M. Fisher and R. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18(2):194--211, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... C is the description logic obtained from ALC by adding the following role constructs: union, chaining, reflexive transitive closure, and identity role over a concept (see [108, 3] C corresponds to the propositional dynamic logic D, which is the original propositional dynamic logic introduced in [56]. All the basic reasoning tasks in C (D) are known to be EXPTIME complete. ALC is obtained from ALC by adding two concept constructs denoting the least fixpoint and the greatest fixpoint of concept expressions (see Chapter 8 for details) Notabily, the fixpoint constructs allow for recursive ....

....basic reasoning task can be (linearly) reformulated as logical implication in a TBox. Namely satisfiability of a concept C can be reformulated as ; 6j= C v , satisfiability of a TBox K as K 6j= v . 2. 2 Propositional dynamic logics We focus on the propositional dynamic logic DI (Converse PDL [56]) which as it turns out corresponds to CI. The abstract syntax of DI is as follows: OE : j j A j OE 1 OE 2 j OE 1 OE 2 j OE 1 ) OE 2 j :OE j r OE j [r]OE r : P j r 1 [ r 2 j r 1 ; r 2 j r The notation (R stands for i repetitions of R i.e. R , and ....

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N. J. Fisher and R. E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18:194--211, 1979.

.... holds. Thus we can celebrate the 25 anniversary of program logics this year. Especially for those who have not a special logical background. 24 A decidable propositional variant of dynamic logic the Propositional Dynamic Logic was suggested by M.J. Fisher and R.E. Ladner in 1977 [13]. A couple of years later K. Segerberg developed a sound and complete axiomatization for this logic. A. Pnueli was the rst to propose the use of temporal logic for reasoning about programs [22] His approach to the speci cation of concurrent and reactive systems is now well developed [19] as ....

Fisher M.J. Ladner R.E. Propositional dynamic logic of regular programs. J. Comput. System Sci., v.18, n.2, 1979, p.194- 211.

....except that some states are declared to be universal and others are declared to be existential. This partition of states is used in the definition of the acceptance condition for an alternating Turing machine. In this subsection we just repeat the definition of an alternating Turing machine from [7]. A one tape alternating Turing machine is a tuple M = E, U, q 0 , #, #, b, #) where E is the set of existential states, U is the set of universal states, q 0 is the initial state, # is the input alphabet, # is the output alphabet, b # # is the blank symbol and # E) L, R is the ....

....every x of 4 length n the length of any configuration # reachable from the configuration q 0 x is at most s(n) 1. ASPACE(s(n) is the class of sets accepted by alternating Turing machines which operates in space s(n) Remark 4 The above definition only slightly di#ers from the definition in [7]. Namely, we require that the initial state is a universal state and that # alternates between the universal and the existential states. For every alternating Turing machine that violates the above requirement it is easy to construct an equivalent (i.e. accepting the same set of strings) ....

M. Fisher and R. Lander. Propositional Dynamic Logic of Regular Programs. Journal of Computer and System Sciences, 18:194-211, 1979.

....for calculus. We prove the correctness and completeness of the calculus and illustrate its features. We also discuss the transformation of the tableaux method (naively NEXPTIME) into an EXPTIME algorithm. 1 Introduction Propositional Dynamic Logics (PDLs) are modal logics introduced in [10] to model the evolution of the computation process by describing the properties of states reached by programs during their execution [15, 24, 27] Over the years, PDLs have been proved to be a valuable formal tool in Computer Science, Logic, Computational Linguistics, and Artificial Intelligence ....

.... [15, 24, 27] Over the years, PDLs have been proved to be a valuable formal tool in Computer Science, Logic, Computational Linguistics, and Artificial Intelligence far beyond their original use for program verification (e.g. 4, 12, 14, 15, 24, 23] In this paper we focus on Converse PDL (CPDL) [10], obtained from the basic logic PDL by adding the converse operator to programs, which is interpreted as the converse of the (input output) relation interpreting the program. A possible use of the converse is for formalizing preconditions, e.g. can be interpreted as before running program ....

[Article contains additional citation context not shown here]

N. Fisher and R. Ladner. Propositional dynamic logic of regular programs. J. of Comp. and System Sci., 18:194--211, 1979.

