157 research outputs found
Probabilistic Model Counting with Short XORs
The idea of counting the number of satisfying truth assignments (models) of a
formula by adding random parity constraints can be traced back to the seminal
work of Valiant and Vazirani, showing that NP is as easy as detecting unique
solutions. While theoretically sound, the random parity constraints in that
construction have the following drawback: each constraint, on average, involves
half of all variables. As a result, the branching factor associated with
searching for models that also satisfy the parity constraints quickly gets out
of hand. In this work we prove that one can work with much shorter parity
constraints and still get rigorous mathematical guarantees, especially when the
number of models is large so that many constraints need to be added. Our work
is based on the realization that the essential feature for random systems of
parity constraints to be useful in probabilistic model counting is that the
geometry of their set of solutions resembles an error-correcting code.Comment: To appear in SAT 1
Mastermind is NP-Complete
In this paper we show that the Mastermind Satisfiability Problem (MSP) is
NP-complete. The Mastermind is a popular game which can be turned into a
logical puzzle called Mastermind Satisfiability Problem in a similar spirit to
the Minesweeper puzzle. By proving that MSP is NP-complete, we reveal its
intrinsic computational property that makes it challenging and interesting.
This serves as an addition to our knowledge about a host of other puzzles, such
as Minesweeper, Mah-Jongg, and the 15-puzzle
Non-causal computation
Computation models such as circuits describe sequences of computation steps
that are carried out one after the other. In other words, algorithm design is
traditionally subject to the restriction imposed by a fixed causal order. We
address a novel computing paradigm beyond quantum computing, replacing this
assumption by mere logical consistency: We study non-causal circuits, where a
fixed time structure within a gate is locally assumed whilst the global causal
structure between the gates is dropped. We present examples of logically
consistent non- causal circuits outperforming all causal ones; they imply that
suppressing loops entirely is more restrictive than just avoiding the
contradictions they can give rise to. That fact is already known for
correlations as well as for communication, and we here extend it to
computation.Comment: 6 pages, 4 figure
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