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35
On the Restraining Power of Guards
 Journal of Symbolic Logic
, 1998
"... Guarded fragments of firstorder logic were recently introduced by Andr'eka, van Benthem and N'emeti; they consist of relational firstorder formulae whose quantifiers are appropriately relativized by atoms. These fragments are interesting because they extend in a natural way many proposit ..."
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Cited by 153 (3 self)
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Guarded fragments of firstorder logic were recently introduced by Andr'eka, van Benthem and N'emeti; they consist of relational firstorder formulae whose quantifiers are appropriately relativized by atoms. These fragments are interesting because they extend in a natural way many propositional modal logics, because they have useful modeltheoretic properties and especially because they are decidable classes that avoid the usual syntactic restrictions (on the arity of relation symbols, the quantifier pattern or the number of variables) of almost all other known decidable fragments of firstorder logic. Here, we investigate the computational complexity of these fragments. We prove that the satisfiability problems for the guarded fragment (GF) and the loosely guarded fragment (LGF) of firstorder logic are complete for deterministic double exponential time. For the subfragments that have only a bounded number of variables or only relation symbols of bounded arity, satisfiability is EXPTI...
Guarded Fixed Point Logic
, 1999
"... Guarded fixed point logics are obtained by adding least and greatest fixed points to the guarded fragments of firstorder logic that were recently introduced by Andr eka, van Benthem and N emeti. Guarded fixed point logics can also be viewed as the natural common extensions of the modal µcalculus an ..."
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Cited by 81 (6 self)
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Guarded fixed point logics are obtained by adding least and greatest fixed points to the guarded fragments of firstorder logic that were recently introduced by Andr eka, van Benthem and N emeti. Guarded fixed point logics can also be viewed as the natural common extensions of the modal µcalculus and the guarded fragments. We prove that the satisfiability problems for guarded fixed point logics are decidable and complete for deterministic double exponential time. For guarded fixed point sentences of bounded width, the most important case for applications, the satisfiability problem is EXPTIMEcomplete.
TwoVariable Logic with Counting is Decidable
, 1996
"... We prove that the satisfiability problem for C² is decidable. C² is firstorder logic with only two variables in the presence of arbitrary counting quantifiers 9 ?m , m ? 1. It considerably extends L², plain firstorder with only two variables, which is known to be decidable by a result of Mort ..."
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Cited by 78 (4 self)
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We prove that the satisfiability problem for C² is decidable. C² is firstorder logic with only two variables in the presence of arbitrary counting quantifiers 9 ?m , m ? 1. It considerably extends L², plain firstorder with only two variables, which is known to be decidable by a result of Mortimer. Unlike L², C² does not have the finite model property. As C² extends L² by expressive means for counting, significant applications arise from the fact that C² embeds corresponding counting extensions of modal logics.
The TwoVariable Guarded Fragment with Transitive Relations
 In Proc. LICS'99
, 1999
"... We consider the restriction of the guarded fragment to the twovariable case where, in addition, binary relations may be specified as transitive. We show that (i) this very restricted form of the guarded fragment without equality is undecidable and that (ii) when allowing nonunary relations to occu ..."
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Cited by 39 (1 self)
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We consider the restriction of the guarded fragment to the twovariable case where, in addition, binary relations may be specified as transitive. We show that (i) this very restricted form of the guarded fragment without equality is undecidable and that (ii) when allowing nonunary relations to occur only in guards, the logic becomes decidable. The latter subclass of the guarded fragment is the one that occurs naturally when translating multimodal logics of the type K4, S4 or S5 into rstorder logic. We also show that the loosely guarded fragment without equality and with a single transitive relation is undecidable.
Simulating reachability using firstorder logic with applications to verification of linked data structures
 In CADE20
, 2005
"... This paper shows how to harness existing theorem provers for firstorder logic to automatically verify safety properties of imperative programs that perform dynamic storage allocation and destructive updating of pointervalued structure fields. One of the main obstacles is specifying and proving the ..."
