Quantifier Elimination

We describe an algorithm for deciding the first-order multisorted theory BAPA, which combines Boolean algebras of sets of uninterpreted elements (BA) and Presburger arithmetic operations (PA). BAPA can express the relationship between integer variables and ...

We present a faithful axiomatization of von Mises' notion of a random sequence, using an abstract independence relation. A byproduct is a quantifier elimination theorem for Friedman's “almost all” quantifier in terms of this independence relation.

Ane size-change analysis has been used for termination analysis of eager functional programming languages. The same style of analysis is also capable of compactly recording and calculating other properties of programs, including their runtime, maximum stack depth, and (relative) path time costs. In this paper we show how precise poly- nomial bounds on such costs may be calculated on programs,

In this paper, we study the verification of dense time properties by discrete time analysis. Interval Duration Logic, (IDL), is a highly expressive dense time logic for specifying properties of real-time systems. Validity checking of IDL formulae is in general undecidable. A corresponding discrete-time logic QDDC has decidable validity. In this paper, we consider a reduction of IDL validity question to QDDC validity using notions of digitization. A new notion of Strong Closure under Inverse Digitization, SCID, is proposed. For all SCID formulae, the dense and the discrete-time validity coincide. Moreover, SCID has good algebraic properties which can be used to conveniently prove that many IDL formulae are SCID. We also give some approximation techniques to strengthen/weaken formulae to SCID form. For SCID formulae, the validity of dense-time IDL formulae can be checked using the validity checker for discrete-time logic QDDC.

SPASS is an automated theorem prover for full first-order logic with equality and a number of non-classical logics. This system description provides an overview of our recent developments in SPASS 3.5 including subterm contextual rewriting, improved split backtracking, a significantly faster FLOTTER implementation with additional control flags, completely symmetric implementation of forward and backward redundancy criteria, faster parsing with improved support for big files, faster and extended sort module, and support for include commands in input files. Finally, SPASS 3.5 can now parse files in TPTP syntax, comes with a new converter tptp2dfg and is distributed under a BSD style license.

Formal set theory is traditionally concerned with pure sets; consequently, the satisfiability problem for fragments of set theory was most often addressed (and in many cases positively solved) in the pure framework. In practical applications, however, it is common to assume the existence of a number of primitive objects (sometimes called atoms ) that can be members of sets but behave differently from them. If these entities are assumed to be devoid of members, the standard extensionality axiom must be revised; then decidability results can sometimes be achieved via reduction to the pure case and sometimes can be based on direct goal-driven algorithms. An alternative approach to modeling atoms that allows one to retain the original formulation of extensionality was proposed by Quine: atoms are self-singletons. In this article we adopt this approach in coping with the satisfiability problem: We show the decidability of this problem relativized to ∃*∀-sentences, and develop a goal-driv...

ABSTRACT. This paper describes an experimental platform for approximate knowledge databases called the Approximate Knowledge Database (AKDB), based on a semantics inspired by rough sets. The implementation is based upon the use of a standard SQL database to store ...

An algorithm is presented which eliminates second{order quantiers over predicate vari- ables in formulae of type 9P1; ... ;Pn where is an arbitrary formula of rst{order pred- icate logic. The resulting formula is equiva- lent to the original formula { if the algorithm terminates. The algorithm can for example be applied to do interpolation, to eliminate the second{order quantiers in

Many problems in control theory can be formulated as formulae in the first-order theory of real closed fields. In this paper we investigate some of the expressive power of this theory. We consider dynamical systems described by polynomial differential equations subjected to constraints on control and system variables and show how to formulate questions in the above framework which can be answered by quantifier elimination. The problems treated in this paper regard stationarity, stability, and following of a polynomially parametrized curve. The software package QEPCAD has been used to solve a number of examples.

