14,008 research outputs found
Studies on modal logics of time and space
This dissertation presents original results in Temporal Logic and Spatial Logic. Part I concerns Branching-Time Logic. Since Prior 1967, two main semantics for Branching-Time Logic have been devised: Peircean and Ockhamist semantics. Zanardo 1998 proposed a general semantics, called Indistinguishability semantics, of which Peircean and Ockhamist semantics are limit cases. We provide a finite axiomatization of the Indistinguishability logic of upward endless bundled trees using a non-standard inference rule, and prove that this logic is strongly complete.
In Part II, we study the temporal logic given by the tense operators F for future and P for past together with the derivative operator , interpreted on the real numbers. We prove that this logic is neither strongly nor Kripke complete, it is PSPACE-complete, and it is finitely axiomatizable.
In Part III, we study the spatial logic given by the derivative operator and the graded modalities {n | n in N}. We prove that this language, call it L, is as expressive as the first-order language Lt of Flum and Ziegler 1980 when interpreted on T3 topological spaces. Then, we give a general definition of modal operator: essentially, a modal operator will be defined by a formula of Lt with at most one free variable. If a modal operator is defined by a formula predicating only over points, then it is called point-sort operator. We prove that L, even if enriched with all point-sort operators, however enriched with finitely many modal operators predicating also on open sets, cannot express Lt on T2 spaces. Finally, we axiomatize the logic of any class between all T1 and all T3 spaces and prove that it is PSPACE-complete.Open Acces
Real-time and Probabilistic Temporal Logics: An Overview
Over the last two decades, there has been an extensive study on logical
formalisms for specifying and verifying real-time systems. Temporal logics have
been an important research subject within this direction. Although numerous
logics have been introduced for the formal specification of real-time and
complex systems, an up to date comprehensive analysis of these logics does not
exist in the literature. In this paper we analyse real-time and probabilistic
temporal logics which have been widely used in this field. We extrapolate the
notions of decidability, axiomatizability, expressiveness, model checking, etc.
for each logic analysed. We also provide a comparison of features of the
temporal logics discussed
Flow Logic
Flow networks have attracted a lot of research in computer science. Indeed,
many questions in numerous application areas can be reduced to questions about
flow networks. Many of these applications would benefit from a framework in
which one can formally reason about properties of flow networks that go beyond
their maximal flow. We introduce Flow Logics: modal logics that treat flow
functions as explicit first-order objects and enable the specification of rich
properties of flow networks. The syntax of our logic BFL* (Branching Flow
Logic) is similar to the syntax of the temporal logic CTL*, except that atomic
assertions may be flow propositions, like or , for
, which refer to the value of the flow in a vertex, and
that first-order quantification can be applied both to paths and to flow
functions. We present an exhaustive study of the theoretical and practical
aspects of BFL*, as well as extensions and fragments of it. Our extensions
include flow quantifications that range over non-integral flow functions or
over maximal flow functions, path quantification that ranges over paths along
which non-zero flow travels, past operators, and first-order quantification of
flow values. We focus on the model-checking problem and show that it is
PSPACE-complete, as it is for CTL*. Handling of flow quantifiers, however,
increases the complexity in terms of the network to , even
for the LFL and BFL fragments, which are the flow-counterparts of LTL and CTL.
We are still able to point to a useful fragment of BFL* for which the
model-checking problem can be solved in polynomial time. Finally, we introduce
and study the query-checking problem for BFL*, where under-specified BFL*
formulas are used for network exploration
Hybrid Branching-Time Logics
Hybrid branching-time logics are introduced as extensions of CTL-like logics
with state variables and the downarrow-binder. Following recent work in the
linear framework, only logics with a single variable are considered. The
expressive power and the complexity of satisfiability of the resulting logics
is investigated.
