By Thomas Zeume
"Small Dynamic Complexity sessions" used to be offered the E.W. Beth Dissertation Prize 2016 for notable dissertations within the fields of good judgment, language, and data. The thesis experiences the principles of question re-assessment after editing a database. It explores the constitution of small dynamic descriptive complexity sessions and gives new equipment for proving reduce bounds during this dynamic context. one of many contributions to the previous point helped to substantiate the conjecture by way of Patnaik and Immerman (1997) that reachability will be maintained by way of first-order replace formulas.
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Example text
When a tuple containing a so far unused element a is inserted into the input database, then a is also inserted into the domain accessible to the update formulas. This setting is a little closer to real database systems but most results in dynamic complexity hold equally for both fixed, finite domains and active domains. Initializations How the initial state of a dynamic program looks like depends on two choices: the set of permissible input databases and the power of the initialization mapping.
Some of the initialization settings are the same. 4 Variants of the Dynamic Complexity Framework 23 one starts from an empty or from a non-empty initial database. This is because the initialization for a non-empty database can be obtained as the auxiliary relations obtained after inserting all tuples of the database into the empty one. Furthermore, it is easy to see that applying an invariant initialization mapping to an empty database is pretty much useless, as, all tuples with the same constants at the same positions are treated in the same way.
The following formula verifies the existence of such p1 and p2 : 32 2 Dynamic Complexity: Definitions and Examples T (x1 , u) ∧ T (v, y1 ) ∧ T (x2 , u) ∧ T (v, y2 ) ∧ ∃z∃z ϕ X →U1 ,Y,U2 (x1 , u, v, y1 , x2 , z, z , y2 ) ∧ E τ (u, v) τ =σ,U1 ,U2 ∈V U1 →τ ∈P U2 →τ ∈P ∧ T (x2 , z) ∧ E τ (z, z ) ∧ (z = u ∨ z = v) ∧ T (z , y2 ) ∧ T (z, u) ∧ ¬T (z , u) Again, in the first line the premises for this case are checked. 2), and it is verified that p1 and p2 are witness paths. The third and forth lines verify that z and z yield an alternative path.