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Register a new userAdapting Real Quantifier Elimination Methods for Conflict Set Computation
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Added by
Maximilian Jaroschek
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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The satisfiability problem in real closed fields is decidable. In the context of satisfiability modulo theories, the problem restricted to conjunctive sets of literals, that is, sets of polynomial constraints, is of particular importance. One of the central problems is the computation of good explanations of the unsatisfiability of such sets, i.e. obtaining a small subset of the input constraints whose conjunction is already unsatisfiable. We adapt two commonly used real quantifier elimination methods, cylindrical algebraic decomposition and virtual substitution, to provide such conflict sets and demonstrate the performance of our method in practice.
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The paper presents our research on quantifier elimination (QE) for compositional reasoning and verification. For compositional reasoning, QE provides the foundation of our approach, serving as the calculus for composition to derive the strongest system-property in a single step, from the given component atomic-properties and their interconnection relation. We first developed this framework for time-independent properties, and later extended it to time-dependent property composition. The extension requires, in addition, shifting the given properties along time to span the time horizon of interest, he least of which for the strongest system-property is no more than the total time horizons of the component level atomic-properties. The system-initial-condition is also composed from atomic-initial-conditions of the components the same way. It is used to verify a desired system-level property, alongside the derived strongest system-property, by way of induction. Our composition approach is uniform regardless of the composition types (cascade/parallel/feedback) for both time-dependent and time-independent properties. We developed a new prototype verifier named ReLIC (Reduced Logic Inference for Composition) that implements our above approaches. We demonstrated it through several illustrative and practical examples. Further, we advanced the $k$-induction based model-checking with QE capabilities, by formulating its base and inductive steps into QE problems where all the variables are universally quantified. Our integration of the QE solver Redlog with the $k$-induction based model-checking tool JKind, shows the successful solving of a non-linear problem that the SMT capable JKind failed to resolve. Finally, we also showcase the recent adoption of our approaches within an industrial V&V tool suite for augmented static analysis of Simulink models and Deep Neural Networks (DNNs).
We study the (hereditary) discrepancy of definable set systems, which is a natural measure for their inherent complexity and approximability. We establish a strong connection between the hereditary discrepancy and quantifier elimination over signatures with only unary relation and function symbols. We prove that set systems definable in theories (over such signatures) with quantifier elimination have constant hereditary discrepancy. We derive that set systems definable in bounded expansion classes, which are very general classes of uniformly sparse graphs, have bounded hereditary discrepancy. We also derive that nowhere dense classes, which are more general than bounded expansion classes, in general do not admit quantifier elimination over a signature extended by an arbitrary number of unary function symbols. We show that the set systems on a ground set $U$ definable on a monotone nowhere dense class of graphs $mathscr C$ have hereditary discrepancy at most $|U|^{c}$ (for some~$c<1/2$) and that, on the contrary, for every monotone somewhere dense class $mathscr C$ and for every positive integer $d$ there is a set system of $d$-tuples definable in $mathscr C$ with discrepancy $Omega(|U|^{1/2})$. We further prove that if $mathscr C$ is a class of graphs with bounded expansion and $phi(bar x;bar y)$ is a first-order formula, then we can compute in polynomial time, for an input graph $Ginmathscr C$, a mapping $chi:V(G)^{|bar x|}rightarrow{-1,1}$ witnessing the boundedness of the discrepancy of the set-system defined by~$phi$, an $varepsilon$-net of size $mathcal{O}(1/varepsilon)$, and an $varepsilon$-approximation of size $mathcal{O}(1/varepsilon)$.
The only C*-algebras that admit elimination of quantifiers in continuous logic are $mathbb{C}, mathbb{C}^2$, $C($Cantor space$)$ and $M_2(mathbb{C})$. We also prove that the theory of C*-algebras does not have model companion and show that the theory of $M_n(mathcal {O_{n+1}})$ is not $forallexists$-axiomatizable for any $ngeq 2$.
We discuss the topic of unsatisfiability proofs in SMT, particularly with reference to quantifier free non-linear real arithmetic. We outline how the methods here do not admit trivial proofs and how past formalisation attempts are not sufficient. We note that the new breed of local search based algorithms for this domain may offer an easier path forward.
Quantifier elimination over the reals is a central problem in computational real algebraic geometry, polynomial system solving and symbolic computation. Given a semi-algebraic formula (whose atoms are polynomial constraints) with quantifiers on some variables, it consists in computing a logically equivalent formula involving only unquantified variables. When there is no alternation of quantifiers, one has a one block quantifier elimination problem. This paper studies a variant of the one block quantifier elimination in which we compute an almost equivalent formula of the input. We design a new probabilistic efficient algorithm for solving this variant when the input is a system of polynomial equations satisfying some regularity assumptions. When the input is generic, involves $s$ polynomials of degree bounded by $D$ with $n$ quantified variables and $t$ unquantified ones, we prove that this algorithm outputs semi-algebraic formulas of degree bounded by $mathcal{D}$ using $O {widetilde{~}}left ((n-s+1) 8^{t} mathcal{D}^{3t+2} binom{t+mathcal{D}}{t} right )$ arithmetic operations in the ground field where $mathcal{D} = 2(n+s) D^s(D-1)^{n-s+1} binom{n}{s}$. In practice, it allows us to solve quantifier elimination problems which are out of reach of the state-of-the-art (up to $8$ variables).
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