--- title: Full-Information Protocol Overview url: https://www.emergentmind.com/topics/full-information-protocol type: topic --- # Full-Information Protocol Overview “Full-information protocol” is not a single formal object across the literature. In distributed computing, it usually denotes a protocol in which each agent’s local state records its entire history and, in each round, it re-sends its whole state to all other agents [2305.06271]. Closely related work studies a *full-information model* in which all past broadcasts are public [1503.01588], a game-theoretic communication primitive by which a player’s entire observation history is revealed to another player [2307.01063], and a quantum Byzantine setting in which the adversary knows the exact joint state of all unmeasured qubits [2409.01707]. In interactive quantum communication, the phrase is used differently again: a full-information or no-forgetting protocol is one with $CRIC(\Pi,\mu)=0$ [1701.02062]. The common theme is maximal exposure or retention of protocol state, but the formal content depends on the problem domain. ## 1. Canonical meanings and formal scope Across the cited literature, the phrase appears in several distinct but related roles. The table summarizes the main meanings that are formalized explicitly. | Context | Formal feature | Source | |---|---|---| | Eventual Byzantine Agreement | each agent records its entire history and rebroadcasts its whole local state | [2305.06271] | | One-round coin-flipping | all past broadcasts, including adversarial ones, are public | [1503.01588] | | Imperfect-information games | communication action reveals the entire observation history of another player | [2307.01063] | | Quantum communication | no-forgetting condition $CRIC(\Pi,\mu)=0$ | [1701.02062] | | Quantum Byzantine Agreement | adversary holds a purifying register of the entire pure global state | [2409.01707] | In the runs-and-systems formulation used for eventual Byzantine agreement, an information-exchange protocol is a tuple \[ E=\langle E_1,\dots,E_n\rangle, \] and a full-information protocol is obtained by taking local states of the form \[ L_i=\{\langle t,init_i,decided_i,\mathit{hist}_i\rangle\}, \] with \(M_i=L_i\) and \(\mu_i(\langle t,\dots,\mathit{hist}\rangle,a)=\langle t,\dots,\mathit{hist}\rangle\), so that agent \(i\) rebroadcasts its entire local state every round [2305.06271]. This is the most literal protocol-level use of the term. The game-synthesis literature generalizes the same intuition. A synchronous imperfect-information game with communication specifies observation functions \[ O_i:S\times C\to \Sigma_i, \] and a communication graph \(R_\sigma\subseteq P\times P\) such that when \((i,j)\in R_\sigma\), player \(i\) “peeks” the full view of \(j\) [2307.01063]. Here, “full information” is not universal transparency; it is selective but history-complete disclosure. A plausible implication is that “full-information” should be read as a family resemblance rather than a single definition. In each usage, the operative object is not merely a transcript, but a recursively enriched state that contains prior observations, communicated knowledge, or retained information. ## 2. Full-information coin-flipping and the adaptive-corruption barrier Goldwasser, Kalai, and Park study one-round coin-flipping in the full-information model introduced by Ben-Or and Linial in 1985 [1503.01588]. In this model there are \(n\) computationally unbounded players, communication takes place over a single reliable broadcast channel, each honest party has private fresh randomness, and all past broadcasts are public. The paper distinguishes two adversaries. A standard adaptive adversary can corrupt new parties based on messages seen so far, but cannot retroactively change honest broadcasts. A strong adaptive adversary first “peeks” at all honest messages that would be sent in a round, then decides which parties to corrupt and can replace or suppress those messages [1503.01588]. The central upper bound is that any one-round full-information coin-flipping protocol secure against \(t\) strong adaptive corruptions must satisfy \[ t=\widetilde O(\sqrt n). \] The same work proves that if a symmetric one-round protocol is secure against \(t\) standard adaptive corruptions, then there exists a symmetric one-round protocol secure against \(\Omega(t)\) strong adaptive corruptions; as a corollary, any symmetric one-round