Existential quantification

Abstract: Because of the “all-or-none” character of nervous activity, neural events and the relations among them can be treated by means of propositional logic. It is found that the behavior of every net can be described in these terms, with the addition of more complicated logical means for nets containing circles; and that for any logical expression satisfying certain conditions, one can find a net behaving in the fashion it describes. It is shown that many particular choices among possible neurophysiological assumptions are equivalent, in the sense that for every net behaving under one assumption, there exists another net which behaves under the other and gives the same results, although perhaps not in the same time. Various applications of the calculus are discussed.

Abstract: PREFACEqLogic is doubtless unshakable, but it cannot withstand a man who wants to live.q Franz Kafka: The TrialThe present book forms a sequel to my theoretical works Intentions in Architecture (1963) and Existence, Space and Architecture (1971). It is also related to my historical study Meaning in Western Architecture (1975). Common to all of them is the view that architecture represents a means to give man an qexistential footholdq. My primary aim is therefore to investigate the psychic implications of architecture rather than its practical side, although I certainly admit that there exists an interrelationship between the two aspects. In Intentions in Architecture the practical, qfunctionalq, dimension was in fact discussed as part of a comprehensive system. At the same time, however, the book stressed that the qenvironment influences human beings, and this implies that the purpose of architecture transcends the definition given by early functionalismq. A thorough discussion of perception and symbolization was therefore included, and it was emphasized that man cannot gain a foothold through scientific understanding alone. He needs symbols, that is, works of art which qrepresent life-situationsq. The conception of the work of art as a qconcretizationq of a life-situation is maintained in the present book. It is one of the basic needs of man to experience his life-situations as meaningful, and the purpose of the work of art is to qkeepq and transmit meanings. The concept of qmeaningq was also introduced in Intentions in Architecture. In general, the early book aimed at understanding architecture in concrete qarchitecturalq terms, an aim which I still consider particularly important. Too much confusion is created today by those who talk about everything else when they discuss architecture! My writings therefore reflect a belief in architecture; I do not accept that architecture, vernacular or monumental, is a luxury or perhapsnsomething which is made qto impress the populaceq (Rapoport). There are not different qkindsq of architecture, but only different situations which require different solutions in order to satisfy man's physical and psychic needs.My general aim and approach has therefore been the same in all the writings mentioned above. As time has passed, however, a certain change in method has become manifest. In Intentions in Architecture art and architecture were analyzed qscientificallyq, that is, by means of methods taken over from natural science. I do not think that this approach is wrong, but today I find other methods more illuminating. When we treat architecture analytically, we miss the concrete environmental character, that is, the very quality which is the object of man's identification, and which may give him a sense of existential foothold. To overcome this lack, I introduced in Existence, Space and Architecture the concept of qexistential spaceq. qExistential spaceq is not a logico-mathematical term, but comprises the basic relationships between man and his environment. The present book continues the search for a concrete understanding of the environment. The concept of existential space is here divided in the complementary terms qspaceq and qcharacterq, in accordance with the basic psychic functions qorientationq and qidentificationq. Space and character are not treated in a purely philosophical way (as has been done by O. F. Bollnow), but are directly related to architecture, following the definition of architecture as a qconcretization of existential spaceq. qConcretizationq is furthermore explained by means of the concepts of qgatheringq and qthingq. The word qthingq originally meant a gathering, and the meaning of anything consists in what it gathers. Thus Heidegger said: qA thing gathers worldq.nThe philosophy of Heidegger has been the catalyst which has made the present book possible and determined its approach. The wish for understanding architecture as a concrete phenomenon, already expressed in Intentions in Architecture, could be satisfied in the present book, thanks to Heidegger's essays on language and aesthetics, which have been collected and admirably translated into English by A. Hofstadter (Poetry, Language, Thought, New York 1971). First of all I owe to Heidegger the concept of dwelling. qExistential footholdq and qdwellingq are synonyms, and qdwellingq, in an existential sense, is the purpose of architecture. Man dwells when he can orientate himself within and identify himself with an environment, or, in short, when he experiences the environment as meaningful. Dwelling therefore implies something more than qshelterq. It implies that the spaces where life occurs are places, in the true sense of the word. A place is a space which has a distinct character. Since ancient times the genius loci, or qspirit of placeq, has been recognized as the concrete reality man has to face and come to terms with in his daily life. Architecture means to visualize the genius loci, and the task of the architect is to create meaningful places, whereby he helps man to dwell.I am well aware of the shortcomings of the present book. Many problems could only be treated in a very sketchy way, and need further elaboration. The book represents, however, a first step towards a qphenomenology of architectureq, that is, a theory which understands architecture in concrete, existential terms.The conquest of the existential dimension is in fact the main purpose of the present book. After decades of abstract, qscientificq theory, it is urgent that we return to a qualitative, phenomenological understanding of architecture.n n n n

