A notion of complexity for an arbitrary structure was defined in the book of Poizat Les petits cailloux (1995): we can define P and NP problems over a differential field K. Using the Witness Theorem of Blum et al., we prove the P-stability of the theory of differential fields: a P problem over a differential field K is still P when restricts to a sub-differential field k of K. As a consequence,...

Hartmanis used Kolmogorov complexity to provide an alternate proof of the classical result of Baker, Gill, and Solovay that there is an oracle relative to which P is not NP. We refine the technique to strengthen the result, constructing an oracle relative to which a conjecture of Lipton is false.

We examine the enumeration problem for essential closed sets of a formal context. Essential closed sets are sets that can be written as the closure of a pseudo-intent. The results for enumeration of essential closed sets are similar to existing results for pseudo-intents, albeit some differences exist. For example, while it is possible to compute the lectically first pseudo-intent in polynomial...

We improve hardness results for the problem of embedding one finite metric into another with minimum distortion. This problem is equivalent to optimally embedding one weighted graph into another under the shortest path metric. We show that unless P = NP, the minimum distortion of embedding one such graph into another cannot be efficiently approximated within a factor less than 9/4 even when the...

In this note we introduce a notion of a generically (strongly generically) NP-complete problem and show that the randomized bounded version of the halting problem is strongly generically NP-complete.

We prove that we may compute the normal form of each term of the simply typed-calculus in a polynomial number of sharable reductions (where the notion of sharing is L evy's \optimal" one). As a simple corollary, we get that P = NP \up to sharing", i.e. up to the computational overhead due to sharing.

Recent works by Ajtai and by Ajtai and Dwork bring to light the old (general) question of whether it is at all possible to base the security of cryptosystems on the assumption that P 6 = NP. We discuss this question and in particular review and extend a two-decade old result of Brassard regarding this question. Our conclusion is that the question remains open.

We study planar drawings of directed graphs in the L-drawing standard. We provide necessary conditions for the existence of these drawings and show that testing for the existence of a planar L-drawing is an NP-complete problem. Motivated by this result, we focus on upwardplanar L-drawings. We show that directed st-graphs admitting an upward(resp. upward-rightward-) planar L-drawing are exactly ...

Minesweeper is a one-person game which looks deceptively easy to play, but where average human performance is far from optimal. Playing the game requires logical, arithmetic and probabilistic reasoning based on spatial relationships on the board. Simply checking a board state for consistency is an NP-complete problem. Given the difficulty of hand-crafting strategies to play this and other games...