10 research outputs found
On the complexity of the closed fragment of Japaridze's provability logic
We consider well-known provability logic GLP. We prove that the
GLP-provability problem for variable-free polymodal formulas is
PSPACE-complete. For a number n, let L^n_0 denote the class of all polymodal
variable-free formulas without modalities , ,... . We show that, for
every number n, the GLP-provability problem for formulas from L^n_0 is in
PTIME.Comment: 12 pages, the results of this work and a proof sketch are in Advances
in Modal Logic 2012 extended abstract on the same nam
Complexity and Expressivity of Branching- and Alternating-Time Temporal Logics with Finitely Many Variables
We show that Branching-time temporal logics CTL and CTL*, as well as
Alternating-time temporal logics ATL and ATL*, are as semantically expressive
in the language with a single propositional variable as they are in the full
language, i.e., with an unlimited supply of propositional variables. It follows
that satisfiability for CTL, as well as for ATL, with a single variable is
EXPTIME-complete, while satisfiability for CTL*, as well as for ATL*, with a
single variable is 2EXPTIME-complete,--i.e., for these logics, the
satisfiability for formulas with only one variable is as hard as satisfiability
for arbitrary formulas.Comment: Prefinal version of the published pape
Сложность константного фрагмента пропозициональной динамической логики
Основной результат работы состоит в том, что
константные фрагменты логик (с K-модальностью и её рефлексивно-транзитивным
замыканием), PDL, а также некоторых других являются
EXPTIME-полными.
Доказательство содержит описание довольно общей идеи построения
полиномиального погружения логик в их фрагменты от
n переменных (и даже в константные фрагменты, как в случае
и PDL). В качестве следствия описанной
конструкции получена EXPTIME-полнота фрагментов от одной
переменной логик знания с оператором всеобщего знания
Intuitionistic implication makes model checking hard
We investigate the complexity of the model checking problem for
intuitionistic and modal propositional logics over transitive Kripke models.
More specific, we consider intuitionistic logic IPC, basic propositional logic
BPL, formal propositional logic FPL, and Jankov's logic KC. We show that the
model checking problem is P-complete for the implicational fragments of all
these intuitionistic logics. For BPL and FPL we reach P-hardness even on the
implicational fragment with only one variable. The same hardness results are
obtained for the strictly implicational fragments of their modal companions.
Moreover, we investigate whether formulas with less variables and additional
connectives make model checking easier. Whereas for variable free formulas
outside of the implicational fragment, FPL model checking is shown to be in
LOGCFL, the problem remains P-complete for BPL.Comment: 29 pages, 10 figure
Complete Additivity and Modal Incompleteness
In this paper, we tell a story about incompleteness in modal logic. The story
weaves together a paper of van Benthem, `Syntactic aspects of modal
incompleteness theorems,' and a longstanding open question: whether every
normal modal logic can be characterized by a class of completely additive modal
algebras, or as we call them, V-BAOs. Using a first-order reformulation of the
property of complete additivity, we prove that the modal logic that starred in
van Benthem's paper resolves the open question in the negative. In addition,
for the case of bimodal logic, we show that there is a naturally occurring
logic that is incomplete with respect to V-BAOs, namely the provability logic
GLB. We also show that even logics that are unsound with respect to such
algebras do not have to be more complex than the classical propositional
calculus. On the other hand, we observe that it is undecidable whether a
syntactically defined logic is V-complete. After these results, we generalize
the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van
Benthem's theme of syntactic aspects of modal incompleteness
The Decision Problem of Provability Logic with Only One Atom