.... checking problem non deterministic graphs deterministic graphs PDL P complete; see e.g. 18] P complete; see e.g. 18] P complete; this P complete; this paper, Corollary 10 paper, Corollary 10 Satis ability problem non deterministic graphs deterministic graphs PDL EXPTIME complete [22, 40] EXPTIME complete [39, 9] PDL with nominals EXPTIME complete [23] EXPTIME complete [23] CPDL EXPTIME complete [22, 40] EXPTIME complete [44] CPDL with nominals EXPTIME complete [20, 7] open EXPTIME complete; this open paper, Theorem 12 Table 1: A summary of results on logical reasoning ....

.... [18] P complete; this P complete; this paper, Corollary 10 paper, Corollary 10 Satis ability problem non deterministic graphs deterministic graphs PDL EXPTIME complete [22, 40] EXPTIME complete [39, 9] PDL with nominals EXPTIME complete [23] EXPTIME complete [23] CPDL EXPTIME complete [22, 40] EXPTIME complete [44] CPDL with nominals EXPTIME complete [20, 7] open EXPTIME complete; this open paper, Theorem 12 Table 1: A summary of results on logical reasoning tasks. 5 The Complexity of Reasoning with Path Constraints In order to characterize the complexity of reasoning problems ....

N.J. Fisher and R.E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18:194-211, 1979.

.... checking problem non deterministic graphs deterministic graphs PDL P complete; see e.g. 18] P complete; see e.g. 18] P complete; this P complete; this paper, Corollary 10 paper, Corollary 10 Satisfiability problem non deterministic graphs deterministic graphs PDL EXPTIME complete [22, 40] EXPTIME complete [39, 9] PDL with nominals EXPTIME complete [23] EXPTIME complete [23] CPDL EXPTIME complete [22, 40] EXPTIME complete [44] CPDL with nominals EXPTIME complete [20, 7] open EXPTIME complete; this open paper, Theorem 12 Table 1: A summary of results on logical reasoning ....

.... [18] P complete; this P complete; this paper, Corollary 10 paper, Corollary 10 Satisfiability problem non deterministic graphs deterministic graphs PDL EXPTIME complete [22, 40] EXPTIME complete [39, 9] PDL with nominals EXPTIME complete [23] EXPTIME complete [23] CPDL EXPTIME complete [22, 40] EXPTIME complete [44] CPDL with nominals EXPTIME complete [20, 7] open EXPTIME complete; this open paper, Theorem 12 Table 1: A summary of results on logical reasoning tasks. Lemma 13 The problem below is in NLOGSPACE in jGj and jtj: instance: a finite L structure G, a path expression t, ....

N.J. Fisher and R.E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18:194--211, 1979.

....modal calculus, due to Pratt and Kozen [57, 79] Propositional calculus has a next time modality and, for expressing temporal properties, least and greatest fixpoints as opposed to classical modalities. This logic is very powerful, since it has been shown to include Propositional Dynamic Logic [32], Hennessy Milner Logic [39] as well as the branching time temporal logics CTL and CTL and the linear time temporal logic TL [28, 24] Moreover, this formalism has the advantage that models are labelled transition systems, which are commonly used to give the semantics of concurrent and reactive ....

....Propositional modal calculus [57, 79] is a very elegant and expressive logic, that has received a great interest in recent years for the verification of concurrent and reactive systems. This logic is very popular for many reasons. Firstly, it has been shown to include Propositional Dynamic Logic [32], Hennessy Milner Logic [39] as well as the linear time logic TL and the branching time temporal logics CTL and CTL [28, 24] Moreover, models of propositional calculus are labelled transition systems, which are commonly used in giving the semantics to concurrent and reactive systems. The main ....

M.J. Fisher and R.E. Ladner. Propositional Dynamic logic of Regular Programs. Theoretical Computer Science, 18(2):194--211, 1979.

.... 8 Model checking problem non deterministic graphs deterministic graphs PDL O(jjGjj j j) 11] O(jjGjj j j) 11] PDL path O(jjGjj j j) this O(jjGjj j j) this paper, Theorem 8 paper, Theorem 8 Satis ability problem non deterministic graphs deterministic graphs PDL EXPTIME complete [14, 22] EXPTIME complete [20, 5] PDL with nominals EXPTIME complete [15] EXPTIME complete [15] CPDL EXPTIME complete [14, 22] EXPTIME complete [25] CPDL with nominals EXPTIME complete [12, 4] open PDL path EXPTIME complete; this open paper, Theorem 10 Implication problem non deterministic graphs ....

