33 research outputs found
The decision problem of modal product logics with a diagonal, and faulty counter machines
In the propositional modal (and algebraic) treatment of two-variable
first-order logic equality is modelled by a `diagonal' constant, interpreted in
square products of universal frames as the identity (also known as the
`diagonal') relation. Here we study the decision problem of products of two
arbitrary modal logics equipped with such a diagonal. As the presence or
absence of equality in two-variable first-order logic does not influence the
complexity of its satisfiability problem, one might expect that adding a
diagonal to product logics in general is similarly harmless. We show that this
is far from being the case, and there can be quite a big jump in complexity,
even from decidable to the highly undecidable. Our undecidable logics can also
be viewed as new fragments of first- order logic where adding equality changes
a decidable fragment to undecidable. We prove our results by a novel
application of counter machine problems. While our formalism apparently cannot
force reliable counter machine computations directly, the presence of a unique
diagonal in the models makes it possible to encode both lossy and
insertion-error computations, for the same sequence of instructions. We show
that, given such a pair of faulty computations, it is then possible to
reconstruct a reliable run from them
Reasoning about obligations in Obligationes : a formal approach.
Despite the appearance of `obligation' in their name, medieval obligational dispu-
tations between an Opponent and a Respondent seem to many to be unrelated to
deontic logic. However, given that some of the example disputations found in me-
dieval texts involve Respondent reasoning about his obligations within the context
of the disputation, it is clear that some sort of deontic reasoning is involved. In this
paper, we explain how the reasoning diers from that in ordinary basic deontic logic,
and dene dynamic epistemic semantics within which the medieval obligations can
be expressed and the examples evaluated. Obligations in this framework are history-
based and closely connected to action, thus allowing for comparisons with, e.g., the
knowledge-based obligations of Pacuit, Parikh, and Cogan, and stit-theory. The con-
tributions of this paper are twofold: The introduction of a new type of obligation into
the deontic logic family, and an explanation of the precise deontic concepts involved
in obligationes
Reasoning about Obligations in Obligationes: A Formal Approach
Despite the appearance of `obligation' in their name, medieval obligational dispu- tations between an Opponent and a Respondent seem to many to be unrelated to deontic logic. However, given that some of the example disputations found in me- dieval texts involve Respondent reasoning about his obligations within the context of the disputation, it is clear that some sort of deontic reasoning is involved. In this paper, we explain how the reasoning diers from that in ordinary basic deontic logic, and dene dynamic epistemic semantics within which the medieval obligations can be expressed and the examples evaluated. Obligations in this framework are history- based and closely connected to action, thus allowing for comparisons with, e.g., the knowledge-based obligations of Pacuit, Parikh, and Cogan, and stit-theory. The con- tributions of this paper are twofold: The introduction of a new type of obligation into the deontic logic family, and an explanation of the precise deontic concepts involved in obligationes
A data complexity and rewritability tetrachotomy of ontology-mediated queries with a covering axiom
Aiming to understand the data complexity of answering conjunctive queries mediated by an axiom stating that a class is covered by the union of two other classes, we show that deciding their first-order rewritability is PSPACE-hard and obtain a number of sufficient conditions for membership in AC0, L, NL, and P. Our main result is a complete syntactic AC0/NL/P/CONP tetrachotomy of path queries under the assumption that the covering classes are disjoint
Horn fragments of the Halpern-Shoham Interval Temporal Logic
We investigate the satisfiability problem for Horn fragments of the Halpern-Shoham interval temporal logic depending on the type (box or diamond) of the interval modal operators, the type of the underlying linear order (discrete or dense), and the type of semantics for the interval relations (reflexive or irreflexive). For example, we show that satisfiability of Horn formulas with diamonds is undecidable for any type of linear orders and semantics. On the contrary, satisfiability of Horn formulas with boxes is tractable over both discrete and dense orders under the reflexive semantics and over dense orders under the irreflexive semantics but becomes undecidable over discrete orders under the irreflexive semantics. Satisfiability of binary Horn formulas with both boxes and diamonds is always undecidable under the irreflexive semantics
Undecidable propositional bimodal logics and one-variable first-order linear temporal logics with counting
First-order temporal logics are notorious for their bad computational
behaviour. It is known that even the two-variable monadic fragment is highly
undecidable over various linear timelines, and over branching time even
one-variable fragments might be undecidable. However, there have been several
attempts on finding well-behaved fragments of first-order temporal logics and
related temporal description logics, mostly either by restricting the available
quantifier patterns, or considering sub-Boolean languages. Here we analyse
seemingly `mild' extensions of decidable one-variable fragments with counting
capabilities, interpreted in models with constant, decreasing, and expanding
first-order domains. We show that over most classes of linear orders these
logics are (sometimes highly) undecidable, even without constant and function
symbols, and with the sole temporal operator `eventually'.
We establish connections with bimodal logics over 2D product structures
having linear and `difference' (inequality) component relations, and prove our
results in this bimodal setting. We show a general result saying that
satisfiability over many classes of bimodal models with commuting linear and
difference relations is undecidable. As a by-product, we also obtain new
examples of finitely axiomatisable but Kripke incomplete bimodal logics. Our
results generalise similar lower bounds on bimodal logics over products of two
linear relations, and our proof methods are quite different from the proofs of
these results. Unlike previous proofs that first `diagonally encode' an
infinite grid, and then use reductions of tiling or Turing machine problems,
here we make direct use of the grid-like structure of product frames and obtain
undecidability by reductions of counter (Minsky) machine problems