The complexity of homomorphisms and renamings for minimal unsatisfiable formulas

We investigate the complexity of deciding whether for minimal unsatisfiable formulas F and H there exists a variable renaming, a literal renaming or a homomorphism such that (F)=H. A variable renaming is a permutation of variables. A literal renaming is a permutation of variables which additionally replaces some of the variables by its complements. A homomorphism can be considered as a literal renaming which can map different literals to one literal.     

Exploiting transition locality in automatic verification of finite-state concurrent systems

In this paper we show that statistical properties of the transition graph of a system to be verified can be exploited to improve memory or time performances of verification algorithms.We show experimentally that protocols exhibit transition locality. That is, with respect to levels of a breadth-first state space exploration, state transitions tend to be between states belonging to close levels of the transition graph. We support our claim by measuring transition locality for the set of protocols included in the Mur verifier distribution .We present a cache-based verification algorithm that exploits transition locality to decrease memory usage and a disk-based verification algorithm that exploits transition locality to decrease disk read accesses, thus reducing the time overhead due to disk usage. Both algorithms have been implemented within the Mur verifier.Our experimental results show that our cache-based algorithm can typically save more than 40% of memory with an average time penalty of about 50% when using (Mur) bit compression and 100% when using bit compression and hash compaction, whereas our disk-based verification algorithm is typically more than ten times faster than a previously proposed disk-based verification algorithm and, even when using 10% of the memory needed to complete verification, it is only between 40 and 530% (300% on average) slower than (RAM) Mur with enough memory to complete the verification task at hand. Using just 300 MB of memory our disk-based Mur was able to complete verification of a protocol with about 10⁹ reachable states. This would require more than 5 GB of memory using standard Mur .     

The Kreisel length-of-proof problem

A first-order system F has theKreisel length-of-proof property if the following statement is true for all formulas(x): If there is ak1 such that for alln0 there is a proof of(¯n) in F with at mostk lines, then there is a proof of x(x) in F. We consider this property for Parikh systems, which are first-order axiomatic systems that contain a finite number of axiom schemata (including individual axioms) and a finite number of rules of inference. We prove that any usual Parikh system formulation of Peano arithmetic has the Kreisel length-of-proof property if the underlying logic of the system is formulated without a schema for universal instantiation in either one of two ways. (In one way, the formula to be instantiated is built up in steps, and in the other way, the term to be substituted is built up in steps.) Our method of proof uses techniques and ideas from unification theory.     

Definability of Polyadic Lifts of Generalized Quantifiers

We study generalized quantifiers on finite structures.With every function : we associate a quantifier Q by letting Q x say there are at least (n) elementsx satisfying , where n is the sizeof the universe. This is the general form ofwhat is known as a monotone quantifier of type < 1 >.We study so called polyadic liftsof such quantifiers. The particular lifts we considerare Ramseyfication, branching and resumption.In each case we get exact criteria fordefinability of the lift in terms of simpler quantifiers.     

Decision procedures for elementary sublanguages of set theory: X. Multilevel syllogistic extended by the singleton and powerset operators

The class of unquantified formulae of set theory involving Boolean operators, the powerset and singleton operators, and the equality and membership predicates is shown to have a solvable satisfiability problem.It is shown that whenever a formula in the above class is satisfiable there exists a hereditarily finite model of , whose rank is bounded by a doubly exponential expression in the number of variables occurring in .This paper was written while the author was a Visiting Professor at Courant Institute of Mathematical Sciences, New York University, U.S.A.     

How to Compute Interval Inclusions of Geodetic Coordinates from Interval Inclusions of Cartesian Coordinates

In satellite geodesy the position of a point P is usually determined by computing its coordinate vector x with respect to an earth-fixed Cartesian coordinate system S. S is chosen such that a rotational ellipsoid E, closely approximating the surface of the earth, has normal form with respect to S. Since the geodetic coordinates of P with respect to E (ellipsoidal latitude , ellipsoidal longitude , and ellipsoidal height h) describe the location of P with respect to the surface of the earth much better than x, a frequently appearing problem is to compute , , and h from x.In practice this problem is solved by iterative methods, the convergence properties of which are not analyzed in detail but for which fast (numerical) convergence is observed for points near to E.In the present paper a theoretically well founded new method is developed, working for all P having a unique nearest point in E.In addition it will be shown that the new method can be implemented such that interval inclusions for , , and h can be computed from interval inclusions of the components of x.     

The relationship between knowledge,belief, and certainty

We consider the relation between knowledge and certainty, where a fact isknown if it is true at all worlds an agent considers possible and iscertain if it holds with probability 1. We identify certainty with belief, interpreted probabilistically. We show that if we assume one fixed probability assignment, then the logic KD45, which has been identified as perhaps the most appropriate for belief, provides a complete axiomatization for reasoning about certainty. Just as an agent may believe a fact although is false, he may be certain that a fact is true although is false. However, it is easy to see that an agent can have such false (probabilistic) beliefs only at a set of worlds of probability 0. If we restrict attention to structures where all worlds have positive probability, then S5 provides a complete axiomatization. If we consider a more general setting, where there might be a different probability assignment at each world, then by placing appropriate conditions on thesupport of the probability function (the set of worlds which have non-zero probability), we can capture many other well-known modal logics, such as T and S4. Finally, we considerMiller's principle, a well-known principle relating higher-order probabilities to lower-order probabilities, and show that in a precise sense KD45 characterizes certainty in those structures satisfying Miller's principle.     

The Use of Certain Equations of Mathematical Physics in Optimization of a Function on a Set. I

The results of application of potential theory to optimization are used to extend the use of (Helmholtz) diffusion and diffraction equations for optimization of their solutions (x, ) with respect to both x, and . If the aim function is modified such that the optimal point does not change, then the function (x, ) is convex in (x, for small . The possibility of using heat conductivity equation with a simple boundary layer for global optimization is investigated. A method is designed for making the solution U(x,t) of such equations to have a positive-definite matrix of second mixed derivatives with respect to x for any x in the optimization domain and any small t < 0 (the point is remote from the extremum) or a negative-definite matrix in x (the point is close to the extremum). For the functions (x, ) and U(x,t) having these properties, the gradient and the Newton–Kantorovich methods are used in the first and second stages of optimization, respectively.     

Decision procedures for elementary sublanguages of set theory

In this paper we prove the decidability of the class of unquantified formulae of set theory involving the operators , , , \, {·}, pred _< and the predicates =, , , Finite, where pred _<(s) denotes the collection of all sets having rank strictly less than the rank of s.This work generalizes and combines earlier results published in the same series of papers.This work has been partially supported by ENI and ENIDATA within the AXL project.