Definition 4 Sample Clauses

Definition 4. The code rate of an authentication scheme, denoted by R, is de- fined as the number of bits that can be authenticated by Xxxxx and Xxx with one bit of their initial, correlated strings. For traditional authentication codes, the code rate R is determined by the length of the source states divided by that of the encoding rules (authentication keys).

Definition 4. 3.4 The minimum 0-1 distance of a code , denoted by d0→1( ), is defined as the smallest value among the 0-1 distances between any two different codewords in C, i.e., d0→1(C) = min d(ci → cj ), where ci, cj ∈ C. The minimum 0-1 distance of any conventional linear code is 0 since the zero code- word always lies in the code. The following theorem shows how to change a conven- tional linear code of Hamming distance d into a code with 0-1 distance d.

Definition 4. 1.1 The grievance shall mean a written complaint by a member of the bargaining unit or the Association that there has been a violation, misinterpretation, or misapplication of any provision(s) of this agreement. The provision(s) grieved shall be so designated.

Definition 4. For a set of data points P and a query point q, RkNN query retrieves every points p ∈ P for which Dist(q,p) ≤ Dist(p,pk) where pk is the kth nearest point to p in P — {q}.

Definition 4. Random set R and the probability spaces (Ω, S, P) and (ΩA, SA, PA)). Fix 0 ≤ pm ≤ 1 for each m ∈ N. We generate a random set R ⊂ N by adding m to R with probability pm, independently for each m. We let (Ω, S, P) be the probability space of the random sets R. More generally, for A ⊂ N, let (ΩA, SA, PA) be the probability space of the random sets R ∩ A. In general, we shall fix absolute constants α > 0 and 0 < δ ≤ 1, and let pm = min{1, αm—1+δ} for all positive integers m. Note that we restrict our probabilities only to the above probabilities, ignoring the case when, say, pm = m—1/2 log m. Covering the remaining cases would not require a new proof technique, but it would be a bit more cumbersome. Readers interested in the details of the construction of the spaces (Ω, S, P) and (ΩA, SA, PA) are encouraged to consult, for example, Xxxxxxxxxx and Xxxx [21, Theorem 13, page 142]. Using the natural correspondence between subsets of N and 0–1 vectors indexed by N, we may identify (Ω, S, P) with the product of the two-point spaces (Ωm, Sm, Pm) (m ∈ N), where Ωm = {0, 1}, Sm = 2Ωm , and Pm({1}) = pm and Pm({0}) = 1 − pm. Thus, S is the σ-algebra generated by the sets C(m) = {R ⊂ N: m ∈ R} (m ∈ N), (4.1) that is, the smallest family of subsets of N that is closed under complementa- tion, finite intersections, and countable unions that contains the sets in (4.1). Furthermore, P(C(m)) = P(m ∈ R) = pm for all m, and this suffices to define P on every member of S uniquely. Similarly, (ΩA, SA, PA) may be identified with the product of the two-point spaces (Ωm, Sm, Pm) (m ∈ A) above. In what follows, we shall often write P instead of PA, as this will not cause any confusion. We will study how dense Sidon sets are contained in R. We introduce the notion of the growth of a set S ⊂ N. We say that S has lower growth at least h(n) if S[n] ≥ h(n) for every sufficiently large n. We also say that S has upper growth at most h(n) if S[n] ≤ h(n) for every sufficiently large n. Let R be a set of N. We will abbreviate the fact that there exists a Sidon subset S ⊂ R with lower growth at least h(n), by writing lgrIS(R) ≥ h(n). Similarly, ugr∀S(R) ≤ h(n) will mean that all Sidon subsets S ⊂ R have upper growth at most h(n). An abridged version of our results of this paper is the following.

Definition 4. Given a formula Ψ, the set of maximal consistent subsets of ecl(Ψ), is denoted Γ. The soundness of these rules are argued for exactly as for the case of ATL ([40]), except the (S) axiom which is slightly different from the super-additivity axiom we know from ATL. We show the soundness of this axiom.

