Definition 7. .1. The set of SF1-expressions in normal form, NSF, is described by the grammar 7NSF e e, ƒ :: = 0 | 1 | a ∈ 7SL | e + ƒ | e · ƒ | e∗ where 7SL is as defined in Definition 5.15. 7 7 ∈ 7 From this description it is immediate that an SF1-term e NSF is formed from terms of SL connected via the regular F1-algebra operators. Hence, F1- expressions formed over the alphabet SL are the same set of terms as NSF. We shall use this observation to prove completeness for NSF with respect to the language model by leveraging completeness of F1. We use the function (—)Π to give a translation between the SF1 semantics of a term in 7NSF and the F1 semantics of that same term: )Π = e
Lemma 7.2. For all e ∈ 7NSF, we have ( e)SF1 )F1 Proof. We proceed by induction on the construction of e. In the base, there are ) three cases to consider. If e = 0, then e 1 = ∅ = e) 1 , and we are done. If SF F )SF1 F1 a ∈ 7SL, we use Lemma 5.17 to obtain a = a. As a ∈ 7SL ⊆ 7SL, we know that e = 1, then ( e )Π = ({ε})Π = {ε} = 1) , and the claim follows. If e = a for ( )Π = ({ a) }) = {( a) )Π } = {a} = {a} = a) , and the claim follows. SL For the inductive step, first consider e = H(e0). ( H(e0) )Π = {ε} if a)SF1 SL F ε ∈ e0)SF1 ∅ H e0 )F1 {ε} ε ∈ e0)F1 otherwise. We also have ( ) = if)SF1 and )0 F1 which we obtain that ε ∈ e0)SF1 e ε ∈ e0)F1 . Hence we can conclude that ∅ otherwise. The induction hypothesis states that ( e0)SF1 ) = e , from (
Definition 7. 1.1 The Cost of the Work shall be the total cost or, to the extent the Project is not completed, the estimated cost to the Owner of all elements of the Project designed or specified by the Architect and the Owner's consultants. The Cost of the Work for This Part of the Project shall be the total cost or estimated cost to the Owner of all elements of the Project designed or specified by the Consultant.
Definition 7. 14 (Randomised polynomial time). The class RP consists of exactly those problems A for which there exists an NPTM M such that for each input x, if x A then M (x) accepts with probability at least 1/2, and if x A then M (x) accepts with probability 0.
Definition 7. The relation = ( )∗ is a structural equivalence of dynamic expressions in dtsiPBC. Thus, two dynamic expressions G and G′ are structurally equivalent, denoted by G G′, if they can be reached from one another by applying the inaction rules in a forward or backward direction.
Definition 7. An output-projective binary output garbling scheme = (Gb, Xx, Ev, De) is garbled-output random if for all sufficiently large security parameters λ, for any polynomial time adversary A, GOutRandAdvG (1λ, A) = negl . In order to achieve this, we modify the scheme of Bal et al. [BMR16] to put the output wire label through the hash function H one more time; the two labels where W and W will thus be Y0 = H(finaloutput, W 0 ) and Y1 = H(finaloutput, W 0 ⊕ R), ⊕ R were the labels in the scheme of Bal et al.
