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. i,j,i j 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. 3.7 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. 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. .1. 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. .1. 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. .3.4 In a given round t ∈ t of a team persuasion game, we say that the team of defenders are winning iff t ∈ Acc (AFon) iff the team of attackers are not winning. In each round the acceptability of the topic may change, and hence the winner can change. We are interested in definitively winning states, as defined in D1 in Section 4.1. We explore the existence of such states in Section 4.4. Since we are modelling the arguments between two teams, each trying to per- suade or dissuade an audience of the topic, we specialise to bipartite argumentation frameworks because no agent should attack an argument of another agent in its own team. Further, the framework is weakly connected because all arguments asserted are relevant to the debate. Further, we assume that every argument has a coun- terargument, and that the topic is not capable of defending itself, so it does not directly attack any argument.
Definition 4. 1 (Constellation). Given the (non-empty) set of agents A, a non-empty set of roles R and a state independent role assignment ρ, the grand constellation of A is the vector Σ NR where Σr = Ar for all r. Unless the parameters are clear from the context, we denote the grand constellation for A with roles ρ, Σ(A, ρ), Given a coalition A ⊆ A, the constellation of a coalition A is σA ≤ Σ(A, ρ) where for every role r, (σA)r = |Ar|. As mentioned, we require that constellations never fail to designate. We ensure this in two different ways. When specifying the language HATL we assume that the role assignment is state independent. We show completeness for a logic based on this assumption. Towards the end of the chapter, in Example 4.30, to illustrate the difference between coalitions and constellations (hence between agents and positions), we relax this assumption and simply require that the grand constellation is invariant in the model, i.e., that the social structure is constant.
Definition 4. 1.1 (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. 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):
Definition 4. 3. A Galois extension E/F of p-adic fields is weakly ramified if the subgroup Γ2 is trivial. A Galois extension L/K of number fields is weakly ramified if for every non-Archimedean place w of L above a place v of K, the extension Lw/Ky is weakly ramified (or equivalently, the group Gw,2 is trivial). This is a strong restriction on extensions and, as a consequence, we will use the following results in the sequel.