.... path O(jjGjj j j) this O(jjGjj j j) this paper, Theorem 8 paper, Theorem 8 Satis ability problem non deterministic graphs deterministic graphs PDL EXPTIME complete [14, 22] EXPTIME complete [20, 5] PDL with nominals EXPTIME complete [15] EXPTIME complete [15] CPDL EXPTIME complete [14, 22] EXPTIME complete [25] CPDL with nominals EXPTIME complete [12, 4] open PDL path EXPTIME complete; this open paper, Theorem 10 Implication problem non deterministic graphs deterministic graphs inclusion constraints PSPACE hard, in EXPTIME; open this paper, Theorem 11 backward constraints ....

N.J. Fisher and R.E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18:194-211, 1979.

.... C is the Description Logic obtained from ALC by adding the following role constructs: union, chaining, reflexive transitive closure, and identity role over a concept (see [53, 2] C corresponds to the Propositional Dynamic Logic PDL, which is the original Propositional Dynamic Logic introduced in [30]. All the usual reasoning tasks in C (PDL) are known to be EXPTIMEcomplete. ALC is obtained from ALC by adding two concept constructs denoting the least fixpoint and the greatest fixpoint of concept expressions. Notably, the fixpoint constructs allow for recursive concept definitions. Observe that ....

N. J. Fisher and R. E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18:194--211, 1979.

....below. In general PST is based on a reduction of the decidability of a propositional program logic to the validity problem of a variant of the Second Order Propositional Dynamic Logic of flowcharts (SOPDL) in a class of Herbrand models. SOPDL is a variant of Propositional Dynamic Logic (PDL) [4] with second order quantifiers (weak as well as strong) and non deterministic monadic flowcharts (or program schemata) Herbrand Models are models of the Second order logic of several (n) monadic Successors (S(n)S) 10] i.e. free semi groups generated from program symbols. Unfortunately SOPDL is ....

Fisher M.J. Ladner R.E. Propositional dynamic logic of regular programs. J. Comput. System Sci., v.18, n.2, 1979, p.194- 211.

....in this state either diverges or terminates in another state where OE holds. Thus we can celebrate 25 th anniversary of program logics this year. A decidable propositional variant of dynamic logic the Propositional Dynamic Logic was suggested by M.J. Fisher and R.E. Ladner in 1997 [13]. A couple of years later K. Segerberg developed a sound and complete axiomatization for this logic. A. Pnueli was the first who proposed to use temporal logic for reasoning about programs [22] His approach for specification of concurrent and reactive systems is now well developed [19] as well ....

Fisher M.J. Ladner R.E. Propositional dynamic logic of regular programs. J. Comput. System Sci., v.18, n.2, 1979, p.194- 211.

....3. 3 is de ned on any modal model M = W; R i ; v) i2I as M;w 3 i there exists a w 0 2 W such that M;w 0 : 6) As usual, its dual :3: is denoted by 2. We denote the expansion of modal logic K with the universal modality by K3, and its satis ability problem by K3sat. It follows from [2] that K3sat is complete for exptime. 7) We will also need an undecidable logic. De ne Tile as the modal logic with two unary modalities hri and hui interpreted on the class of frames (W; R hri ; R hui ) where R hri and R hui are commuting total functions. We usually denote such a frame by (W; r; ....

M. Fisher and R. Ladner. Propositional dynamic logic of regular programs. J. Comput. Syst. Sci., 18(2):194-211, 1979.

....[12, 13] PST is based on a reduction of the decidability of a propositional program logic to the validity problem for a variant of Second Order Propositional Dynamic Logic of program schemata (SOPDL) in a variant of Herbrand Models. SOPDL is a variant of Propositional Dynamic Logic (PDL) [4] with nondeterministic monadic program schemata [6] and with second order quantifiers (weak as well as strong) Herbrand Models without converse (HM) are models for Second Order Theory of Monadic Successors S(n)S [11] Unfortunately SOPDL is undecidable, but the validity in HM is decidable with ....

Fisher M.J. Ladner R.E. Propositional dynamic logic of regular programs. J. Comput. System Sci., v.18, n.2, 1979, p.194- 211.

