Hierarchy theorems for two-way finite state transducers

Summary For a family of languages , CAL() is defined as the family of images of under nondeterministic two-way finite state transducers, while FINITE · VISIT() is the closure of under deterministic two-way finite state transducers; CAL⁰()= and for n0, CAL ⁿ⁺¹()=CAL ⁿ (CAL()). For any semiAFL , if FINITE · VISIT() CAL(), then CAL ⁿ () forms a proper hierarchy and for every n0, FINITE · VISIT(CALⁿ()) CAL ⁿ⁺¹() FINITE · VISIT(CAL ⁿ⁺¹()). If is a SLIP semiAFL or a weakly k-iterative full semiAFL or a semiAFL contained in any full bounded AFL, then FINITE · VISIT() CAL() and in the last two cases, FINITE · VISIT(). If is a substitution closed full principal semiAFL and FINITE · VISIT(), then FINITE · VISIT() CAL(). If is a substitution closed full principal semiAFL generated by a language without an infinite regular set and ₁ is a full semiAFL, then is contained in CAL^m(₁) if and only if it is contained in ₁. Among the applications of these results are the following. For the following families , CAL ⁿ () forms a proper hierarchy: =INDEXED, =ETOL, and any semiAFL contained in CF. The family CF is incomparable with CAL ^m (NESA) where NESA is the family of one-way nonerasing stack languages and INDEXED is incomparable with CAL ^m (STACK) where STACK is the family of one-way stack languages.This work was supported in part by the National Science Foundation under Grants No. DCR74-15091 and MCS-78-04725     

Separating complexity classes related to bounded alternating ω-branching programs

We develop a theory of communication within branching programs that provides exponential lower bounds on the size of branching programs that are bounded alternating. Our theory is based on the algebraic concept of -branching programs, : , a semiring homomorphism, that generalizes ordinary branching programs, -branching programs [M2] andMOD _p-branching programs [DKMW].Due to certain exponential lower and polynomial upper bounds on the size of bounded alternating -branching programs we are able to separate the corresponding complexity classesN _ba ,co-N _{ba ba} , andMOD _p - _ba ,p prime, from each other, and from that classes corresponding to oblivious linear length-bounded branching programs investigated in the past.     

A note on classes of complements and the LBA problem

Summary The complements of an AFL form an AFL if and only if is closed under length-preserving universal quantification. The complements of the context-sensitive languages form a principal AFL with a hardest set L ₁. The context-sensitive languages are closed under complementation if and only if L ₁ is context-sensitive.This research was supported in part by the National Science Foundation under Grants MCS76-10076 and DCR74-15091     

Iterated linear control and iterated one-turn pushdowns

For a class of languages, an-controlled linear grammarK consists of a linear context-free grammarG and a control languageH in, where the terminals ofH are interpreted as the labels of rules ofG. The language generated byK is obtained by derivations ofG such that the corresponding strings of labels of the rules applied are control strings inH. The control of linear grammars can be iterated by starting with and by taking the result of thekth step as class of control languages for the (k + 1)st step. The language class obtained by thekth step is denoted by CTRL_k(). Denote by(S) the language class accepted by nondeterministic one-wayS automata, whereS is a storage type. We prove that for anyS, CTRL_k((S)) = (P _1t ^k (S)), whereP _1t ^k (S) is the storage type the configurations of which consist ofk-iterated one-turn pushdowns ofS-configurations. We establish a strong connection between iterated linear control and iterated one-turn pushdowns. In particular, we characterize CTRL _k ( _CF), where _CF is the class of context-free languages, by iterated one-turn pushdown automata in which the innermost pushdown is unrestricted.The work of the author has been supported by the Netherlands Organization for the advancement of pure research (Z.W.O.).     

