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118
A Logic for Reasoning about Probabilities
 In: Information and Computation 87
, 1990
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Modeling and Verifying Systems using a Logic of Counter Arithmetic with Lambda Expressions and Uninterpreted Functions
, 2002
"... In this paper, we present the logic of Counter arithmetic with Lambda expressions and Uninterpreted functions (CLU). CLU generalizes the logic of equality with uninterpreted functions (EUF) with constrained lambda expressions, ordering, and successor and predecessor functions. In addition to mod ..."
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Cited by 156 (43 self)
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In this paper, we present the logic of Counter arithmetic with Lambda expressions and Uninterpreted functions (CLU). CLU generalizes the logic of equality with uninterpreted functions (EUF) with constrained lambda expressions, ordering, and successor and predecessor functions. In addition to modeling pipelined processors that EUF has proved useful for, CLU can be used to model many infinitestate systems including those with infinite memories, finite and infinite queues including lossy channels, and networks of identical processes. Even with this richer expressive power, the validity of a CLU formula can be efficiently decided by translating it to a propositional formula, and then using Boolean methods to check validity. We give theoretical and empirical evidence for the efficiency of our decision procedure. We also describe verification techniques that we have used on a variety of systems, including an outoforder execution unit and the loadstore unit of an industrial microprocessor.
An Algorithmic Theory of Lattice Points in Polyhedra
, 1999
"... We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations to classical and higherdimensional Dedekind sums, complexity of the Presburger arithmetic, efficien ..."
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Cited by 128 (7 self)
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We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations to classical and higherdimensional Dedekind sums, complexity of the Presburger arithmetic, efficient computations with rational functions, and others. Although the main slant is algorithmic, structural results are discussed, such as relations to the general theory of valuations on polyhedra and connections with the theory of toric varieties. The paper surveys known results and presents some new results and connections.
Towards Exact Geometric Computation
, 1994
"... Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues&quo ..."
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Cited by 97 (6 self)
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Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues". To reconcile theory and practice, authors have suggested that theoretical algorithms ought to be redesigned to become robust under fixedprecision arithmetic. We suggest that in many cases, implementors should make robustness a nonissue by computing exactly. The advantages of exact computation are too many to ignore. Many of the presumed difficulties of exact computation are partly surmountable and partly inherent with the robustness goal. This paper formulates the theoretical framework for exact computation based on algebraic numbers. We then examine the practical support needed to make the exact approach a viable alternative. It turns out that the exact computation paradigm encomp...
Decidability of Model Checking for InfiniteState Concurrent Systems
 Acta Informatica
"... We study the decidability of the model checking problem for linear and branching time logics, and two models of concurrent computation, namely Petri nets and Basic Parallel Processes. 1 Introduction Most techniques for the verification of concurrent systems proceed by an exhaustive traversal of the ..."
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Cited by 64 (1 self)
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We study the decidability of the model checking problem for linear and branching time logics, and two models of concurrent computation, namely Petri nets and Basic Parallel Processes. 1 Introduction Most techniques for the verification of concurrent systems proceed by an exhaustive traversal of the state space. Therefore, they are inherently incapable of considering systems with infinitely many states. Recently, some new methods have been developed in order to at least palliate this problem. Using them, several verification problems for some restricted infinitestate models have been shown to be decidable. These results can be classified into those showing the decidability of equivalence relations [8, 9, 24, 26], and those showing the decidability of model checking for different modal and temporal logics. In this paper, we contribute to this second group. The model checking problem has been studied so far for three infinitestate models: contextfree processes, pushdown processes, and...
Counting in Trees for Free
, 2004
"... In [22], it was shown that MSO logic for ordered unranked trees becomes undecidable if Presburger constraints are allowed at children of nodes. We now show that a decidable logic is obtained if we use a a modal fixpoint logic instead. We present an automata theoretic characterization of this logi ..."
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Cited by 48 (2 self)
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In [22], it was shown that MSO logic for ordered unranked trees becomes undecidable if Presburger constraints are allowed at children of nodes. We now show that a decidable logic is obtained if we use a a modal fixpoint logic instead. We present an automata theoretic characterization of this logic by means of deterministic Presburger tree automata (PTA) and show how it can be used to express numerical document queries. Surprisingly, the complexity of satisfiability for the extended logic is asymptotically the same as for the original logic. The nonemptiness for PTAs is in general pspacecomplete which is moderate given that it is already pspacehard to test whether the complement of a regular expression is nonempty. We also identify a subclass of PTAs with a tractable nonemptiness problem. Further, to decide whether a tree t satisfies a formula # is polynomial in the size of # and linear in the size of t.
