Analyzing restricted fragments of the theory of linear arithmetic

The theory of linear arithmetic (TLA), also known as the Theory of Rationals, is an extremely well-studied theory. It has been widely applied to a number of domains, including program verification and constraint specification. This paper discusses the computational complexities of two broad fragments of TLA, namely Quantified Linear Programs (QLPs) and Quantified Linear Implications (QLIs). These fragments are ideal for expressing specifications in real-time scheduling, and for modeling reactive systems. In this paper, we study the computational complexities of several variants of QLPs and QLIs. Our principal result shows that there exists a one-to-one correspondence between alternations in a class of QLIs and the complexity classes comprising the polynomial hierarchy PH. In other words, for each class of the PH, there exists a class of QLIs that is complete for that class. Our work mirrors the work in [L.J. Stockmeyer, The Polynomial-Time Hierarchy, Theoretical Computer Science, 3:1-22, 1977] which established the connection between the classes of PH and quantifier alternations in a Quantified Boolean Formula. Our results are surprising, since the variables in the fragments that we consider are continuous, as opposed to discrete.

[1]  Volker Weispfenning,et al.  The Complexity of Linear Problems in Fields , 1988, Journal of symbolic computation.

[2]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[3]  Birgit Pfitzmann,et al.  A model for asynchronous reactive systems and its application to secure message transmission , 2001, Proceedings 2001 IEEE Symposium on Security and Privacy. S&P 2001.

[4]  L. Khachiyan.  Polynomial algorithms in linear programming , 1980 .

[5]  Hans Kleine Büning,et al.  Resolution for Quantified Boolean Formulas , 1995, Inf. Comput..

[6]  Peter W. Shor,et al.  Stretchability of Pseudolines is NP-Hard , 1990, Applied Geometry And Discrete Mathematics.

[7]  Wayne F. Bialas,et al.  Two-Level Linear Programming , 1984 .

[8]  Marcus Schaefer.  Completeness in the Polynomial Time Hierarchy , 2001 .

[9]  Marie-Françoise Roy,et al.  Existential Theory of the Reals , 2003 .

[10]  Lane A. Hemaspaandra,et al.  SIGACT news complexity theory comun 37 , 2002, SIGA.

[11]  David Harel,et al.  A Grand Challenge: Full Reactive Modeling of a Multi-cellular Animal , 2003, HSCC.

[12]  Birgit Pfitzmann,et al.  Composition and integrity preservation of secure reactive systems , 2000, CCS.

[13]  David Harel,et al.  The immune system as a reactive system: modeling T cell activation with statecharts , 2001, Proceedings IEEE Symposia on Human-Centric Computing Languages and Environments (Cat. No.01TH8587).

[14]  M. A. Reichert,et al.  Computing the Decomposition of Primes p and p-adic Absolute Values in Semi-simple Algebras over Q , 1987, J. Symb. Comput..

[15]  Lane A. Hemaspaandra.  SIGACT news complexity theory column 38 , 2002, SIGA.

[16]  Jeanne Ferrante,et al.  A Decision Procedure for the First Order Theory of Real Addition with Order , 1975, SIAM J. Comput..

[17]  Richard Gerber,et al.  Parametric Dispatching of Hard Real-Time Tasks , 1995, IEEE Trans. Computers.

[18]  M. Schaefer,et al.  Completeness in the Polynomial-Time Hierarchy A Compendium ∗ , 2008 .

[19]  K. Subramani,et al.  On quantified linear implications , 2014, Annals of Mathematics and Artificial Intelligence.

[20]  Thomas Sturm,et al.  A New Approach for Automatic Theorem Proving in Real Geometry , 1998, Journal of Automated Reasoning.

[21]  Thomas Sturm,et al.  Real Quantifier Elimination in Practice , 1997, Algorithmic Algebra and Number Theory.

[22]  K. Subramani,et al.  On a decision procedure for quantified linear programs , 2007, Annals of Mathematics and Artificial Intelligence.

[23]  Eduardo D. Sontag,et al.  Real Addition and the Polynomial Hierarchy , 1985, Inf. Process. Lett..

[24]  G. Sacks.  A DECISION METHOD FOR ELEMENTARY ALGEBRA AND GEOMETRY , 2003 .

[25]  George E. Collins,et al.  Partial Cylindrical Algebraic Decomposition for Quantifier Elimination , 1991, J. Symb. Comput..

[26]  Thomas Sturm,et al.  REDLOG: computer algebra meets computer logic , 1997, SIGS.

[27]  Christopher W. Brown.  QEPCAD B: a program for computing with semi-algebraic sets using CADs , 2003, SIGS.

[28]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[29]  John F. Canny,et al.  Some algebraic and geometric computations in PSPACE , 1988, STOC '88.

[30]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

[31]  James H. Davenport,et al.  Real Quantifier Elimination is Doubly Exponential , 1988, J. Symb. Comput..

[32]  A. Sangiovanni-Vincentelli,et al.  A formal approach to reactive system design: unmanned aerial vehicle flight management system design example , 1999, Proceedings of the 1999 IEEE International Symposium on Computer Aided Control System Design (Cat. No.99TH8404).

[33]  Zohar Manna,et al.  The calculus of computation - decision procedures with applications to verification , 2007 .

[34]  Ashok K. Agrawala,et al.  Dynamic Dispatching of Cyclic Real-Time Tasks with Relative Timing Constraints , 2000, Real-Time Systems.

[35]  Stefan Ratschan,et al.  Efficient solving of quantified inequality constraints over the real numbers , 2002, TOCL.