Chains of full AFL's

If a full AFL is not closed under substitution, then ô , the result of substituting members of into, is not substitution closed and hence generates an infinite hierarchy of full AFL's. If ₁ and ₂ are two incomparable full AFL's, then the least full AFL containing ₁ and ₂ is not substitution closed. In particular, the substitution closure of any full AFL properly contained in the context-free languages is itself properly contained in the context-free languages. If any set of languages generates the context-free languages, one of its members must do so. The substitution closure of the one-way stack languages is properly contained in the nested stack languages. For eachn, there is a class of full context-free AFL's whose partial ordering under inclusion is isomorphic to the natural partial ordering onn-tuples of positive integers.This paper was completed while the author was in the Division of Engineering and Applied Physics of Harvard University. Research sponsored in part by the Air Force Cambridge Research Laboratories, Office of Aerospace Research, USAF, under contracts F-1962870C0023 and F-1962868C0029, and by the Air Force Office of Scientific Research, Office of Aerospace Research, USAF, under AFOSR Grant No. AF-AFOSR-1203-67A and the Division of Engineering and Applied Physics of Harvard University.     

What holds in a context?

In order to establish that [A] = [B] follows from a set of assumptions often one provesA =B and then invokes the principle of substitution of equals for equals. It has been observed that in the ancillary proof ofA =B one is allowed to use, in addition to those assumptions of which are free for , certain (open) sentencesP which may not be part of and may not follow from , but are related to the context . We show that in an appropriate formal system there is a closed form solution to the problem of determining precisely what sentencesP can be used. We say that those sentenceshold in the context under the set of assumptions . We suggest how the solution could be exploited in an interactive theorem prover.     

On the sequence of authorization policy transformations

In [2, 3], we proposed a model-based approach to specify the transformation of authorizations based on the principle of minimal change [10] and its application in database systems. Nevertheless, there were some limitations in this approach. Firstly, we could not represent a sequence of transformations. Secondly, default authorizations could not be expressed. In this paper, we propose two high-level formal languages, ^s and ^sd, to specify a sequence of authorization transformations and default authorizations. Our work starts with ^s, a simple, but expressive, language to specify certain sequence of authorization transformations. Furthermore, ^sd has more powerful expressiveness than ^s in the sense that constraints, causal and inherited authorizations, and general default authorizations can be specified.     

Bicentres de langages algébriques

Resume Les notions de bicentre et bicentre strict d'un langage, définies par A. De Luca, A. Restivo et S. Salemi généralisent la notion de centre d'un langage définie par M. Nivat. L'objet du présent papier est de répondre á la question suivante lorsque désigne la famille des langages algébriques ou l'une de ses sous-familles classiques:Si L appartient à , le bicentre de L (respectivement le bicentre strict de L) appartient-il à ?Le principal résultat est une réponse positive à cette question lorsqu'il s'agit de la notion de bicentre et que est un full-AFL uniforme de langages algébriques.

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Bicenters of context-free languages

Summary The notions of bicenter and strict bicenter of a language have been defined by A. De Luca, A. Restivo and S. Salemi and are a generalisation of the notion of center of a language, defined by M, Nivat. This paper deals with the following question, when is the family of context-free languages or one of its classical subfamilies:when L is in , is the bicenter (resp. the strict bicenter) of L also in ?Concerning the notion of bicenter, the main result of the paper is a positive answer when is a uniform full-AFL of context-free languages.

     

Sound and complete Hoare-like calculi based on copy rules

Summary This paper presents a uniform approach to known and new results on relative completeness of Hoare-like calculi for languages of ALGOL-like programs with procedures as procedure parameters. First the notion of a copy rule is introduced. It provides a uniform framework for dealing with different variants of semantics reaching from dynamic to static scope. Then for each copy rule a Hoare-like calculus () is presented, the soundness of which is shown by using an approximating semantics. The key to the completeness results lies in a general completeness theorem on the calculi () which has these results as corollaries. Finally, a new type of theorem on Hoare-like calculi is proved by which the notion of formal provability in () is completely characterized. This characterization theorem is the main result of the paper. It covers both soundness and completeness of the calculi () and additionally gives an idea of what the limits of presently known Hoare-like proof techniques for programming languages with procedures are.     

