94 research outputs found
Complexity Issues in Justification Logic
Justification Logic is an emerging field that studies provability, knowledge, and belief via explicit proofs or justifications that are part of the language. There exist many justification logics closely related to modal epistemic logics of knowledge and belief. Instead of modality □ in pure justification logics, or in addition to modality □ in hybrid logics, which has an existential epistemic reading \u27there exists a proof of F,\u27 all justification logics use constructs t:F, where a justification term t represents a blueprint of a Hilbert-style proof of F. The first justification logic, LP, introduced by Sergei Artemov, was shown to be a justification counterpart of modal logic S4 and serves as a missing link between S4 and Peano arithmetic, thereby solving a long-standing problem of provability semantics for S4 and Int.
The machinery of explicit justifications can be used to analyze well-known epistemic paradoxes, e.g. Gettier\u27s examples of justified true belief that can hardly be considered knowledge, and to find new approaches to the concept of common knowledge. Yet another possible application is the Logical Omniscience Problem, which reflects an undesirable property of knowledge as described by modality when an agent knows all the logical consequences of his/her knowledge. The language of justification logic opens new ways to tackle this problem.
This thesis focuses on quantitative analysis of justification logics. We explore their decidability and complexity of Validity Problem for them. A closer analysis of the realization phenomenon in general and of one procedure in particular enables us to deduce interesting corollaries about self-referentiality for several modal logics. A framework for proving decidability of various justification logics is developed by generalizing the Finite Model Property. Limitations of the method are demonstrated through an example of an undecidable justification logic. We study reflected fragments of justification logics and provide them with an axiomatization and a decision procedure whose complexity (the upper bound) turns out to be uniform for all justification logics, both pure and hybrid. For many justification logics, we also present lower and upper complexity bounds
An Instantiation-Based Approach for Solving Quantified Linear Arithmetic
This paper presents a framework to derive instantiation-based decision
procedures for satisfiability of quantified formulas in first-order theories,
including its correctness, implementation, and evaluation. Using this framework
we derive decision procedures for linear real arithmetic (LRA) and linear
integer arithmetic (LIA) formulas with one quantifier alternation. Our
procedure can be integrated into the solving architecture used by typical SMT
solvers. Experimental results on standardized benchmarks from model checking,
static analysis, and synthesis show that our implementation of the procedure in
the SMT solver CVC4 outperforms existing tools for quantified linear
arithmetic
The undecidability of arbitrary arrow update logic
Arbitrary Arrow Update Logic is a dynamic modal logic with a modality to quantify over arrow updates. Some properties of this logic have already been established, but until now it remained an open question whether the logic's satisfiability problem is decidable. Here, we show by a reduction of the tiling problem that the satisfiability problem of Arbitrary Arrow Update Logic is co-RE hard, and therefore undecidable
A Decidable Quantified Fragment of Set Theory Involving Ordered Pairs with Applications to Description Logics
We present a decision procedure for a quantified fragment of
set theory involving ordered pairs and some operators to manipulate them. When our decision procedure is applied to formulae in this fragment whose quantifier prefixes have length bounded by a fixed constant, it runs in nondeterministic polynomial-time.
Related to this fragment, we also introduce a description logic which
provides an unusually large set of constructs, such as, for instance,
Boolean constructs among roles. The set-theoretic nature of the description logics semantics yields a straightforward reduction of the
knowledge base consistency problem to the satisfiability problem for formulae of our fragment with quantifier prefixes of length at most 2, from which the NP-completeness of reasoning in this novel description logic follows. Finally, we extend this reduction to cope also with SWRL rules
A decidable quantified fragment of set theory with ordered pairs and some undecidable extensions
In this paper we address the decision problem for a fragment of set theory
with restricted quantification which extends the language studied in [4] with
pair related quantifiers and constructs, in view of possible applications in
the field of knowledge representation. We will also show that the decision
problem for our language has a non-deterministic exponential time complexity.
