10705506
Linear logic proof games and optimization
§ 1. Introduction . Perhaps the most surprising recent development in complexity theory is the discovery that the class NP can be characterized using a form of randomized proof checker that only examines a constant number of bits of the “proof” that a string is in a language [6, 5, 31, 3, 4]. More specifically, writing ∣ x ∣ for the length of a string x , a language L in the class NP of languages recognizable in Nondeterministic polynomial time is traditionally given by a polynomial p and a polynomial-time predicate P such that a string x is in L iff there is some string y satisfying P (x, y), where ∣ y ∣ ≤ p (∣ x ∣). Intuitively, we can think of a string y as a possible proof that x ϵ L , with the predicate P some kind of proof checker that distinguishes good proofs from bad ones. A string x is therefore in a language L ϵ NP if there is a short proof that x ϵ L , and not in L otherwise. The surprising discovery is that the deterministic proof checker that reads the entire input and proof can be replaced by a probabilistic verifier that on input x and possible proof y , where y is presented in a certain way, flips at most O (log ∣ x ∣) coins and reads at most a constant number of bits of x and y . Obviously, a probabilistic verifier that does not read the whole proof will sometimes make mistakes. However, the surprising and essentially non-intuitive mathematical fact is that for each language L in NP, it is possible to find a polynomial q and verifier V flipping a logarithmic number of coins and reading a constant number of bits such that, for any x ϵ L , there exists y with ∣ y ∣ ≤ q(∣ x ∣) and with V (x, y) accepting with probability 1 and, for x ∉ L , there is no y with ∣ y ∣ ≤ q (∣ x ∣) and with V (x, y) accepting with probability ≥ 1/4. This idea canalsobeextended to PSPACE [10, 9], using a game that alternates between two players instead of a proof presented by a “single player.”
3062362
Managing Digital Rights using Linear Logic
Digital music players protect songs by enforcing licenses that convey specific rights for individual songs or groups of songs. For licenses specified in industry, we show that deciding whether a license authorizes a sequence of actions is NP-complete, with a restricted version of the problem solvable efficiently using a reduction to maximum network flow. The authorization algorithm used in industry is online, deciding which rights to exercise as actions occur, but we show that all online algorithms are necessarily non-monotonic: each allows actions under one license that it does not allow under a more flexible license. In one approach to achieving monotonicity, we exhibit the unique maximal set of licenses on which there exists a monotonic online algorithm. This set of well-behaved licenses induces an approximation algorithm by replacing each license with a well-behaved license. In a second approach, we consider allowing the player to revise its past decisions about which rights to exercise while still ensuring compliance with the license. We propose an efficient algorithm based on linear logic, with linear negation used to revise past decisions. We prove our algorithm monotonic, live, and sound with respect to the semantics of licenses
9031501
Design of a role-based trust-management framework
We introduce the RT framework, a family of role-based trust management languages for representing policies and credentials in distributed authorization. RT combines the strengths of role-based access control and trust-management systems and is especially suitable for attribute-based access control. Using a few simple credential forms, RT provides localized authority over roles, delegation in role definition, linked roles, and parameterized roles. RT also introduces manifold roles, which can be used to express threshold and separation-of-duty policies, and delegation of role activations. We formally define the semantics of credentials in the RT framework by presenting a translation from credentials to Datalog rules. This translation also shows that this semantics is algorithmically tractable.
215952437
Beyond proof-of-compliance: security analysis in trust management
Trust management is a form of distributed access control that allows one principal to delegate some access decisions to other principals. While the use of delegation greatly enhances flexibility an...
18708753
Unification and ML-Type Reconstruction
We study the complexity of type reconstruction for a core fragment of ML with lambda abstraction, function application, and the polymorphic let declaration. We derive exponential upper and lower bounds on recognizing the typable core ML expressions. Our primary technical tool is uniication of succinctly represented type expressions. After observing that core ML expressions, of size n, can be typed in DTIME(2 n), we exhibit two diierent families of programs whose principal types grow exponentially. We show how to exploit the expressiveness of the let-polymorphism in these constructions to derive lower bounds on deciding typability: one leads naturally to NP-hardness and the other to DTIME(2 n k)-hardness for each integer k 1. Our generic simulation of any exponential time Turing Machine by ML type reconstruction may be viewed as a nonstandard way of computing with types. Our worst-case lower bounds stand in contrast to practical experience, which suggests that commonly used algorithms for type reconstruction do not slow compilation substantially.
16397953
Parametricity and Variants of Girard's J Operator
Abstract The Girard–Reynolds polymorphic λ -calculus is generally regarded as a calculus of parametric polymorphism in which all well-formed terms are strongly normalizing with respect to β -reductions. Girard demonstrated that the addition of a simple “non-parametric” operation, J , to the calculus allows the definition of a non-normalizing term. Since the type of J is not inhabited by any closed term, one might suspect that this may play a role in defining a non-normalizing term using it. We demonstrate that this is not the case by giving a simple variant, J′ , of J whose type is otherwise inhabited and which causes normalization to fail. It appears that impredictivity is essential to the argument; predictive variants of the polymorphic λ -calculus admit non-parametric operations without sacrificing normalization.
