2,043 research outputs found
PSPACE Reasoning for Graded Modal Logics
We present a PSPACE algorithm that decides satisfiability of the graded modal
logic Gr(K_R)---a natural extension of propositional modal logic K_R by
counting expressions---which plays an important role in the area of knowledge
representation. The algorithm employs a tableaux approach and is the first
known algorithm which meets the lower bound for the complexity of the problem.
Thus, we exactly fix the complexity of the problem and refute an
ExpTime-hardness conjecture. We extend the results to the logic Gr(K_(R \cap
I)), which augments Gr(K_R) with inverse relations and intersection of
accessibility relations. This establishes a kind of ``theoretical benchmark''
that all algorithmic approaches can be measured against
Alternating register automata on finite words and trees
We study alternating register automata on data words and data trees in
relation to logics. A data word (resp. data tree) is a word (resp. tree) whose
every position carries a label from a finite alphabet and a data value from an
infinite domain. We investigate one-way automata with alternating control over
data words or trees, with one register for storing data and comparing them for
equality. This is a continuation of the study started by Demri, Lazic and
Jurdzinski. From the standpoint of register automata models, this work aims at
two objectives: (1) simplifying the existent decidability proofs for the
emptiness problem for alternating register automata; and (2) exhibiting
decidable extensions for these models. From the logical perspective, we show
that (a) in the case of data words, satisfiability of LTL with one register and
quantification over data values is decidable; and (b) the satisfiability
problem for the so-called forward fragment of XPath on XML documents is
decidable, even in the presence of DTDs and even of key constraints. The
decidability is obtained through a reduction to the automata model introduced.
This fragment contains the child, descendant, next-sibling and
following-sibling axes, as well as data equality and inequality tests
The satisfiability threshold for random linear equations
Let be a random matrix over the finite field with
precisely non-zero entries per row and let be a random vector
chosen independently of . We identify the threshold up to which the
linear system has a solution with high probability and analyse the
geometry of the set of solutions. In the special case , known as the
random -XORSAT problem, the threshold was determined by [Dubois and Mandler
2002, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], and the proof
technique was subsequently extended to the cases [Falke and Goerdt
2012]. But the argument depends on technically demanding second moment
calculations that do not generalise to . Here we approach the problem from
the viewpoint of a decoding task, which leads to a transparent combinatorial
proof
- …