However, for the restricted case of formulae whose quantifier prefixes have
length bounded by a constant, the decision problem becomes NP-complete. We also
observe that in spite of such restriction, several useful set-theoretic
constructs, mostly related to maps, are expressible. Finally, we present some
undecidable extensions of our language, involving any of the operators domain,
range, image, and map composition.
[4] Michael Breban, Alfredo Ferro, Eugenio G. Omodeo and Jacob T. Schwartz
(1981): Decision procedures for elementary sublanguages of set theory. II.
Formulas involving restricted quantifiers, together with ordinal, integer, map,
and domain notions. Communications on Pure and Applied Mathematics 34, pp.
177-195Comment: In Proceedings GandALF 2012, arXiv:1210.202
Querying Schemas With Access Restrictions
We study verification of systems whose transitions consist of accesses to a
Web-based data-source. An access is a lookup on a relation within a relational
database, fixing values for a set of positions in the relation. For example, a
transition can represent access to a Web form, where the user is restricted to
filling in values for a particular set of fields. We look at verifying
properties of a schema describing the possible accesses of such a system. We
present a language where one can describe the properties of an access path, and
also specify additional restrictions on accesses that are enforced by the
schema. Our main property language, AccLTL, is based on a first-order extension
of linear-time temporal logic, interpreting access paths as sequences of
relational structures. We also present a lower-level automaton model,
Aautomata, which AccLTL specifications can compile into. We show that AccLTL
and A-automata can express static analysis problems related to "querying with
limited access patterns" that have been studied in the database literature in
the past, such as whether an access is relevant to answering a query, and
whether two queries are equivalent in the accessible data they can return. We
prove decidability and complexity results for several restrictions and variants
of AccLTL, and explain which properties of paths can be expressed in each
restriction.Comment: VLDB201
A decidable multi-modal logic of context
We give a logic for formulas Á¡± Ã, with the informal reading
”à is true in the context described by Á”. These are interpreted
as binary modalities, by quantification over an enumerable set of
unary modalities c¡± Ã, meaning ”à is true in context c”. The
logic allows arbitrary nesting of contexts.
A corresponding axiomatic presentation is given, and proven
to be decidable, sound, and complete.
Previously, quantificational logic of context restricted the
nesting of contexts, and was only known to be decidable in very
special cases
Action, Time and Space in Description Logics
Description Logics (DLs) are a family of logic-based knowledge representation (KR) formalisms designed to represent and reason about static conceptual knowledge in a semantically well-understood way. On the other hand, standard action formalisms are KR formalisms based on classical logic designed to model and reason about dynamic systems. The largest part of the present work is dedicated to integrating DLs with action formalisms, with the main goal of obtaining decidable action formalisms with an expressiveness significantly beyond propositional. To this end, we offer DL-tailored solutions to the frame and ramification problem. One of the main technical results is that standard reasoning problems about actions (executability and projection), as well as the plan existence problem are decidable if one restricts the logic for describing action pre- and post-conditions and the state of the world to decidable Description Logics. A smaller part of the work is related to decidable extensions of Description Logics with concrete datatypes, most importantly with those allowing to refer to the notions of space and time
Why Propositional Quantification Makes Modal Logics on Trees Robustly Hard?
International audienceAdding propositional quantification to the modal logics K, T or S4 is known to lead to undecid-ability but CTL with propositional quantification under the tree semantics (QCTL t) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of QCTL t as well as of the modal logic K with propositional quantification under the tree semantics. More specifically, we show that QCTL t restricted to the temporal operator EX is already Tower-hard, which is unexpected as EX can only enforce local properties. When QCTL t restricted to EX is interpreted on N-bounded trees for some N ≥ 2, we prove that the satisfiability problem is AExp pol-complete; AExp pol-hardness is established by reduction from a recently introduced tiling problem, instrumental for studying the model-checking problem for interval temporal logics. As consequences of our proof method, we prove Tower-hardness of QCTL t restricted to EF or to EXEF and of the well-known modal logics K, KD, GL, S4, K4 and D4, with propositional quantification under a semantics based on classes of trees
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