The ultimate in undecidability

Undecidability in number theory1

Does the equation x+y+z = 29 have a solution in integers? Yes: (3, 1, 1), for instance. How about x+y+z = 30? Again yes, although this was not known until 1999: the smallest solution is (−283059965,−2218888517, 2220422932). And how about x+y+z = 33? This is an unsolved problem. Of course, number theory does not end with the study of cubic equations in three variables: one might ask also about x...

Undecidability in quantum mechanics

The logical structure of quantum mechanics is investigated. After some remarks on formal systems and interpretations, it is shown that Bohr’s complementarity can be seen as an undecidable proposition. The flowing of the time prevents the birth of contradictions in nature, because it poses a cut between two different, but complementary aspects of the reality. PACS-03.65.Bz Foundations, theory of...

Undecidability in Epistemic Planning

Dynamic epistemic logic (DEL) provides a very expressive framework for multi-agent planning that can deal with nondeterminism, partial observability, sensing actions, and arbitrary nesting of beliefs about other agents’ beliefs. However, as we show in this paper, this expressiveness comes at a price. The planning framework is undecidable, even if we allow only purely epistemic actions (actions ...

Undecidability in number theory

D oes the equation x + y + z = 29 have a solution in integers? Yes: (3,1,1), for instance. How about the equation x + y + z = 30? Again yes, although this was not known until 1999: the smallest solution is (−283059965,−2218888517, 2220422932). And how about x + y + z = 33? This is an unsolved problem. Of course, number theory does not end with the study of cubic equations in three variables: on...

Undecidability in number theory