On generators and generative capacity of EOL forms

Summary In this paper we study the generative capacity of EOL forms from two different points of view. On the one hand, we consider the generative capacity of special EOL forms which one could call linear like and context free like, establishing the existence of a rich variety of non-regular sub-EOL language families. On the other hand, we propose the notion of a generator L of a language family We mean by this that any synchronized EOL system generating L generates — if understood as an EOL form — all languages of . We characterize the generators of the family of regular languages, and prove that other well known language families do not have generators.Partially supported under NSE Research Council of Canada, grant No. A-7700     

Graphs,causality, and stabilizability: Linear,shift-invariant systems on ℒ2[0, ∞)

This paper presents a number of basic elements for a system theory of linear, shift-invariant systems on ₂[0, ). The framework is developed from first principles and considers a linear system to be a linear (possibly unbounded) operator on ₂[0, ). The properties of causality and stabilizability are studied in detail, and necessary and sufficient conditions for each are obtained. The idea of causal extendibility is discussed and related to operators defined on extended spaces. Conditions for w-stabilizability and w-stability are presented. The graph of the system (operator) plays a unifying role in the definitions and results. We discuss the natural partial order on graphs (viewed as subspaces) and its relevance to systems theory.Supported in part by the NSF and the AFOSR, U.S.A. and the Nuffield Foundation, U.K.     

Lengths of Semigraphoid Inferences

A subset of a semigraphoid K over n elements is constructed in such a way that starting from it is necessary to apply semigraphoid axioms recursively 2 ^n–2–1 times to arrive at K. This is first known example of exponentially long semigraphoid inference. A comparison is made between local and global inferences. Graphoids and their duals are discussed as well.     

Langages saturés et cônes décroissants langages et cônes bifidèles

Résumé C() (resp.C ^d (–),C ^f (–)) est la plus petite famille de langages contenant et fermée par homomorphisme (resp. homomorphisme alphabétique, homomorphisme non effaçant), homomorphisme inverse (resp. homomorphisme strictement alphabétique inverse, homomorphisme inverse) et par intersection rationnelle.C ^d (–) etC ^f (–) sont donc des cônes restreints deC(–). Nous donnons des relations entre cônes restreints deC(–) etC(–). Ce qui nous permet de construire une hiérarchie infinie de langages algébriques non effaçables (L est effaçable C({L})=C ^f ({L})).

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Saturated languages and decreasing cones bifaithful languages and bifaithful cones

Summary C() (resp.C ^d (–),C ^f (–)) is the smallest family of languages containing and closed under morphism (resp. alphabetic morphism, non erasing morphism), inverse morphism (resp. inverse alphabetic non erasing morphism, inverse morphism) and intersection with regular sets.C ^d (–) andC ^f (–) are restricted cones ofC(). We give relations between restricted cones ofC() andC(), that allows us to construct an infinite hierarchy of no-erasable context-fee languages (L is erasableC({L})=C ^f ({L}))

     

An interpretation of typed OOP in a language with state

In this paper we give semantics toLoop, an expressive typed object-oriented programming language with updatable instance variables.Loop has a rich type system that allows for the typing of methods operating over an open-ended self type. We prove the type system given is sound;i.e., well-typed programs do not experience message not understood errors. The semantics ofLoop is given by a translation into a state-based language,Soop, that contains reference cells, records, and a form of F-bounded polymorphic type.Partially supported by NSF grant CCR-9109070Partially supported by AFOSR grant F49620-93-1-0169     

A theory of nonmonotonic rule systems I

We introduce here the study of generalnonmonotonic rule systems. These deal with situations where a conclusion is drawn from a system of beliefsS (and seen to be inS), basedboth on some premises being inS and on some restraints not being inS. In the monotone systems of traditional logic there are no restraints, conclusions are drawn solely based on premises being inS. Nonmonotonic rule systems capture the essential syntactic, semantic, and algorithmic features of many nonmonotone systems such as default logic, negation as failure, truth maintenance, autoepistemic logic, and also important combinatorial questions from mathematics such as the marriage problem. This reveals semantics and syntax and proof procedures and algorithms for computing belief sets in many cases where none were previously available and entirely uniformly. In particular, we introduce and study deductively closed sets, extensions and weak extensions. Semantics of nonmonotonic rule systems is studied in part II of this paper and extensions to predicate classical, intuitionistic, and modal logics are left to a later paper.Work partially supported by NSF grant RII-8610671 and Kentucky EPSCoR program and ARO contract DAAL03-89-K-0124.Work partially supported by NSF grant DMS-8902797 and ARO contract DAAG629-85-C-0018.Work partially supported by NSF grant DMS-8702473.     