full-information coin-flipping protocol tolerates at most \(\widetilde O(\sqrt n)\) standard adaptive corruptions [1503.01588]. The proof architecture is structurally important. The paper gives a robust-set characterization \[ R^{\Pi_n}(b,t)=\bigl\{\vec r\in{\mathcal R}^n:\forall\,\vec r'\text{ with }{\rm Hamming}(\vec r,\vec r')\le t,\;\Pi_n(\vec r)=\Pi_n(\vec r')=b\bigr\}, \] and shows that strong-adaptive security is equivalent to lower bounds on the probability of landing in \(R^{\Pi_n}(0,t)\) and \(R^{\Pi_n}(1,t)\). It then reduces arbitrary message length to \(O(\log^{1+\delta}n)\) bits per player, then to single-bit players, and finally applies the Lichtenstein–Linial–Saks bound [1503.01588]. One recurring misconception is that longer one-round messages might overcome the adaptive-corruption barrier. The paper rules this out: increased message length does not help in the strong-adaptive setting, and in the symmetric case it does not help against ordinary adaptive corruptions either [1503.01588]. The full-information constraint is therefore not merely a semantic choice; in one-round coin-flipping it induces a quantitative resilience barrier. ## 3. Eventual Byzantine agreement, knowledge tests, and optimality In eventual Byzantine agreement with omission failures, Alpturer, Halpern, and van der Meyden separate a protocol into an information-exchange component \(E\) and an action protocol \(P\), allowing optimality to be studied relative to a fixed exchange discipline [2305.06271]. Under a fixed \(E\), one says \(P'\le_\gamma P\) if in every corresponding run with the same initial states and failure pattern, whenever \(P_i\) first decides in round \(m\), \(P'_i\) does not decide at any earlier round. A protocol is optimal if no other EBA protocol strictly dominates it [2305.06271]. Their knowledge-based program \(P^0\) has the following structure: if an agent has already decided, it performs \(\mathit{noop}\); if \(init_i=0\) or \(K_i(\vee_{j\in Agt}\,j\!\mathit{decided}_j=0)\), it executes \(\mathit{decide}_i(0)\); if \(K_i(\forall_{j\in Agt}\neg(\mathit{deciding}_j=0))\), it executes \(\mathit{decide}_i(1)\); otherwise it performs \(\mathit{noop}\) [2305.06271]. The intended epistemic reading is asymmetric: deciding \(0\) is triggered as soon as a \(0\)-decision is known to exist, while deciding \(1\) requires knowledge that no agent can be in the process of deciding \(0\). Optimality is tied to a safety condition on the information-exchange protocol. Safety rules out “spurious” knowledge: if an agent has not received a \(0\)-chain, then there must exist an indistinguishable point where all initial values are \(1\); and if an agent lacks knowledge that no one is about to decide \(0\), there must exist an indistinguishable point where some nonfaulty agent is indeed about to decide \(0\) [2305.06271]. The theorem is then exact: if \(\gamma=(E,SO(t),\pi)\) is an EBA context and \(P^0\) is safe with respect to \(\gamma\), every implementation of \(P^0\) is optimal in \(\gamma\) [2305.06271]. The same paper gives two limited-exchange implementations. In \(E_{\min}(n)\), each agent sends exactly one non-null bit, namely its decision, to all \(n\) agents, for \(n^2\) bits total; if some agent starts with \(0\), all decide by round \(2\), while if all start with \(1\), they decide \(1\) at round \(t+2\) [2305.06271]. In \(E_{\mathit{basic}}(n)\), undecided agents additionally send \((\mathit{init},1)\), yielding \(O(n^2 t)\) bits total and allowing all-1 failure-free runs to decide by round \(2\) [2305.06271]. The full-information exchange is stronger but not automatically optimal. The paper therefore adds common-knowledge tests and defines \(P^1\), whose leading rules are \(K_i\,C_N(t\land no_N(1)\land \exists0)\) for deciding \(0\) and \(K_i\,C_N(t\land no_N(0)\land \exists1)\) for deciding \(1\), before falling back to the \(P^0\) tests [2305.06271]. Every FIP implementation of \(P^1\) is correct, and the added common-knowledge rules make it optimal among all FIP protocols. The usual graph-based full-information implementation costs \(O(n^4 t^2)\) bits, but the action protocol can still be implemented in polynomial time by tracking a compact communication graph [2305.06271]. ## 4. Full-information protocols as a synthesis target in games The synthesis perspective treats full-information protocols as a communication discipline inside imperfect-information