Abstract: Semantic analysis of programs is essential in optimizing compilers and program verification systems. It encompasses data flow analysis, data type determination, generation of approximate invariant assertions, etc. Several recent papers (among others Cousot & Cousot[77a], Graham & Wegman[76], Kam & Ullman[76], Kildall[73], Rosen[78], Tarjan[76], Wegbreit[75]) have introduced abstract approaches to program analysis which are tantamount to the use of a program analysis framework (A,t,a) where A is a lattice of (approximate) assertions, t is an (approximate) predicate transformer and a is an often implicit function specifying the meaning of the elements of A. This paper is devoted to the systematic and correct design of program analysis frameworks with respect to a formal semantics. Preliminary definitions are given in Section 2 concerning the merge over all paths and (least) fixpoint program-wide analysis methods. In Section 3 we briefly define the (forward and backward) deductive semantics of programs which is later used as a formal basis in order to prove the correctness of the approximate program analysis frameworks. Section 4 very shortly recall the main elements of the lattice theoretic approach to approximate semantic analysis of programs. The design of a space of approximate assertions A is studied in Section 5. We first justify the very reasonable assumption that A must be chosen such that the exact invariant assertions of any program must have an upper approximation in A and that the approximate analysis of any program must be performed using a deterministic process. These assumptions are shown to imply that A is a Moore family, that the approximation operator (wich defines the least upper approximation of any assertion) is an upper closure operator and that A is necessarily a complete lattice. We next show that the connection between a space of approximate assertions and a computer representation is naturally made using a pair of isotone adjoined functions. This type of connection between two complete lattices is related to Galois connections thus making available classical mathematical results. Additional results are proved, they hold when no two approximate assertions have the same meaning. In Section 6 we study and examplify various methods which can be used in order to define a space of approximate assertions or equivalently an approximation function. They include the characterization of the least Moore family containing an arbitrary set of assertions, the construction of the least closure operator greater than or equal to an arbitrary approximation function, the definition of closure operators by composition, the definition of a space of approximate assertions by means of a complete join congruence relation or by means of a family of principal ideals. Section 7 is dedicated to the design of the approximate predicate transformer induced by a space of approximate assertions. First we look for a reasonable definition of the correctness of approximate predicate transformers and show that a local correctness condition can be given which has to be verified for every type of elementary statement. This local correctness condition ensures that the (merge over all paths or fixpoint) global analysis of any program is correct. Since isotony is not required for approximate predicate transformers to be correct it is shown that non-isotone program analysis frameworks are manageable although it is later argued that the isotony hypothesis is natural. We next show that among all possible approximate predicate transformers which can be used with a given space of approximate assertions there exists a best one which provides the maximum information relative to a program-wide analysis method. The best approximate predicate transformer induced by a space of approximate assertions turns out to be isotone. Some interesting consequences of the existence of a best predicate transformer are examined. One is that we have in hand a formal specification of the programs which have to be written in order to implement a program analysis framework once a representation of the space of approximate assertions has been chosen. Examples are given, including ones where the semantics of programs is formalized using Hoare[78]'s sets of traces. In Section 8 we show that a hierarchy of approximate analyses can be defined according to the fineness of the approximations specified by a program analysis framework. Some elements of the hierarchy are shortly exhibited and related to the relevant literature. In Section 9 we consider global program analysis methods. The distinction between "distributive" and "non-distributive" program analysis frameworks is studied. It is shown that when the best approximate predicate transformer is considered the coincidence or not of the merge over all paths and least fixpoint global analyses of programs is a consequence of the choice of the space of approximate assertions. It is shown that the space of approximate assertions can always be refined so that the merge over all paths analysis of a program can be defined by means of a least fixpoint of isotone equations. Section 10 is devoted to the combination of program analysis frameworks. We study and examplify how to perform the "sum", "product" and "power" of program analysis frameworks. It is shown that combined analyses lead to more accurate information than the conjunction of the corresponding separate analyses but this can only be achieved by a new design of the approximate predicate transformer induced by the combined program analysis frameworks.

Abstract: Model-checking is a method of verifying concurrent systems in which a state-transition graph model of the system behavior is compared with a temporal logic formula. This paper extends model-checking for the branching-time logic CTL to the analysis of real-time systems, whose correctness depends on the magnitudes of the timing delays. For specifications, we extend the syntax of CTL to allow quantitative temporal operators such as ?? <5, meaning "possibly within 5 time units." The formulas of the resulting logic, Timed CTL (TCTL), are interpreted over continuous computation trees, trees in which paths are maps from the set of nonnegative reals to system states. To model finite-state systems we introduce timed graphs-state-transition graphs annotated with timing constraints. As our main result, we develop an algorithm for model-checking, for determining the truth of a TCTL-formula with respect to a timed graph. We argue that choosing a dense domain instead of a discrete domain to model time does not significantly blow up the complexity of the model-checking problem. On the negative side, we show that the denseness of the underlying time domain makes the validity problem for TCTL ?11-hard. The question of deciding whether there exists a timed graph satisfying a TCTL-formula is also undecidable.