablity problem for K is known to be EXPTIME hard, see e.g. [17, 31]. Our reduction only uses a restricted form of the PDL satis ability problem, viz. restricted to the modal operators ha 0 i and ha 0 i. We use the spy point technique as described in [10] by adapting the proof of [7, Theorem 2] As part of this technique, we introduce a node (the spy ....

E. Hemaspaandra. The price of universality. Notre Dame Journal of Formal Logic, 37(2):173-203, 1996.

....the lower bound, we use a reduction from the global satisfiability problem for the standard modal logic K. A formula is globally satisfiable if there exists a model such that is satisfied in every state of the model. The global satisfiablity problem for K is known to be EXPTIME hard, see e.g. [17, 31]. Our reduction only uses a restricted form of the PDL satisfiability problem, viz. restricted to the modal operators ha 0 i and ha 0 i. We use the spy point technique as described in [10] by adapting the proof of [7, Theorem 2] As part of this technique, we introduce a node (the spy ....

E. Hemaspaandra. The price of universality. Notre Dame Journal of Formal Logic, 37(2):173--203, 1996.

....(1) and (2) Our contribution. In this paper, we show that the satis ability problem for the logic SIM is ExpTime complete. The ExpTime lower bound is a consequence of more general results since SIM contains a universal modal connective with a family of B modal connectives (see e.g. Spa93,CL94,Hem96] The ExpTime upper bound is established by an exponential reduction into the emptiness problem for B uchi automata on in nite trees that is known to be in PTime (see e.g. VW86,EJ88] As mentioned previously, this technique is nowadays standard for logics of programs, but it has never been ....

....tree for respecting the SIM consistent global information G 0 . We are now in the position to establish the main result of the paper. Theorem 11. The satis ability problem for the logic SIM is ExpTime complete. Proof. The lower bound is by an easy veri cation from the results in [CL94] and [Hem96, Theorem 5.1] Let us establish the ExpTime upper bound. Lemma 9 together with Lemma 10 implies that every SIM formula is SIM satis able i A accepts at least one tree. Since card(SYMB( j j 2 j j j j and card(GCONS( is in 2 , A has 2 states. Moreover, card( is ....

E. Hemaspaandra. The price of universality. Notre Dame Journal of Formal Logic, 37(2):173-203, 1996.

....about regular languages in order to analyze the computational complexity of regular grammar logics. Related modal logics. Although the grammar logics may seem arti cial, many polymodal logics containing fragments that are regular 3 grammar logics can be found in the literature (see e.g. [FL79, Cat89, HM92, Gas94, FdCH95, Hem96, HM97] to quote a few examples) More importantly, Description Logics (DLs) that are used to represent terminological knowledge (see e.g. SSS91] are strongly related to grammar logics. A current line of research in DLs community consists in studying more and more expressive description logics as soon ....

....be the set of monomodal formulae for which there is a [resp. re exive] Kripke structure M = hW; R; V i satisfying M j= K GSAT [resp. T GSAT ] stands for the global satis ability problem for the standard modal logic K [resp. T] that is known to be EXPTIME hard [CL94, Theorem 1] see also [Hem96]) As in the proof of Theorem 32, we distinguish two cases according to = or not. 2 . Let us de ne a logarithmic space transformation from K GSAT into m ) Indeed, one can show that 2 K GSAT i ] The details are similar to Case 1 in the proof of Theorem 32. Case 2: ....

[Article contains additional citation context not shown here]

E. Hemaspaandra. The price of universality. Notre Dame Journal of Formal Logic, 37(2):173-203, 1996.

....whenever jLj 1. The EXPTIME upper bound is a corollary of the EXPTIME completeness of CPDL with nominals [12, page 98] As to the lower bound, the proof (omitted here) is by reducing the global satis ability problem for the standard modal logic K (known to be EXPTIME hard, see e.g. [10, 17]) to PDL path satis ability restricted to the 6 modal connectives ha 0 i and ha 1 0 i. To do so, we can take advantage of the spy point technique from [6] and adapt the proof of [4, Theorem 2] The only diculty is to use the spy point technique and simultaneously encode the proposition ....

E. Hemaspaandra. The price of universality. Notre Dame Journal of Formal Logic, 37(2):173-203, 1996.

....well as the one in the language of 4, actually coincides with L. Using the uniform version of Solovay s theorem, Smory nski [1985] showed that CS is the minimal bimodal provability logic, i.e. it coincides with PRL T;U for a certain pair of finite extensions T; U of Peano arithmetic. Beklemishev [1992] showed that there is even a pair of provability predicates for Peano arithmetic itself for which the corresponding bimodal provability logic coincides with CS. Such predicates can be called independent in the sense that they know as little about each other as is possible in principle. It should ....

....In his review of these and some consecutive papers Beklemishev [1993b] rightly remarked that the changing of the orders of the proofs in Guaspari Solovay style interferes with the order induced by the function g and makes some of the results somewhat less clear than one might wish. Montagna [1992] applied the results on provable fixed points in a study of metamathematical rules, i.e. rules like Pr T (p ff q) ff that can be considered as realizations of modal logical rules (in case: 2A=A) He classified these rules into two types: rules giving only polynomial speed up in proofs in ....

[Article contains additional citation context not shown here]

On the proofs of arithmetical completeness of interpretability logic, Notre Dame Journal of Formal Logic, 35, pp. 542--551.

....for K (K4, L) In fact, the Kripke models for K4 (resp. L) are exactly the ones that validate the formulas derivable, respectively in K4 and L. One says that K4 and L characterize these classes of models. For the main concepts of modal logic, see e.g. Chellas [1980] Hughes and Cresswell [1984]. Something stronger is true: in K, K4 and L one can derive all the formulas that are valid in their respective model classes (modal completeness) The standard method in modal logic for proving completeness is to construct the necessary countermodels by taking maximal consistent sets of the ....