games. Berwanger, Mathew, and van den Bogaard model a synchronous game \[ G=(P,(A^i)_{i\in P},C,S,s_0,\Delta,O,\Omega), \] with one active player, passive observers, explicit communication labels, deterministic transition function \(\Delta\), and perfect-recall observation functions \(O_i:S\times C\to\Sigma_i\) [2307.01063]. The key communication action is that player \(i\) may receive the entire observation history of player \(j\). The reduction schema is classical in outline but technically specialized. One first forms the information tree \(U(G)\) of information sets \([\tau]_{\sim}\). To obtain a finite perfect-information game, the paper constructs a rectangular morphism \(h:\Gamma^*\to P\) of finite index satisfying rectangularity, morphism, and refinement conditions. In the full-information setting with \(n\) observers, \(h(\tau)\) is built as a vector of knowledge-sets indexed by all coalitions \(J\ni 0\), and the recursive update is driven by the synchronization operator \(\sync_J(c)\) [2307.01063]. The resulting complexity is sharp. With \(n\) observers, full-information protocols can express indistinguishability relations not expressible with \(n-1\) observers, giving a strict hierarchy in expressiveness and complexity; the strategy-synthesis problem is \((n+1)\)-EXPTIME-complete [2307.01063]. For fixed parity or \(\omega\)-regular objectives, one can decide and synthesize a winning strategy in an FIP with \(n\) observers in time \((n+1)\)-EXPTIME, and reachability already yields the matching lower bound [2307.01063]. This suggests that full-information communication is not algorithmically trivial merely because it discloses entire views. The disclosure induces higher-order knowledge structures—knowledge of others’ views, of coalitional synchronization, and of recursively communicated histories—that are expensive to represent but still regular enough to admit finite quotients. ## 5. Quantum reinterpretations: no-forgetting protocols and full-information adversaries In two-party interactive quantum communication, Laurière and Touchette use “full-information protocol” in a different technical sense: a protocol that never forgets information about the other party’s input [1701.02062]. For classical inputs, they define the classical-input information cost \(CIC\), Holevo information cost \(HIC\), and reverse information cost \(CRIC\), and prove the exact identities \[ HIC=CIC-CRIC,\qquad QIC=CIC+CRIC,\qquad CIC\le QIC\le 2\,CIC. \] A protocol is full-information or no-forgetting precisely when \[ CRIC(\Pi,\mu)=0\quad\Longleftrightarrow\quad QIC(\Pi,\mu)=HIC(\Pi,\mu)=CIC(\Pi,\mu) \] [1701.02062]. The conceptual tool is the Information-Flow Lemma, which tracks the net change in conditional mutual information across an interactive process: \[ I(\bar E:\bar B_{r+1}\mid \bar F)_{\rho_{r+1}}-I(\bar E:\bar B_0\mid\bar F)_{\rho_0} = \sum_{i=1}^r\Bigl[ I(\bar E:\bar C_i\mid \bar F\,\bar B_i)_{\rho_i}-I(\bar E:\bar D_i\mid \bar F\,\bar B_i)_{\rho_i} \Bigr] \] [1701.02062]. Operationally, \(QIC\) decomposes into transmitted-input cost and forgotten-input cost. In classical reversible protocols, \(CRIC=0\); in quantum protocols, the paper argues that forgetting is often necessary. The lower-bound consequences are explicit. Any \(r\)-round quantum protocol for \(DISJ_n\) that never forgets information satisfies \[ QCC^{r,NF}(DISJ_n,\eps)\in\Omega(n), \] so the quadratic quantum speedup is lost [1701.02062]. At zero error, the Inner Product function has \[ QIC_0(IP_n,\nu)=n, \] and for a random Boolean function \(f\), with overwhelming probability, \[ QIC_0(f,\nu)=n(1-o(1)) \] [1701.02062]. A second quantum usage concerns full-information *adversaries* rather than no-forgetting protocols. In the quantum Byzantine agreement model of [2409.01707], players exchange qubits and the adversary holds a purifying register of the entire pure global state; equivalently, in round \(k\) the adversary knows the exact joint state of all unmeasured qubits. The paper proves a black-box lifting theorem: any non-erasing classical private-channel BA protocol \(\Pi\) with resilience \(t\), round complexity \(\mathrm{RC}(\Pi)\), and communication cost \(\mathrm{CC}(\Pi)\) yields a quantum full-information protocol \(\Sigma\) with the same \(t\), \(\mathrm{RC}\), and \(\mathrm{CC}\) [2409.01707]. This lifting has concrete consequences. The quantum full-information model admits \(O(1)\)-round agreement for \(t