We present for the first-order theory of atomic Boolean algebras of sets with linear cardinality constraints a quantifier elimination algorithm. In the case of atomic Boolean algebras of sets, this is a new generalization of Boole’s well-known variable elimination method for conjunctions of Boolean equality constraints. We also explain the connection of this new logical result with the evaluation of relational calculus queries on constraint databases that contain Boolean linear cardinality constraints.

Decomposable Negation Normal Form formulae (DNNFs) form an interesting propositional fragment, both for efficiency and succinctness reasons. A famous subclass of the DNNF fragment is the OBDD fragment which offers many polytime queries and transformations, including quantifier eliminations (under some ordering restrictions). Nevertheless, the decomposable AND nodes at work in OBDDs enable only sequential decisions: clusters of variables are never assigned "in parallel" like in full DNNFs. This is an serious drawback since succinctness for the full DNNF fragment relies on such a "parallelization property". This is why we suggest to go a step further, from (sequentially) ordered decision diagrams to (partially) ordered, decomposable decision graphs, in which any decomposable AND node is allowed, and not only assignment ones. We show that, like the OBDD fragment, such a new class offers many tractable queries and transformations, including quan- tifier eliminations ...

The problem of characterizing in a guaranteed way the set of all feasible set-points of a control problem is known to be difficult. In the present work, the problem to be solved involves non-linear equality constraints with variables affected by logical quantifiers. This problem is not solvable by current symbolic methods like quantifier elimination, which is commonly used for solving this class of problems. We propose the utilization of guaranteed set-computation techniques based on interval analysis, in particular a solver referred to as Quantified Set Inversion (QSI). As an application example, the problem of simultaneously controlling the speed and the orientation of a sailboat is presented. For this purpose, the combination of QSI solver and feedback linearization techniques is employed.

This paper presents a new algorithm for the absolute factorization of parametric multivariate polynomials over the field of rational numbers. This algorithm decomposes the parameters space into a finite number of constructible sets. The absolutely irreducible factors of the input parametric polynomial are given uniformly in each constructible set. The algorithm is based on a parametric version of Hensel's lemma and an algorithm for quantifier elimination in the theory of algebraically closed field in order to reduce the problem of finding absolute irreducible factors to that of representing solutions of zero-dimensional parametric polynomial systems. The complexity of this algorithm is single exponential in the number n of the variables of the input polynomial, its degree d w.r.t. these variables and the number r of the parameters.

This report introduces Q EPCAD , B, a program for computing with real algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD B both extends and improves upon the Q EPCAD system for quantifier elimination by partial cylindrical algebraic decomposition written by Hoon Hong in the early 1990s. This paper briefly discusses some of the improvements in the implementation of CAD and quantifier elimination via CAD, and provides somewhat more detail on extensions to the system that go beyond quantifier elimination. The author is responsible for most of the extended features of Q EPCAD B, but improvements to the basic CAD implementation and to the SACLIB library on which Q EPCAD is based are the results of many people's work, including: George E. Collins, Mark J. Encarnación, Hoon Hong, Jeremy Johnson, Werner Krandick, Richard Liska, Scott McCallum, Nicolas Robidoux, and Stanly Steinberg. Source code, documentation and installation instructions for QEPCAD B are all availab...

Tarski (1948) gave a quantifier elimination procedure for elementary algebra and geometry which, when implemented, was found to be impractical. Collins (1975) gave a different and far more efficient algorithm, which has revived interest in the feasibility of quantifier elimination as a means of solving nontrivial problems in elementary algebra and geometry. Collins' algorithm has recently been implemented (1981). Experience so far obtained with the software indicates that, while some worthwhile problems can now be solved, it is desirable to find methods for improving performance. In this paper we report on several such methods, that we have developed in the course of applying the Collins algorithm to the Kahan ellipse problem. These methods have yielded substantial performance improvements; in fact, they have made feasible computations relating to the Kahan problem that initially could not be carried out at all in our computing environment. We are pursuing the further development of these techniques, with the goal of eventually producing a solution to the Kahan problem using the Collins algorithm.