2,221 research outputs found
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
An Objection to Naturalism and Atheism from Logic
I proffer a success argument for classical logical consequence. I articulate in what sense that notion of consequence should be regarded as the privileged notion for metaphysical inquiry aimed at uncovering the fundamental nature of the world. Classical logic breeds necessitism. I use necessitism to produce problems for both ontological naturalism and atheism
A Galois connection between classical and intuitionistic logics. I: Syntax
In a 1985 commentary to his collected works, Kolmogorov remarked that his
1932 paper "was written in hope that with time, the logic of solution of
problems [i.e., intuitionistic logic] will become a permanent part of a
[standard] course of logic. A unified logical apparatus was intended to be
created, which would deal with objects of two types - propositions and
problems." We construct such a formal system QHC, which is a conservative
extension of both the intuitionistic predicate calculus QH and the classical
predicate calculus QC.
The only new connectives ? and ! of QHC induce a Galois connection (i.e., a
pair of adjoint functors) between the Lindenbaum posets (i.e. the underlying
posets of the Lindenbaum algebras) of QH and QC. Kolmogorov's double negation
translation of propositions into problems extends to a retraction of QHC onto
QH; whereas Goedel's provability translation of problems into modal
propositions extends to a retraction of QHC onto its QC+(?!) fragment,
identified with the modal logic QS4. The QH+(!?) fragment is an intuitionistic
modal logic, whose modality !? is a strict lax modality in the sense of Aczel -
and thus resembles the squash/bracket operation in intuitionistic type
theories.
The axioms of QHC attempt to give a fuller formalization (with respect to the
axioms of intuitionistic logic) to the two best known contentual
interpretations of intiuitionistic logic: Kolmogorov's problem interpretation
(incorporating standard refinements by Heyting and Kreisel) and the proof
interpretation by Orlov and Heyting (as clarified by G\"odel). While these two
interpretations are often conflated, from the viewpoint of the axioms of QHC
neither of them reduces to the other one, although they do overlap.Comment: 47 pages. The paper is rewritten in terms of a formal meta-logic (a
simplified version of Isabelle's meta-logic
Undecidable First-Order Theories of Affine Geometries
Tarski initiated a logic-based approach to formal geometry that studies
first-order structures with a ternary betweenness relation (\beta) and a
quaternary equidistance relation (\equiv). Tarski established, inter alia, that
the first-order (FO) theory of (R^2,\beta,\equiv) is decidable. Aiello and van
Benthem (2002) conjectured that the FO-theory of expansions of (R^2,\beta) with
unary predicates is decidable. We refute this conjecture by showing that for
all n>1, the FO-theory of monadic expansions of (R^2,\beta) is \Pi^1_1-hard and
therefore not even arithmetical. We also define a natural and comprehensive
class C of geometric structures (T,\beta), where T is a subset of R^2, and show
that for each structure (T,\beta) in C, the FO-theory of the class of monadic
expansions of (T,\beta) is undecidable. We then consider classes of expansions
of structures (T,\beta) with restricted unary predicates, for example finite
predicates, and establish a variety of related undecidability results. In
addition to decidability questions, we briefly study the expressivity of
universal MSO and weak universal MSO over expansions of (R^n,\beta). While the
logics are incomparable in general, over expansions of (R^n,\beta), formulae of
weak universal MSO translate into equivalent formulae of universal MSO.
This is an extended version of a publication in the proceedings of the 21st
EACSL Annual Conferences on Computer Science Logic (CSL 2012).Comment: 21 pages, 3 figure
Decidability in the logic of subsequences and supersequences
We consider first-order logics of sequences ordered by the subsequence
ordering, aka sequence embedding. We show that the \Sigma_2 theory is
undecidable, answering a question left open by Kuske. Regarding fragments with
a bounded number of variables, we show that the FO2 theory is decidable while
the FO3 theory is undecidable
Complete Deductive Systems for Probability Logic with Application in Harsanyi Type Spaces
Thesis (PhD) - Indiana University, Mathematics, 2007These days, the study of probabilistic systems is very
popular not only in theoretical computer science but also in
economics. There is a surprising concurrence between game theory and
probabilistic programming. J.C. Harsanyi introduced the notion of
type spaces to give an implicit description of beliefs in games with
incomplete information played by Bayesian players. Type functions on
type spaces are the same as the stochastic kernels that are used to
interpret probabilistic programs. In addition to this semantic
approach to interactive epistemology, a syntactic approach was
proposed by R.J. Aumann. It is of foundational importance to develop
a deductive logic for his probabilistic belief logic.
In the first part of the dissertation, we develop a sound
and complete probability logic for type spaces in a
formal propositional language with operators which means
``the agent 's belief is at least " where the index is a
rational number between 0 and 1. A crucial infinitary inference rule
in the system captures the Archimedean property about
indices. By the Fourier-Motzkin's elimination method in linear
programming, we prove Professor Moss's conjecture that the
infinitary rule can be replaced by a finitary one. More importantly,
our proof of completeness is in keeping with the Henkin-Kripke
style. Also we show through a probabilistic system with
parameterized indices that it is decidable whether a formula
is derived from the system . The second part is on its
strong completeness. It is well-known that is not
strongly complete, i.e., a set of formulas in the language may be
finitely satisfiable but not necessarily satisfiable. We show that
even finitely satisfiable sets of formulas that are closed under the
Archimedean rule are not satisfiable. From these results, we
develop a theory about probability logic that is parallel to the
relationship between explicit and implicit descriptions of belief
types in game theory. Moreover, we use a linear system about
probabilities over trees to prove that there is no strong
completeness even for probability logic with finite indices. We
conclude that the lack of strong completeness does not depend on the
non-Archimedean property in indices but rather on the use of
explicit probabilities in the
syntax.
We show the completeness and some properties of the
probability logic for Harsanyi type spaces. By adding knowledge
operators to our languages, we devise a sound and complete
axiomatization for Aumann's semantic knowledge-belief systems. Its
applications in labeled Markovian processes and semantics for
programs are also discussed
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