Querying datalog with arrays: Design and implementation issues

This paper deals with the implementation of logic queries where array structures are manipulated. Both top-down and bottom-up implementations of the presented logic language, called Datalog ^A , are considered. Indeed, SLD-resolution is generalized to realize Datalog ^A top-down query answering. Further, a fixpoint based evaluation of Datalog ^A queries is introduced, which forms the basis for efficient bottom-up implementation of queries obtained by generalizing rewriting techniques such as magic set method to the case of Datalog ^A programs.Work partially supported by a European Union grant under the EC-US project DEUS EX MACHINA: nondeterminism for deductive databases and by a MURST grant (40% share) under the project Sistemi formali e strumenti per basi di dati evolute.     

Parallel complexity of logical query programs

We consider the parallel time complexity of logic programs without function symbols, called logical query programs, or Datalog programs. We give a PRAM algorithm for computing the minimum model of a logical query program, and show that for programs with the polynomial fringe property, this algorithm runs in time that is logarithmic in the input size, assuming that concurrent writes are allowed if they are consistent. As a result, the linear and piecewise linear classes of logic programs are inN C. Then we examine several nonlinear classes in which the program has a single recursive rule that is an elementary chain. We show that certain nonlinear programs are related to GSM mappings of a balanced parentheses language, and that this relationship implies the polynomial fringe property; hence such programs are inN C Finally, we describe an approach for demonstrating that certain logical query programs are log space complete forP, and apply it to both elementary single rule programs and nonelementary programs.Supported by NSF Grant IST-84-12791, a grant of IBM Corporation, and ONR contract N00014-85-C-0731.     

Unextendible Product Bases and Locally Unconvertible Bound Entangled States

Mutual convertibility of bound entangled states under local quantum operations and classical communication (LOCC) is studied. We focus on states associated with unextendible product bases (UPB) in a system of three qubits. A complete classification of such UPBs is suggested. We prove that for any pair of UPBs S and T the associated bound entangled states _S and _T cannot be converted to each other by LOCC, unless S and T coincide up to local unitaries. More specifically, there exists a finite precision (S,T) > 0 such that for any LOCC protocol mapping _S into a probabilistic ensemble (p, ), the fidelity between _T and any possible final state satisfies F(_T, ) = 1 - (S,T).PACS: 03.65.Bz; 03.67.-a; 89.70+c.     

Maintaining the visibility map of spheres while moving the viewpoint on a circle at infinity

We investigate three-dimensional visibility problems for scenes that consist ofn non-intersecting spheres. The viewing point moves on a flightpath that is part of a circle at infinity given by a planeP and a range of angles {(t)¦t[01]} [02]. At timet, the lines of sight are parallel to the ray inP, which starts in the origin ofP and represents the angle(t) (orthographic views of the scene). We give an algorithm that computes the visibility graph at the start of the flight, all time parameters at which the topology of the scene changes, and the corresponding topology changes. The algorithm has running time0(n + k + p) logn), wheren is the number of spheres in the scene;p is the number of transparent topology changes (the number of different scene topologies visible along the flight path, assuming that all spheres are transparent); andk denotes the number of vertices (conflicts) which are in the (transparent) visibility graph at the start and do not disappear during the flight.The second author was supported by the ESPRIT II Basic Research Actions Program, under Contract No. 3075 (project ALCOM).     

Relativization of questions about log space computability

A notion of log space Turing reducibility is introduced. It is used to define relative notions of log space, ^A , and nondeterministic log space, . These classes are compared with the classes and which were originally defined by Baker, Gill, and Solovay [BGS]. It is shown that there exists a computable setA such that . Furthermore, there exists a computable setA such that and . Also a notion of log space truth table reducibility is defined and shown to be equivalent to the notion of log space Turing reducibility.Research supported by NSF Grant No. GJ 43264.Research supported by NSF Grant No. DCR 92373.