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Cited by 39 (7 self)
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This paper shows how to harness existing theorem provers for firstorder logic to automatically verify safety properties of imperative programs that perform dynamic storage allocation and destructive updating of pointervalued structure fields. One of the main obstacles is specifying and proving the (absence) of reachability properties among dynamically allocated cells. The main technical contributions are methods for simulating reachability in a conservative way using firstorder formulas—the formulas describe a superset of the set of program states that can actually arise. These methods are employed for semiautomatic program verification (i.e., using programmersupplied loop invariants) on programs such as markandsweep garbage collection and destructive reversal of a singly linked list. (The markandsweep example has been previously reported as being beyond the capabilities of ESC/Java.) 1
The Boundary between Decidability and Undecidability for TransitiveClosure Logics
 In Computer Science Logic (CSL
, 2004
"... To reason effectively about programs, it is important to have some version of a transitiveclosure operator so that we can describe such notions as the set of nodes reachable from a program's variables. On the other hand, with a few notable exceptions, adding transitive closure to even very tam ..."
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Cited by 38 (6 self)
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To reason effectively about programs, it is important to have some version of a transitiveclosure operator so that we can describe such notions as the set of nodes reachable from a program's variables. On the other hand, with a few notable exceptions, adding transitive closure to even very tame logics makes them undecidable. In this paper, we explore...
A Logic of Reachable Patterns in Linked DataStructures
, 2007
"... We define a new decidable logic for expressing and checking invariants of programs that manipulate dynamicallyallocated objects via pointers and destructive pointer updates. The main feature of this logic is the ability to limit the neighborhood of a node that is reachable via a regular expression ..."
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Cited by 33 (5 self)
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We define a new decidable logic for expressing and checking invariants of programs that manipulate dynamicallyallocated objects via pointers and destructive pointer updates. The main feature of this logic is the ability to limit the neighborhood of a node that is reachable via a regular expression from a designated node. The logic is closed under boolean operations (entailment, negation) and has a finite model property. The key technical result is the proof of decidability. We show how to express preconditions, postconditions, and loop invariants for some interesting programs. It is also possible to express properties such as disjointness of datastructures, and lowlevel heap mutations. Moreover, our logic can express properties of arbitrary datastructures and of an arbitrary number of pointer fields. The latter provides a way to naturally specify postconditions that relate the fields on the entry of a procedure to the field on the exit of a procedure. Therefore, it is possible to use the logic to automatically prove partial correctness of programs performing lowlevel heap mutations.
Complexity Results for FirstOrder TwoVariable Logic with Counting
, 2000
"... Let C 2 p denote the class of first order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) i", for i p, and let C 2 be the union of C 2 p taken over all integers p. We prove that the satisfiability problem for C 2 1 sentences is NEXPTIM ..."
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Cited by 32 (1 self)
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Let C 2 p denote the class of first order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) i", for i p, and let C 2 be the union of C 2 p taken over all integers p. We prove that the satisfiability problem for C 2 1 sentences is NEXPTIMEcomplete. This strengthens the results by E. Grädel, Ph. Kolaitis and M. Vardi [15] who showed that the satisfiability problem for the first order twovariable logic L 2 is NEXPTIMEcomplete and by E. Grädel, M. Otto and E. Rosen [16] who proved the decidability of C 2 . Our result easily implies that the satisfiability problem for C 2 is in nondeterministic, doubly exponential time. It is interesting that C 2 1 is in NEXPTIME in spite of the fact, that there are sentences whose minimal (and only) models are of doubly exponential size. It is worth noticing, that by a recent result of E. Gradel, M. Otto and E. Rosen [17], extensions of twovariables logic L 2 by a week access to car...
Why Are Modal Logics So Robustly Decidable?
"... Modal logics are widely used in a number of areas in computer science, in particular ..."
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Cited by 31 (1 self)
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Modal logics are widely used in a number of areas in computer science, in particular