Symmetry Protected Quantum Computation

We consider a model of quantum computation using qubits where it is possible to measure whether a given pair are in a singlet (total spin 0) or triplet (total spin 1) state. The physical motivation is that we can do these measurements in a way that is protected against revealing other information so long as all terms in the Hamiltonian are SU(2)-invariant. We conjecture that this model is equivalent to BQP. Towards this goal, we show: (1) this model is capable of universal quantum computation with polylogarithmic overhead if it is supplemented by single qubit X and Z gates. (2) Without any additional gates, it is at least as powerful as the weak model of "permutational quantum computation" of Jordan [14, 18]. (3) With postselection, the model is equivalent to PostBQP.

Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

In a previous paper we defined the black and white SAT problem which has exactly two solutions, where each variable is either true or false. We showed that black and white 2-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonboglár model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a black and white SAT problem. We prove a powerful theorem, the so called transitions theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as black and white SAT problems. We show that the Balatonboglár model is between the strong and the weak model, and it generates 3-SAT problems, so the Balatonboglár model represents strongly connected communication graphs as black and white 3-SAT problems. Our motivation to study these models is the following: The strong model generates a 2-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonboglár model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonboglár model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: the SAT problem and directed graphs.

Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

In a previous paper we defined the Black-and-White SAT problem which has exactly two solutions, where each variable is either true or false. We showed that Black-and-White $2$-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonbogl\'{a}r model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a Black-and-White SAT problem. We prove a powerful theorem, the so called Transitions Theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as Blask-and-White SAT problems. We show that the Balatonbogl\'{a}r model is between the strong and the weak model, and it generates $3$-SAT problems, so the Balatonbogl\'{a}r model represents strongly connected communication graphs as Black-and-White $3$-SAT problems. Our motivation to study these models is the following: The strong model generates a $2$-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonbogl\'{a}r model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonbogl\'{a}r model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: SAT problem and directed graphs.

A Logic Framework for P2P Deductive Databases

This paper presents a logic framework for modeling the interaction among deductive databases in a peer-to-peer (P2P) environment. Each peer joining a P2P systemprovides or imports datafrom its neighbors by using a set ofmapping rules, that is, a set of semantic correspondences to a set of peers belonging to the same environment. By using mapping rules, as soon as it enters the system, a peer can participate and access all data available in its neighborhood, and through its neighborhood it becomes accessible to all the other peers in the system. A query can be posed to any peer in the system and the answer is computed by using locally stored data and all the information that can be consistently imported from the neighborhood. Two different types of mapping rules are defined: mapping rules allowing to import a maximal set of atoms not leading to inconsistency (calledmaximal mapping rules) and mapping rules allowing to import a minimal set of atoms needed to restore consistency (calledminimal mapping rules). Implicitly, the use of maximal mapping rules statesit is preferable to import as long as no inconsistencies arise; whereas the use of minimal mapping rules states thatit is preferable not to import unless a inconsistency exists. The paper presents three different declarative semantics of a P2P system: (i) theMax Weak Model Semantics, in which mapping rules are used to importas much knowledge as possiblefrom a peer’s neighborhood without violating local integrity constraints; (ii) theMin Weak Model Semantics, in which the P2P system can be locally inconsistent and the information provided by the neighbors is used to restore consistency, that is, to only integrate the missing portion of a correct, but incomplete database; (iii) theMax-Min Weak Model Semanticsthat unifies the previous two different perspectives captured by the Max Weak Model Semantics and Min Weak Model Semantics. This last semantics allows to characterize each peer in the neighborhood as a resource used either to enrich (integrate) or to fix (repair) the knowledge, so as to define a kind ofintegrate–repairstrategy for each peer. For each semantics, the paper also introduces an equivalent and alternative characterization, obtained by rewriting each mapping rule into prioritized rules so as to model a P2P system as a prioritized logic program. Finally, results about the computational complexity of P2P logic queries are investigated by consideringbraveandcautiousreasoning.