7524589
Sweeping simple polygons with a chain of guards
We consider the problem of locating a continuously-moving target using a group of guards moving inside a simple polygon. Our guards always form a simple polygonal chain within the polygon such that consecutive guards along the chain are mutually visible. We develop algorithms that sweep such a chain of guards through a polygon to locate the target. Our two main results are the following: 1. an algorithm to compute the minimum number r∗ of guards needed to sweep an n-vertex polygon that runs in O(n) time and uses O(n) working space, and 2. a faster algorithm, using O(n log n) time and O(n) space, to compute an integer r such that max(r − 16, 2) ≤ r∗ ≤ r and P can be swept with a chain of r guards. We develop two other techniques to approximate r∗. Using O(n) time and space, we show how to sweep the polygon using at most r∗ +2 guards. We also show that any polygon can be swept by a number of guards equal to two more than the link radius of the polygon. As a key component of our exact algorithm, we introduce the notion of the link diagram of a polygon, which encodes the link distance between all pairs of points on the boundary of the polygon. We prove that the link diagram has size Θ(n) and can be constructed in Θ(n) time. We also show link diagram provides a data structure for optimal two-point link-distance queries, matching an earlier result of Arkin et al. As a key component of our O(n log n)-time approximation algorithm, we introduce the notion of the “link width” of a polygon, which may have independent interest, as it captures important structural properties of simple polygons. Computer Science Dept., Stanford University, 353 Serra Mall, Stanford CA 94305. Email: alon@cs.stanford.edu. Supported by a Rothschild Fellowship and by NSF research grant CCR-96-23851. Computer Science Dept., Stanford University, 353 Serra Mall, Stanford CA 94305. Email: guibas@cs.stanford.edu. Partially supported by DARPA grant DAAE07-98-C-L027 and by ARO MURI grant DAAH04-96-1-007. School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. Email: sariel@math.tau.ac.il. Computer Science Dept., Stanford University, 353 Serra Mall, Stanford CA 94305. Email: dlin@cs.stanford.edu. Supported by DARPA grant DAAE07-98-C-L027 and by ARO MURI grant DAAH04-96-1-007. Dept. Applied Math, University at Stony Brook, Stony Brook, NY 11794-3600. Email: jsbm@ams.sunysb.edu. Partially supported by NSF (CCR-9732220), and grants from Hughes Research Labs, ISX Corporation, Seagull Technologies, NASA, and Sun Microsystems. Computer Science Dept., Stanford University, 353 Serra Mall, Stanford CA 94305. Email: murali@cs.stanford.edu. Supported by DARPA grant DAAE07-98-C-L027.
15492927
Counting Convex Polygons in Planar Point Sets
Given a set S of n points in the plane, we compute in time O(n3) the total number of convex polygons whose vertices are a subset of S. We give an O(m · n3) algorithm for computing the number of convex k-gons with vertices in S, for all values k = 3,…, m; previously known bounds were exponential (O(nk/2). We also compute the number of empty convex polygons (resp.,k-gons, k m) with vertices in S in time O(n3) (resp., O(m · n3)).
14267496
Recognizing polygonal parts from width measurements
Abstract Automatic recognition of parts is an important problem in many industrial applications. One model of the problem is: given a finite set of polygonal parts, use a set of “width” measurements taken by a parallel-jaw gripper to determine which part is present. We study the problem of computing efficient strategies (“grasp plans”), with the goal to minimize the number of measurements necessary in the worst case. We show that finding a minimum length grasp plan is NP -hard, and give a polynomial time approximation algorithm that is simple and produces a solution that is within a log factor from optimal.
207159283
Minimum-cost coverage of point sets by disks
We consider a class of geometric facility location problems in which the goal is to determine a set <i>X</i> of disks given by their centers <i>(t<sub>j</sub>)</i> and radii <i>(r<sub>j</sub>)</i> that cover a given set of demand points <i>Y∈R</i><sup>2</sup> at the smallest possible cost. We consider cost functions of the form Ε<i><sub>j</sub>f(r<sub>j</sub>)</i>, where <i>f(r)=r</i><sup>α</sup> is the cost of transmission to radius <i>r</i>. Special cases arise for α=1 (sum of radii) and α=2 (total area); power consumption models in wireless network design often use an exponent α>2. Different scenarios arise according to possible restrictions on the transmission centers <i>t<sub>j</sub></i>, which may be constrained to belong to a given discrete set or to lie on a line, etc.We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points <i>t<sub>j</sub></i> on a given line in order to cover demand points <i>Y∈R</i><sup>2</sup>; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover <i>Y</i>; (c) a proof of NP-hardness for a discrete set of transmission points in <i>R<sup>2</sup></i> and any fixed α>1; and (d) a polynomial-time approximation scheme for the problem of computing a <i>minimum cost covering tour</i> (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.
16007270
Point Probe Decision Trees for Geometric Concept Classes
A fundamental problem in model-based computer vision is that of identifying to which of a given set of concept classes of geometric models an observed model belongs. Considering a “probe” to be an oracle that tells whether or not the observed model is present at a given point in an image, we study the problem of computing efficient strategies (“decision trees”) for probing an image, with the goal to minimize the number of probes necessary (in the worst case) to determine in which class the observed model belongs. We prove a hardness result and give strategies that obtain decision trees whose height is within a log factor of optimal.