1,262 research outputs found
Reflexive insensitive modal logics
We analyze a class of modal logics rendered insensitive to reflexivity by way
of a modification to the semantic definition of the modal operator. We explore
the extent to which these logics can be characterized, and prove a general
completeness theorem on the basis of a translation between normal modal logics
and their reflexive-insensitive counterparts. Lastly, we provide a sufficient
semantic condition describing when a similarly general soundness result is also
available
A New Arithmetically Incomplete First- Order Extension of Gl All Theorems of Which Have Cut Free Proofs
Reference [12] introduced a novel formula to formula translation tool (“formulators”) that enables syntactic metatheoretical investigations of first-order modal logics, bypassing a need to convert them first into Gentzen style logics in order to rely on cut elimination and the subformula property. In fact, the formulator tool, as was already demonstrated in loc. cit., is applicable even to the metatheoretical study of logics such as QGL, where cut elimination is (provably, [2]) unavailable. This paper applies the formulator approach to show the independence of the axiom schema _A ! _8xA of the logics M3 and ML3 of [17, 18, 11, 13]. This leads to the conclusion that the two logics obtained by removing this axiom are incomplete, both with respect to their natural Kripke structures and to arithmetical interpretations. In particular, the so modified ML3 is, similarly to QGL, an arithmetically incomplete first-order extension of GL, but, unlike QGL, all its theorems have cut free proofs. We also establish here, via formulators, a stronger version of the disjunction property for GL and QGL without going through Gentzen versions of these logics (compare with the more complex proofs in [2, 8]).This research was partially supported by NSERC grant No. 8250
Designing Normative Theories for Ethical and Legal Reasoning: LogiKEy Framework, Methodology, and Tool Support
A framework and methodology---termed LogiKEy---for the design and engineering
of ethical reasoners, normative theories and deontic logics is presented. The
overall motivation is the development of suitable means for the control and
governance of intelligent autonomous systems. LogiKEy's unifying formal
framework is based on semantical embeddings of deontic logics, logic
combinations and ethico-legal domain theories in expressive classic
higher-order logic (HOL). This meta-logical approach enables the provision of
powerful tool support in LogiKEy: off-the-shelf theorem provers and model
finders for HOL are assisting the LogiKEy designer of ethical intelligent
agents to flexibly experiment with underlying logics and their combinations,
with ethico-legal domain theories, and with concrete examples---all at the same
time. Continuous improvements of these off-the-shelf provers, without further
ado, leverage the reasoning performance in LogiKEy. Case studies, in which the
LogiKEy framework and methodology has been applied and tested, give evidence
that HOL's undecidability often does not hinder efficient experimentation.Comment: 50 pages; 10 figure
Kripke’s Modal Logic: A Historical Study
In a very short time Saul Kripke provided a suitable and rigorous semantics for different axiomatic modal systems and established a series of related results. Many key ideas were already in the air in the late Fifties, but it was Kripkean articles’ merit to system atically introduce comprehensive devices and solutions. Later on, the spreading of possible-worlds semantics massively changed the approach to modal logic, which enormously increased in popularity after that. Since Kripke’s work in modal logic is central to the development of the discipline, the aim of this essay is to present the fundamental results published between 1959 and 1965. Indeed, it was in such a brief and early phase of his career that Kripke was able to conceive the main novelties that would become central to the subsequent academic debates about modality. Here, their presentation will follow the original historical progressive introduction. Particular attention will be given to the interconnection between articles, their similarities in structure and the unified analysis produced by means of them. It actually appears quite impressive that, already in 1959, Kripke seemed to have planned all the developments he would present, one after the other, in the following years. First, an overview of the background where Kripke’s ideas start to rise is given. Then, each text’s results are individually briefly analysed
On Sub-Propositional Fragments of Modal Logic
In this paper, we consider the well-known modal logics ,
, , and , and we study some of their
sub-propositional fragments, namely the classical Horn fragment, the Krom
fragment, the so-called core fragment, defined as the intersection of the Horn
and the Krom fragments, plus their sub-fragments obtained by limiting the use
of boxes and diamonds in clauses. We focus, first, on the relative expressive
power of such languages: we introduce a suitable measure of expressive power,
and we obtain a complex hierarchy that encompasses all fragments of the
considered logics. Then, after observing the low expressive power, in
particular, of the Horn fragments without diamonds, we study the computational
complexity of their satisfiability problem, proving that, in general, it
becomes polynomial
Completeness for a First-order Abstract Separation Logic
Existing work on theorem proving for the assertion language of separation
logic (SL) either focuses on abstract semantics which are not readily available
in most applications of program verification, or on concrete models for which
completeness is not possible. An important element in concrete SL is the
points-to predicate which denotes a singleton heap. SL with the points-to
predicate has been shown to be non-recursively enumerable. In this paper, we
develop a first-order SL, called FOASL, with an abstracted version of the
points-to predicate. We prove that FOASL is sound and complete with respect to
an abstract semantics, of which the standard SL semantics is an instance. We
also show that some reasoning principles involving the points-to predicate can
be approximated as FOASL theories, thus allowing our logic to be used for
reasoning about concrete program verification problems. We give some example
theories that are sound with respect to different variants of separation logics
from the literature, including those that are incompatible with Reynolds's
semantics. In the experiment we demonstrate our FOASL based theorem prover
which is able to handle a large fragment of separation logic with heap
semantics as well as non-standard semantics.Comment: This is an extended version of the APLAS 2016 paper with the same
titl
A first-order Temporal Logic for Actions
We present a multi-modal action logic with first-order modalities, which
contain terms which can be unified with the terms inside the subsequent
formulas and which can be quantified. This makes it possible to handle
simultaneously time and states. We discuss applications of this language to
action theory where it is possible to express many temporal aspects of actions,
as for example, beginning, end, time points, delayed preconditions and results,
duration and many others. We present tableaux rules for a decidable fragment of
this logic
Querying the Guarded Fragment
Evaluating a Boolean conjunctive query Q against a guarded first-order theory
F is equivalent to checking whether "F and not Q" is unsatisfiable. This
problem is relevant to the areas of database theory and description logic.
Since Q may not be guarded, well known results about the decidability,
complexity, and finite-model property of the guarded fragment do not obviously
carry over to conjunctive query answering over guarded theories, and had been
left open in general. By investigating finite guarded bisimilar covers of
hypergraphs and relational structures, and by substantially generalising
Rosati's finite chase, we prove for guarded theories F and (unions of)
conjunctive queries Q that (i) Q is true in each model of F iff Q is true in
each finite model of F and (ii) determining whether F implies Q is
2EXPTIME-complete. We further show the following results: (iii) the existence
of polynomial-size conformal covers of arbitrary hypergraphs; (iv) a new proof
of the finite model property of the clique-guarded fragment; (v) the small
model property of the guarded fragment with optimal bounds; (vi) a
polynomial-time solution to the canonisation problem modulo guarded
bisimulation, which yields (vii) a capturing result for guarded bisimulation
invariant PTIME.Comment: This is an improved and extended version of the paper of the same
title presented at LICS 201
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