Definition 5. An F1-algebra [26] is a tuple (A, +, ·,∗ , 0, 1,H) where A is a set, ∗ is a unary operator, + and · are binary operators and 0 and 1 are constants, and such that for all e, f, g ∈ A the following axioms are satisfied: e + (f + g)= (e + f )+ g e + f = f + e e +0 = e e + e = e e · 1= e =1 · e e · 0=0=0 · e e · (f · g)= (e · f ) · g e∗ =1 + e · e∗ =1 + e∗ · e (e + f ) · g = e · g + f · g e · (f + g)= e · f + e · g Additionally, the loop tightening and unique fixpoint axiom hold: (e + 1)∗ = e∗ H(f )=0 ∧ e + f · g = g =⇒ f∗ · e = g Lastly, we have the following axioms for H:
Definition 5. The General Partitioning Mechanism). For an aggregate function f : D → R, a dataset D with n records of n individual users, and a privacy preference φ = (s1, . . . , sn) (s1 ≤ . . . ≤ sn). Let Partition(D, φ, k) be a procedure that partitions the original dataset D into k partitions (D1, . . . , Dk). The partitioning mechanism is defined as PM = B(DPf (D ), . . . , DPf (D )) where DPf is any target s -differentially private s1 1 sk k si i aggregate mechanism for f , B is an ensemble algorithm. The partitioning mechanisms have no privacy risk because it is computed directly from public information, privacy budget of each record. The target aggregate mecha- nism guarantees si-DP for each partition, with si as the minimum privacy parameter value of the records in that partition.
Definition 5. 7. A finite difference method for a P.D.E. is convergent if its solution Un converges to the solution u(j∆x, n∆t) of the corresponding P.D.E..
Definition 5. An oracle, Πi , is fresh at the end of its execution if: U2
Definition 5. Let C = {a ∈ A˜ | a = a∗, (ψ, [Ð, a]ψ) ≤ 0, ∀ψ ∈ H}. Two states ω, ωr ∈ S(A˜) are causally related i.e. ω ≤ ωr iff for any a ∈ C, one has ω(a) ≤ ωr(a). (5.2.8) In this case P(A˜) is the union of M0 := M× {0} and M1 := M× {1}, hence the name of two sheet spacetime. Thus, one may think of having two sheets of four- dimensional Minkowski spacetimes embedded in a five-dimensional one. Since we are interested in the causal relation between points on M0 and M1, we consider a particular type of mixed states ωx,ξ ∈ N (A˜) := M × [0, 1] ⊂ S(A˜) defined by ωx,ξ(a ⊕ b) = ξa(x) + (1 — ξ)b(x), (5.2.9) for a, b ∈ C0∞(M). Such states ωx,ξ can be considered as covering the area between the two sheets. Note that a pure state can be recovered with ξ = 0 or ξ = 1. Theorem 5.2.2. The two states ωx,ξ, ωy,η ∈ N (A) are causally related if and only if x ≤ y on M and
Definition 5. A Value-Based Argumentation Framework (V AF) is defined as (AR, A, V, η), where (AR, A) is an argumentation framework, V is a set of k values which represent the types of arguments and η: AR → V is a mapping that associates a value η(x) ∈ V with each argument x ∈ AR In section 4, the set of values will be defined as the different types of ontology mis- match, which we use to define the categories of arguments and to assign to each argu- ment one category.
Definition 5. Let T = (τi)i∈I be a finite family of transitions over K and Q, and let A be an associative array with keys in K. We define: Q[T ] = {(τi)1 | i ∈ I} ∪ {(τi)2 | i ∈ I} M [T ] = {(τi)3 | i ∈ I} C[T, A] = E × I × SA δ[T, A] : Q × SA × CA → Q × SA × M δ[T ](p, s, (e, i, u)) = (pˇ, sˇ, m) τi =(p, pˇ, m, Xxxxx, Copt, Crole, Cdata) ∧ Cmand ⊆ keys(u) ∧ keys(u) ⊆ Cmand ∪ Copt ∧ ∃r ∈ Crole(role(e, r)) ∧ sˇ = merge(u, s) where keys(u) = {k | ∃v((k, v) ∈ u)}. Now, given an initial state qinit ∈ Q[T ] and a set of final states F ⊆ Q[T ], we define the process J ) T, A, qinit, F = (Q[T ], A, C[T, A], M [T ], δ[T, A], qinit, F ).
Definition 5. Let Pfail acc be the probability that conditional 2-cast based on Q (as computed by the corresponding protocol VerifSetup) does not achieve 2- cast, given that one correct player (s, r0, or r1) accepted at the end of protocol VerifSetup. Protocol 4: Conditional 2-Cast send
Definition 5. .1. The amoeba f of a Laurent polynomial f (x) (or of the algebraic hypersurface f (x) = 0) is defined to be the image of the set f−1(0) under the map Log : (x1, . . . , xn) '→ (log |x1|, . . . , log |xn|). Let A(ϕ) be the amoeba of the singularity of the hypergeometric system Horn(ϕ). }
Definition 5. The recession xxxx XX of a convex set B ⊂ Rn is defined as CB = s ∈ Rn : u + λs ∈ B ∀ u ∈ B, λ ≥ 0 (see [9], 4). Hence the recession cone of a convex set is the maximal element (with respect to inclusion) in the family of cones whose shifts are contained in this set. The following theorem (compare with the results in [7] for the GKZ system) shows that for every vertex of the Xxxxxx polygon of the singularity of a bivariate hypergeometric function there is a basis of the solution space of the corresponding Horn system. This basis consists of hypergeometric series which converge in the pre-image of the connected component of the complement of the amoeba corre- sponding to that vertex.