As main result, the satisfiability problem for the hybrid versions of several
branching-time logics is proved to be 2EXPTIME-complete. These branching-time
logics range from strict fragments of CTL to extensions of CTL that can talk
about the past and express fairness-properties. The complexity gap relative to
CTL is explained by a corresponding succinctness result.
To prove the upper bound, the automata-theoretic approach to branching-time
logics is extended to hybrid logics, showing that non-emptiness of alternating
one-pebble Buchi tree automata is 2EXPTIME-complete.Comment: An extended abstract of this paper was presented at the International
Workshop on Hybrid Logics (HyLo 2007
Karma Theory, Determinism, Fatalism and Freedom of Will
The so-called theory of karma is one of the distinguishing aspects of Hinduism and other non-Hindu south-Asian traditions. At the same time that the theory can be seen as closely connected with the freedom of will and action that we humans supposedly have, it has many times been said to be determinist and fatalist. The purpose of this paper is to analyze in some deepness the relations that are between the theory of karma on one side and determinism, fatalism and free-will on the other side. In order to do that, I shall use what has been described as the best formal approach we have to indeterminism: branching time theory. More specifically, I shall introduce a branching time semantic framework in which, among other things, statements such as “state of affairs e is a karmic effect of agent a”, “a wills it to be the case that e” and “e is inevitable” could be properly represented
A Theory of Presentism
Also appears in: (1) L.N.Oaklander and E.Magalhaes (eds.) Presentism: A Reader (Rowman & Littlefield, 2010) (2) L.N. Oaklander (ed.) Routledge Major Works: The Philosophy of Time: Critical Concepts in Philosophy (London: Routledge, 2008)Most of us would want to say that it is true that Socrates taught Plato. According to realists about past facts, this is made true by the fact that there is, located in the past, i.e., earlier than now, at least one real event that is the teaching of Plato by Socrates. Presentists, however, in denying that past events and facts exist cannot appeal to such facts to make their past-tensed statements true. So what is a presentist to do? There are at least three conditions that would ideally be met in a satisfactory solution to this problem: (1) It must preserve our views about which statements are true and which false; (2) It must be transparent what the truthmakers are for those statements; (3) It must accommodate the truth-value links between various times. I shall survey two different families of proposals for the presentist's truthmakers and show that they fail at least one of these three conditions. This is not entirely negative, for it shows us what an adequate solution to the problem would look like. I go on to show where presentists can find suitable objects that satisfy these conditions, and in this way give a clear statement of presentism, something that is lacking in the literature.Peer reviewe
On the Hybrid Extension of CTL and CTL+
The paper studies the expressivity, relative succinctness and complexity of
satisfiability for hybrid extensions of the branching-time logics CTL and CTL+
by variables. Previous complexity results show that only fragments with one
variable do have elementary complexity. It is shown that H1CTL+ and H1CTL, the
hybrid extensions with one variable of CTL+ and CTL, respectively, are
expressively equivalent but H1CTL+ is exponentially more succinct than H1CTL.
On the other hand, HCTL+, the hybrid extension of CTL with arbitrarily many
variables does not capture CTL*, as it even cannot express the simple CTL*
property EGFp. The satisfiability problem for H1CTL+ is complete for triply
exponential time, this remains true for quite weak fragments and quite strong
extensions of the logic
From Linear to Branching-Time Temporal Logics: Transfer of Semantics and Definability
This paper investigates logical aspects of combining linear orders as semantics for modal and temporal logics, with modalities for possible paths, resulting in a variety of branching time logics over classes of trees. Here we adopt a unified approach to the Priorean, Peircean and Ockhamist semantics for branching time logics, by considering them all as fragments of the latter, obtained as combinations, in various degrees, of languages and semantics for linear time with a modality for possible paths. We then consider a hierarchy of natural classes of trees and bundled trees arising from a given class of linear orders and show that in general they provide different semantics. We also discuss transfer of definability from linear orders to trees and introduce a uniform translation from Priorean to Peircean formulae which transfers definability of properties of linear orders to definability of properties of all paths in tree
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