....Nepomniaschy V.A. in 1983 88 [16, 17, 18] PST is based on a reduction of the decidability of a propositional program logic to the validity problem in Herbrand Models of Second Order Propositional Dynamic Logic of program schemata (SOPDL) SOPDL is a variant of Propositional Dynamic Logic (PDL) [23] with second order quantifiers (weak as well as strong) and non deterministic monadic (Ianov) program schemata [28, 27, 29] Unfortunately SOPDL is undecidable, but the validity in Herbrand Models (i.e. models for Second Order Theory of Monadic Successors [30] is decidable with exponential upper ....

....2 and seq 2 j= T (X s ) 5. seq j= T (j Uw ) iff either seq i j= T j for every finite i jseqj, or seq j j= T for some finite j jseqj and seq i j= T j for every finite i j. The Second Order Propositional Dynamic Logic (SOPDL) 20] is an extension of Propositional Dynamic Logic (PDL) [23, 24, 29, 25, 26] with quantifiers over propositional variables. The syntax of SOPDL is constructed from the same alphabets B and P as above and from an additional finite alphabet A of action variables. The syntax consists of (non deterministic monadic program (Ianov) schemata S and (logical) formulae FSO . A ....

Fisher M.J. Ladner R.E. Propositional dynamic logic of regular programs. J. Comput. System Sci., v.18, n.2, 1979, p.194- 211.

.... without any complexity cost (up to a polynomial) Furthermore, it shows that for arbitrary frames, the guarding strategy (that is, using instead of E) pays o#: adding the somewhere modality E (and no nominals) to ordinary uni modal logic results in an exptime hard satisfiability problem (see [19, 24]) But these results are of limited temporal interest for two reasons. First, while the purely future fragment is useful in certain applications, we often want the past operator too. Moreover, if we think of states in a Kripke model as time points,and view R as the temporal precedence relation, ....

M. Fisher and R. Ladner. Propositional dynamic logic of regular programs. J. of Computer and System Sciences, 18(2):194--211, 1979.

....to test emptiness of a Rabin automaton in time O( mn) 3n ) where m is the number of states and n is the number of pairs of the automaton. This gives an exponential satisfiability testing procedure for the calculus. The lower bound for this problem follows from the lower bound for PDL proved in [6]. Theorem 4.4 (Decidability) The problem of deciding whether a given formula of the calculus is satisfiable is EXPTIME complete in the size of the formula. In [14] the question was raised whether it is possible to characterise validity in the same way as satisfiability. Definition 4.5 ....

M.J. Fisher and R.E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18:194--211, 1979.

....modalities i and a, the converse axioms for i and a, the Segerberg axioms, Lob axiom for i and axioms for the constants sink and source. The inference rules are Modus Ponens, Universal Generalization and Substitution. 2. 2 Canonical Models The canonical model construction is also the standard one [FL79, Gol92, BdRV94, BMdR96]. We first define the Fisher and Lander Closure CFL ( Gamma) for a set Gamma of formulas and the atoms of GammaAt( Gamma) And then prove the atom existence lemma. Lemma (Atom Existence) Let Gamma be a set of formulas. If A 2 CFL ( Gamma) and A is consistent then there exists an A 2 ....

M. J. Fisher and R. F. Lander. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18, 1979.

.... without any complexity cost (up to a polynomial) Furthermore, it shows that for arbitrary frames, the guarding strategy (that is, using instead of E) pays o : adding the somewhere modality E (and no nominals) to ordinary uni modal logic results in an exptime hard satis ability problem (see [FL79, HM92] But these results are of limited temporal interest for two reasons. First, while the purely future fragment is useful in certain applications, we often want the past operator too. Moreover, if we think of states in a Kripke model as time points, and view R as the temporal precedence ....

M. Fisher and R. Ladner. Propositional dynamic logic of regular programs. J. of Computer and System Sciences, 18(2):194-211, 1979.

No context found.

M. Fisher and R. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18(2):194--211, 1979.

No context found.

N.J. Fisher and R.E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18:194-211, 1979.

No context found.

M. J. Fisher and R. E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18(2):194--211, 1979.

No context found.

M. J. Fisher and R. E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18(2):194--211, 1979.

No context found.

N.J. Fisher and R.E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18:194-211, 1979.

No context found.

M. Fisher and R. Ladner. Propositional dynamic logic of regular programs. J. Comput. Syst. Sci., 18(2):194--211, 1979.

No context found.

Fisher M.J. Ladner R.E. Propositional dynamic logic of regular programs. J. Comput. System Sci., v.18, n.2, 1979, p.194- 211.

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