....( Assume 0 S A. Then a fortiori A is not derivable in L from the reflection principles for its boxed subformulas. The result then immediately follows by applying theorem 2.4. a An elegant formulation of the semantics of S in terms of infinite models, so called tail models is given in Visser [1984]. The Logic of Provability 481 The well known normal modal system S4 that is obtained by adding the scheme 2A A to K4 plays a role in section 10. It can be shown that S4 is modally complete with respect to the (finite) reflexive, transitive Kripke models. 3. Proof of Solovay s theorems We rely ....

[Article contains additional citation context not shown here]

The predicate modal logic of provability, Notre Dame Journal of Formal Logic, 25, pp. 179--189.

....= W Theta W and R a b = Ra R b for all a; b 2 M. An model M = W; Ra ) a2M[fUg ; m) is a structure such that F = W; Ra ) a2M ) is an frame and m is a mapping m : For 0 P(W ) M is said to be based on F . r Observe that the modal operator [U] is the standard universal modal operator [GP92, Hem96]. The satisfiability relation j= and the associated notions of L satisfiability and L validity are defined as usual for polymodal logics. In what follows, by an logic L we understand a pair hFor ; X M i such that X M is the class of all models based on a given class of ....

E. Hemaspaandra. The price of universality. Notre Dame Journal of Formal Logic, 37(2):173--203, 1996.

.... University Since the work of Kallick [10] resolution based decision procedures for subclasses of firstorder logic have drawn continuous attention [3, 5, 9] There are two research areas where decidability issues likewise played a prominent role: extended modal logics and description logics [4, 7, 11]. Although is is not difficult to see that most of the logics under consideration can be translated to first order logic, the exact relation to decidable subclasses of first order logic and in particular to subclasses decidable by resolution is still under investigation. A recent important result ....

E. Hemaspaandra. The price of universality. Notre Dame Journal of Formal Logic, 37(2):174--203, 1996.

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The predicate modal logic of provability, Notre Dame Journal of Formal Logic, 25, pp. 179--189.

....the consequences according to agent b of what b thinks is in the theory of agent a. This may, after all, have direct applications in multi agent systems. Even single agent epistemic logic with a substructural base is little investigated and likely to be more difficult than might be imagined. See [8] for an account of how even a simple modal logic like S4 raises hard problems in the substructural case. The effects obtainable by extending the logics of this paper with multiagent epistemic operators are yet unknown. ....

E. Mares and R.K. Meyer, The Admissibility of Gamma in R4. Notre Dame Journal of Formal Logic 33 (1992) 197-206.

....well as the one in the language of 4, actually coincides with L. Using the uniform version of Solovay s theorem, Smory nski [1985] showed that CS is the minimal bimodal provability logic, i.e. it coincides with PRL T;U for a certain pair of finite extensions T; U of Peano arithmetic. Beklemishev [1992] showed that there is even a pair of provability predicates for Peano arithmetic itself for which the corresponding bimodal provability logic coincides with CS. Such predicates can be called independent in the sense that they know as little about each other as is possible in principle. It should ....

....In his review of these and some consecutive papers Beklemishev [1993b] rightly remarked that the changing of the orders of the proofs in Guaspari Solovay style interferes with the order induced by the function g and makes some of the results somewhat less clear than one might wish. Montagna [1992] applied the results on provable fixed points in a study of metamathematical rules, i.e. rules like Pr T (p ff q) ff that can be considered as realizations of modal logical rules (in case: 2A=A) He classified these rules into two types: ones giving only polynomial speed up in proofs in ....

[Article contains additional citation context not shown here]

On the proofs of arithmetical completeness of interpretability logic, Notre Dame Journal of Formal Logic, 35, pp. 542--551.

....for K (K4, L) In fact, the Kripke models for K4 (resp. L) are exactly the ones that validate the formulas derivable, respectively in K4 and L. One says that K4 and L characterize these classes of models. For the main concepts of modal logic, see e.g. Chellas [1980] Hughes and Cresswell [1984]. Something stronger is true: in K, K4 and L one can derive all the formulas that are valid in their respective model classes (modal completeness) The standard method in modal logic for proving completeness is to construct the necessary countermodels by taking maximal consistent sets of the ....

....( Assume 0 S A. Then a fortiori A is not derivable from the reflection principles for its boxed subformulas. The result then immediately follows by applying theorem 2.4. a An elegant formulation of the semantics of S in terms of infinite models, so called tail models is given in Visser [1984]. The well known normal modal system S4 that is obtained by adding the scheme 2A A to K4 plays a role in secion 10. It can be shown that S4 is modally complete with respect to the (finite) reflexive, transitive Kripke models. Draft April 17, The Logic of Provability 365 3. Proof of Solovay s ....

[Article contains additional citation context not shown here]

The predicate modal logic of provability, Notre Dame Journal of Formal Logic, 25, pp. 179--189.

No context found.

E. Hemaspaandra. The price of universality. Notre Dame Journal of Formal Logic, 37(2):173-203, 1996.

No context found.

E. Hemaspaandra. The price of universality. Notre Dame Journal of Formal Logic, 37(2):173-203, 1996.

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E. Hemaspaandra. The price of universality. Notre Dame Journal of Formal Logic, 37(2):173203, 1996.

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Free Logic and the Concept of Existence, Lambert, K.; Notre Dame Journal of Formal Logic, Vol.8, nn.1-2, 1967.

No context found.

The theory of homogeneous simple types as a second order logic, Notre Dame Journal of Formal Logic, vol. 20, 505-524.

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