Salsa: Combining Constraint Solvers with BDDs for Automatic Invariant Checking
, 2000
"... . Salsa is an invariant checker for specifications in SAL (the SCR Abstract Language). To establish a formula as an invariant without any user guidance Salsa carries out an induction proof that utilizes tightly integrated decision procedures, currently a combination of BDD algorithms and a const ..."
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Cited by 38 (12 self)
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. Salsa is an invariant checker for specifications in SAL (the SCR Abstract Language). To establish a formula as an invariant without any user guidance Salsa carries out an induction proof that utilizes tightly integrated decision procedures, currently a combination of BDD algorithms and a constraint solver for integer linear arithmetic, for discharging the verification conditions. The user interface of Salsa is designed to mimic the interfaces of model checkers; i.e., given a formula and a system description, Salsa either establishes the formula as an invariant of the system (but returns no proof) or provides a counterexample. In either case, the algorithm will terminate. Unlike model checkers, Salsa returns a state pair as a counterexample and not an execution sequence. Also, due to the incompleteness of induction, users must validate the counterexamples. The use of induction enables Salsa to combat the state explosion problem that plagues model checkers  it can handle...
Deciding QuantifierFree Presburger Formulas Using Finite Instantiation Based on Parameterized Solution Bounds
 In Proc. 19 th LICS. IEEE
, 2003
"... Given a formula # in quantifierfree Presburger arithmetic, it is well known that, if there is a satisfying solution to #, there is one whose size, measured in bits, is polynomially bounded in the size of #. In this paper, we consider a special class of quantifierfree Presburger formulas in which m ..."
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Cited by 35 (6 self)
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Given a formula # in quantifierfree Presburger arithmetic, it is well known that, if there is a satisfying solution to #, there is one whose size, measured in bits, is polynomially bounded in the size of #. In this paper, we consider a special class of quantifierfree Presburger formulas in which most linear constraints are separation (di#erencebound) constraints, and the nonseparation constraints are sparse. This class has been observed to commonly occur in software verification problems. We derive a new solution bound in terms of parameters characterizing the sparseness of linear constraints and the number of nonseparation constraints, in addition to traditional measures of formula size. In particular, the number of bits needed per integer variable is linear in the number of nonseparation constraints and logarithmic in the number and size of nonzero coe#cients in them, but is otherwise independent of the total number of linear constraints in the formula. The derived bound can be used in a decision procedure based on instantiating integer variables over a finite domain and translating the input quantifierfree Presburger formula to an equisatisfiable Boolean formula, which is then checked using a Boolean satisfiability solver. We present empirical evidence indicating that this method can greatly outperform other decision procedures.
An Improved Lower Bound for the Elementary Theories of Trees
, 1996
"... . The firstorder theories of finite and rational, constructor and feature trees possess complete axiomatizations and are decidable by quantifier elimination [15, 13, 14, 5, 10, 3, 20, 4, 2]. By using the uniform inseparability lower bounds techniques due to Compton and Henson [6], based on repr ..."
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Cited by 29 (3 self)
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. The firstorder theories of finite and rational, constructor and feature trees possess complete axiomatizations and are decidable by quantifier elimination [15, 13, 14, 5, 10, 3, 20, 4, 2]. By using the uniform inseparability lower bounds techniques due to Compton and Henson [6], based on representing large binary relations by means of short formulas manipulating with high trees, we prove that all the above theories, as well as all their subtheories, are nonelementary in the sense of Kalmar, i.e., cannot be decided within time bounded by a k story exponential function 1 exp k (n) for any fixed k. Moreover, for some constant d ? 0 these decision problems require nondeterministic time exceeding exp 1 (bdnc) infinitely often. 1 Introduction Trees are fundamental in Computer Science. Different tree structures are used as underlying domains in automated theorem proving, term rewriting, functional and logic programming, constraint solving, symbolic computation, knowledge re...
Definable Relations and FirstOrder Query Languages over Strings
"... We study analogs of classical relational calculus in the context of strings. We start by studying string logics. Taking a classical modeltheoretic approach, we fix a set of string operations and look at the resulting collection of definable relations. These form an algebra  a class of nary relati ..."
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Cited by 28 (7 self)
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We study analogs of classical relational calculus in the context of strings. We start by studying string logics. Taking a classical modeltheoretic approach, we fix a set of string operations and look at the resulting collection of definable relations. These form an algebra  a class of nary relations for every n, closed under projection and Boolean operations. We show that by choosing the string vocabulary carefully, we get string logics that have desirable properties: computable evaluation and normal forms. We identify five distinct models and study the differences in their modeltheory and complexity of evaluation. We identify a subset of these models which have additional attractive properties, such as finite VC dimension and quantifier elimination. Once you have a logic,