Definition 4. Given a Team Persuasion Game, the set of arguments that are on in round t is Aon := {a ∈ A | St(a) = on} ∪{t}. The induced argument framework is AFon := ⟨Aon, Xxx⟩, where Xxx := R ∩ [Aon × Aon].

Definition 4. The free energy of a set of solutions (configurations) is defined as

Definition 4. An encryption scheme is a triple Γ = ( , , ) of probabilis- tic polynomial-time algorithms. takes as input the security parameter 1η and produces a key pair (pk, sk) where pk is the public encryption key and sk is the private decryption key. S takes as input a public key pk and a plaintext m and outputs a ciphertext. Ð takes as input a private key sk and a ciphertext and out- puts a plaintext or ⊥. It is required that P[(pk, sk) → K(1η); c → S (pk, m); m′ → Ð(sk, c) : m = m′] = 1. We write { m }pk to denote S (pk, m).

Definition 4. 11 (Local Consistency). A labeled tree (T ,V ) is said to be locally consis- tent if, for each non-leaf node t ∈ T, • if ((σ )) Ⓧ φ ∈ V (t), there is an σ-vote vσ , such for every v ∈ ext(q, vσ ), we have φ ∈ V (t · v), and • if ¬((σ )) Ⓧ η ∈ V (t), then for every σ-vote vσ , there is a v ∈ ext(q, vσ ), for which ¬η ∈ V (t · v). The construction of the locally consistent simple labeled tree for an (appropriate) set of formulas, will depend on the following lemma. The lemma is similar to [40, Lemma 31 (Disjoint Coalition Consistency)], but applies to composable constellations rather than disjoint coalitions. Lemma 4.12 (Composable Constellation Consistency). Let {((σ1)) Ⓧ φ1,..., ((σk)) Ⓧ φk, ¬((σ )) Ⓧ η} be a consistent set of formulas. If σ1 + ··· + σk ≤ σ ≤ Σ, then {φ1,..., φk, ¬η} is consistent. {(( )) Ⓧ (( )) Ⓧ ¬(( )) Ⓧ } Proof. Let σ1 φ1,..., σk φk, σ η be a consistent set. Since (σ1 + ··· ∧ ( )) Ⓧ → (( ··· )) Ⓧ ∧··· ∧ ··· ≤ ··· ≤ ≤ (( )) Ⓧ ∧ + σk) σ Σ, we have, from repeated applications of (S) that ( σ1 φ1 σk φk) σ1 + + σk (φ1 φk). Since σ1 + + σk σ , by (( )) Ⓧ ∧··· ∧ constellation monotonicity, σ (φ1 φk) is consistent. by (( ))Ⓧ (( )) Ⓧ ∧ ··· ∧ ( )) Ⓧ → (( )) Ⓧ · · · ∧ → (( )) Ⓧ ∧··· ∧ → (( )) Ⓧ { ¬ } ∧ Assume towards a contradiction that φ1,..., φk, η is not consistent. Then (φ1 φk) η is a tautology. It would follow that σ (φ1 φk) σ η σ -monotonicity. Hence ( σ1 φ1 σk φk) σ η. This contradicts the assumption. ⊕ g Before we state and prove the claim that we can construct locally consistent trees for appropriate consistent sets of formulas, Φ, we define some auxiliary notation. Let Φ be a finite consistent set of formulas, and Φ and Φ be the sets of positive and negative Ⓧ | | | | ≤ | | | |⊕ g ⊕ g -formulas as discussed earlier, such that Φ + Φ 2 Ψ + ( Ψ + 1)2. | | | |⊕g All agents, regardless of role assignment, will have Ψ +( Ψ + 1)2 actions avail- able, so the vote of a role will have a profile in the form (depening on the argument set of formulas Φ satisfying the above mentioned inequality):