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Register a new userGenerating Non-Linear Interpolants by Semidefinite Programming
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Dai Liyun
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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Interpolation-based techniques have been widely and successfully applied in the verification of hardware and software, e.g., in bounded-model check- ing, CEGAR, SMT, etc., whose hardest part is how to synthesize interpolants. Various work for discovering interpolants for propositional logic, quantifier-free fragments of first-order theories and their combinations have been proposed. However, little work focuses on discovering polynomial interpolants in the literature. In this paper, we provide an approach for constructing non-linear interpolants based on semidefinite programming, and show how to apply such results to the verification of programs by examples.
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Invariant synthesis plays a central role in the verification of programs. In this paper, we propose a novel approach to synthesize basic semialgebraic invariants using semidefinite programming (SDP) that combines advantages of both symbolic constraint solving methods and numeric constraint solving methods. The advantages of our approach are threefold: first, it can deal with arbitrary templates as symbolic computation based techniques; second, it uses SDP instead of computationally intensive symbolic subroutines and is therefore efficient enough as other numeric computation based techniques; lastly, there are some (although weaker) theoretical guarantees on its completeness, which previously can only be provided by symbolic computation based techniques.
The tendency of semidefinite programs to compose perfectly under product has been exploited many times in complexity theory: for example, by Lovasz to determine the Shannon capacity of the pentagon; to show a direct sum theorem for non-deterministic communication complexity and direct product theorems for discrepancy; and in interactive proof systems to show parallel repetition theorems for restricted classes of games. Despite all these examples of product theorems--some going back nearly thirty years--it was only recently that Mittal and Szegedy began to develop a general theory to explain when and why semidefinite programs behave perfectly under product. This theory captured many examples in the literature, but there were also some notable exceptions which it could not explain--namely, an early parallel repetition result of Feige and Lovasz, and a direct product theorem for the discrepancy method of communication complexity by Lee, Shraibman, and Spalek. We extend the theory of Mittal and Szegedy to explain these cases as well. Indeed, to the best of our knowledge, our theory captures all examples of semidefinite product theorems in the literature.
Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method was developed to find asymptotic results for very large graphs, so it seems that the method is not suitable for finding small Ramsey numbers. But this intuition is wrong, and we will develop a technique to do just that in this paper. We find new upper bounds for many small graph and hypergraph Ramsey numbers. As a result, we prove the exact values $R(K_4^-,K_4^-,K_4^-)=28$, $R(K_8,C_5)= 29$, $R(K_9,C_6)= 41$, $R(Q_3,Q_3)=13$, $R(K_{3,5},K_{1,6})=17$, $R(C_3, C_5, C_5)= 17$, and $R(K_4^-,K_5^-;3)= 12$. We hope that this technique will be adapted to address other questions for smaller graphs with the flag algebra method.
Semidefinite Programming (SDP) is a class of convex optimization programs with vast applications in control theory, quantum information, combinatorial optimization and operational research. Noisy intermediate-scale quantum (NISQ) algorithms aim to make an efficient use of the current generation of quantum hardware. However, optimizing variational quantum algorithms is a challenge as it is an NP-hard problem that in general requires an exponential time to solve and can contain many far from optimal local minima. Here, we present a current term NISQ algorithm for SDP. The classical optimization program of our NISQ solver is another SDP over a smaller dimensional ansatz space. We harness the SDP based formulation of the Hamiltonian ground state problem to design a NISQ eigensolver. Unlike variational quantum eigensolvers, the classical optimization program of our eigensolver is convex, can be solved in polynomial time with the number of ansatz parameters and every local minimum is a global minimum. Further, we demonstrate the potential of our NISQ SDP solver by finding the largest eigenvalue of up to $2^{1000}$ dimensional matrices and solving graph problems related to quantum contextuality. We also discuss NISQ algorithms for rank-constrained SDPs. Our work extends the application of NISQ computers onto one of the most successful algorithmic frameworks of the past few decades.
We give a categorical semantics for a call-by-value linear lambda calculus. Such a lambda calculus was used by Selinger and Valiron as the backbone of a functional programming language for quantum computation. One feature of this lambda calculus is its linear type system, which includes a duplicability operator ! as in linear logic. Another main feature is its call-by-value reduction strategy, together with a side-effect to model probabilistic measurements. The ! operator gives rise to a comonad, as in the linear logic models of Seely, Bierman, and Benton. The side-effects give rise to a monad, as in Moggis computational lambda calculus. It is this combination of a monad and a comonad that makes the present paper interesting. We show that our categorical semantics is sound and complete.
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