The original publication is available at www.springerlink.com. The decision problem for provability logic remains PSPACE-complete even if the number of propositional atoms is restricted to one. In some cases the set of all tautologies of a modal logic is in coNP. An example of a logic like that is the well-known S5. However, most of the traditional modal systems, including S4 and T, have PSPACE-complete decision problem. So one can say that adding modalities to the language of classical propositional logic does increase algorithmic complexity — not a surprising paradigm. The methods for constructing a polynomial space decision procedure and for proving PSPACE-completeness of a modal logic can be learnt from R. Ladner’s paper [Lad77]. Provability logic GL is not mentioned in [Lad77], but it is not difficult to verify that GL has PSPACE-complete decision problem as well. In this paper we go farther and use Ladner’s methods to show that the decision problem of GL is PSPACE-complete even if the number of propositional atoms used to build modal formulas is restricted to one. This fact can be interpreted as saying that, in case of provability logic, allowing more than one atom does not increase the expressive power of the language. The structure of the present paper is similar to that of our [ ˇ Sve03] where an alternative simple proof of R. Statman’s result concerning PSPACE-completeness of intuitionistic propositional logic is presented. Modal formulas are built up from propositional atoms and the symbol ⊥ for falsity using logical connectives and a unary symbol ✷ for necessity. We use ✸
Explicit fixed-points in provability logic
Smyslem této diplomové práce je prozkoumat explicitní výpoty pevn ých bod v logice dokazatelnosti GL. Vta o pevných bodech zní: Pro kadou modální formuli A(p) v ní kadý výskyt atomu p je vázán modálním operátorem ¤, existuje formule D obsahující pouze výrokové atomy obsaené v A(p), neobsahující výrokový atom p, a taková, e v GL je dokazatelné D ' A(D). Formule D je navíc ur- ena a na dokazatelnou ekvivalenci jednoznan. Nejprve vyslovíme nkolik speciálních pípad vty o pevných bodech a poté podrobnji prozkoumáme vtu v plném znní. Dále ukáeme jednu sémantickou a dv syntaktické konstrukce pevných bod a dokáeme jejich korektnost. V práci se zabýváme také nkterými sloitostními aspekty konstrukce, pedevím uvádíme jednoduché horní odhady délky a modální sloitosti získaných pevných bod.The aim of this diploma thesis is to discuss the explicit calculations of xed-points in provability logic GL. The xed-point theorem reads: For every modal formula A(p) such that each occurrence of p is under the scope of ¤, there is a formula D containing only sentence letters contained in A(p), not containing the sentence letter p, such that GL proves D ' A(D). Moreover, D is unique up to the provable equivalence. Firstly, we establish some special cases of the theorem and then we will look more closely at the full theorem. We show one semantic and two syntactic full xed-point constructions and prove their correctness. We also discuss some complexity aspects connected with the constructions and present basic upper bounds on length and modal depth of the constructed xed-points.Katedra logikyDepartment of LogicFaculty of ArtsFilozofická fakult
Die Komplexität der Formelauswertung in intuitionistischen Logiken
Der Intuitionismus ist eine Denkweise, die auf Intuition und Konstruktivismus basiert. Mathematisch gesehen werden hier nur konstruktive Beweise anerkannt und der Begriff der Wahrheit wird durch "beweisbar" ersetzt. Die intuitionistische Logik greift diesen Gedanken in der Form auf, dass das Gesetz des ausgeschlossenen Dritten nicht gültig ist. Eine Aussage "A oder nicht A" gilt nur dann als wahr, wenn entweder A oder das Gegenteil von A bewiesen werden kann. Für die intuitionistische Aussagenlogik gibt es eine Semantik, die jener der Modallogik sehr ähnlich ist. In diesem Sinne kann man sie auch als spezielle Modallogik auffassen. Wir beschäftigen uns in dieser Arbeit im Wesentlichen mit der Formelauswertung in intuitionistischen Logiken und untersuchen ihre Komplexität. Dabei betrachten wir Fragmente, die durch verschiedene Einschränkungen entstehen. Auf der semantischen Seite beschränken wir die zugelassenen Modelle. Auf der syntaktischen Seite kann man die Zahl der Variablen einschränken oder nur bestimmte Operatoren zulassen.
Unsere ersten Ergebnisse beziehen sich auf Logiken, bei denen es nur endlich viele paarweise nicht äquivalente Formeln gibt. Hier zeigen wir, dass das Formelauswertungsproblem, das Erfüllbarkeitsproblem und das Tautologieproblem sehr einfach zu lösen sind. Weiter betrachten wir die Logik, bei der nur eine Variable zugelassen ist. Für diese Logik zeigen wir, dass die Formelauswertung AC1-vollständig ist. Dies ermöglicht eine neue Sicht auf die Klasse AC1, da es das erste vollständige natürliche Problem für diese Klasse ist. Außerdem untersuchen wir, für welche Logiken das Formelauswertungsproblem die maximale Komplexität erreicht. Hier geht es insbesondere um die genaue Abgrenzung - also um die Frage, welche Freiheitsgrade man in einer Logik mindestens braucht, damit die Formelauswertung derart komplex ist. Am Ende betrachten wir noch einige Modallogiken, die Begleiter von intuitionistischen Logiken sind