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Terminating tableau calculi for modal logic K with global counting operators
"... This paper presents the first systematic treatment of tableau calculi for modal logic K with global counting operators. Using a recently introduced tableau synthesis framework we establish two terminating tableau calculi for the logic. Whereas the first calculus is a prefix tableau calculus, the sec ..."
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This paper presents the first systematic treatment of tableau calculi for modal logic K with global counting operators. Using a recently introduced tableau synthesis framework we establish two terminating tableau calculi for the logic. Whereas the first calculus is a prefix tableau calculus, the second is a refinement that internalises the semantics of the logic without using nominals. We prove the finite model property for the logic and show that adding the unrestricted blocking mechanism does not break soundness and completeness of the calculi and ensures termination in both cases. We have successfully implemented the prefix tableau calculus in the MetTeL2 tableau prover generation platform. Keywords: property. modal logic, hybrid logic, tableau, counting operators, finite model 1
Satisfiability Modulo Counting: A New Approach for Analyzing Privacy Properties
"... Applications increasingly derive functionality from sensitive personal information, forcing developers who wish to preserve some notion of privacy or confidentiality to reason about partial information leakage. New definitions of privacy and confidentiality, such as differential privacy, address t ..."
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Applications increasingly derive functionality from sensitive personal information, forcing developers who wish to preserve some notion of privacy or confidentiality to reason about partial information leakage. New definitions of privacy and confidentiality, such as differential privacy, address this by offering precise statements of acceptable disclosure that are useful in common settings. However, several recent published accounts of flawed implementations have surfaced, highlighting the need for verification techniques. In this paper, we pose the problem of modelcounting satisfiability, and show that a diverse set of privacy and confidentiality verification problems can be reduced to instances of it. In this problem, constraints are placed on the outcome of modelcounting operations, which occur over formulas containing parameters. The object is to find an assignment to the parameters that satisfies the modelcounting constraints, or to demonstrate unsatisfiability. We present a logic for expressing these problems, and an abstract decision procedure for modelcounting satisfiability problems fashioned after CDCLbased SMT procedures, encapsulating functionality specific to the underlying logic in which counting occurs in a small set of blackbox routines similar to those required of theory solvers in SMT. We describe an implementation of this procedure for linearinteger arithmetic, as well as an effective strategy for generating lemmas. We conclude by applying our decision procedure to the verification of privacy properties over programs taken from a wellknown privacypreserving compiler, demonstrating its ability to find flaws or prove correctness sometimes in a matter of seconds.
Beyond Regularity for Presburger Modal Logics
"... Satisfiability problem for modal logic K with quantifierfree Presburger and regularity constraints (EML) is known to be pspacecomplete. In this paper, we consider its extension with nonregular constraints, and more specifically those expressed by visibly pushdown languages (VPL). This class of lan ..."
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Satisfiability problem for modal logic K with quantifierfree Presburger and regularity constraints (EML) is known to be pspacecomplete. In this paper, we consider its extension with nonregular constraints, and more specifically those expressed by visibly pushdown languages (VPL). This class of languages behaves nicely, in particular when combined with Propositional Dynamic Logic (PDL). By extending EML, we show that decidability is preserved if we allow at most one positive VPLconstraint at each modal depth. However, the presence of two VPLcontraints or the presence of a negative occurrence of a single VPLconstraint leads to undecidability. These results contrast with the decidability of PDL augmented with VPLconstraints. Keywords: Presburger constraint, contextfree constraint, decidability
Reasoning about CTL ∗ with Graded Path Modalities
"... Abstract—Graded path modalities count the number of paths satisfying a property, and generalize the existential (E) and universal (A) path modalities of CTL∗. The resulting logic is denoted GCTL∗, and is a very powerful logic since (as we show) it is equivalent to monadic path logic. We settle the ..."
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Abstract—Graded path modalities count the number of paths satisfying a property, and generalize the existential (E) and universal (A) path modalities of CTL∗. The resulting logic is denoted GCTL∗, and is a very powerful logic since (as we show) it is equivalent to monadic path logic. We settle the complexity of the satisfiability problem of GCTL∗, i.e., 2EXPTIMECOMPLETE, and the complexity of the model checking problem of GCTL∗, i.e., PSPACECOMPLETE. The lower bounds already hold for CTL∗, and so we supply the upper bounds. The significance of this work is twofold: GCTL ∗ is much more expressive than CTL ∗ as it adds to it a form of quantitative reasoning, and this is done at no extra cost in computational complexity. I.