6,556 research outputs found
Algorithmic Structuring of Cut-free Proofs
The problem of algorithmic structuring of proofs in the sequent calculi LK and LKB ( LK where blocks of quantifiers can be introduced in one step) is investigated, where a distinction is made between linear proofs and proofs in tree form. In this framework, structuring coincides with the introduction of cuts into a proof. The algorithmic solvability of this problem can be reduced to the question of k-l-compressibility: "Given a proof of length k , and l ≤ k : Is there is a proof of length ≤ l ?" When restricted to proofs with universal or existential cuts, this problem is shown to be (1) undecidable for linear or tree-like LK-proofs (corresponds to the undecidability of second order unification), (2) undecidable for linear LKB-proofs (corresponds to the undecidability of semi-unification), and (3) decidable for tree-like LKB -proofs (corresponds to a decidable subprob-
lem of semi-unification)
Undecidability of the unification and admissibility problems for modal and description logics
We show that the unification problem `is there a substitution instance of a
given formula that is provable in a given logic?' is undecidable for basic
modal logics K and K4 extended with the universal modality. It follows that the
admissibility problem for inference rules is undecidable for these logics as
well. These are the first examples of standard decidable modal logics for which
the unification and admissibility problems are undecidable. We also prove
undecidability of the unification and admissibility problems for K and K4 with
at least two modal operators and nominals (instead of the universal modality),
thereby showing that these problems are undecidable for basic hybrid logics.
Recently, unification has been introduced as an important reasoning service for
description logics. The undecidability proof for K with nominals can be used to
show the undecidability of unification for boolean description logics with
nominals (such as ALCO and SHIQO). The undecidability proof for K with the
universal modality can be used to show that the unification problem relative to
role boxes is undecidable for Boolean description logic with transitive roles,
inverse roles, and role hierarchies (such as SHI and SHIQ)
(Un)decidable Problems about Reachability of Quantum Systems
We study the reachability problem of a quantum system modelled by a quantum
automaton. The reachable sets are chosen to be boolean combinations of (closed)
subspaces of the state space of the quantum system. Four different reachability
properties are considered: eventually reachable, globally reachable, ultimately
forever reachable, and infinitely often reachable. The main result of this
paper is that all of the four reachability properties are undecidable in
general; however, the last three become decidable if the reachable sets are
boolean combinations without negation
Undecidability of Two-dimensional Robot Games
Robot game is a two-player vector addition game played on the integer lattice
. Both players have sets of vectors and in each turn the vector
chosen by a player is added to the current configuration vector of the game.
One of the players, called Eve, tries to play the game from the initial
configuration to the origin while the other player, Adam, tries to avoid the
origin. The problem is to decide whether or not Eve has a winning strategy. In
this paper we prove undecidability of the robot game in dimension two answering
the question formulated by Doyen and Rabinovich in 2011 and closing the gap
between undecidable and decidable cases
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