2. The Interactive Communication Problem
Background and Related Work
3. Finite-State Protocols
4. Basic Concepts of the Coding Schemes
4.1. Vertical Simulation
- Assume that Alice and Bob have and .
- Alice uses to calculate according to (6).
- Alice encodes using a block code with rate , and sends it to Bob over the channel, using times. This code will be referred to as a vertical block code.
- Bob decodes the output of the channel and obtains .
- Alice (resp. Bob) uses and (resp. and ) to calculate (resp. ) according to (7).
- Alice and Bob advance j by one.
4.2. Efficient State Lookahead
- For every block, the last state can be calculated by both parties given the first state, without knowing the entire transcript of the block, using only (clean) bits exchanged between the parties.
- The bits required for this calculation for the entire protocol, can be reliably exchanged over the noisy channels at a strictly positive rate.
4.3. Efficient Exhaustive Simulation
- At every block, the transcripts associated with all possible M initial states, can be encoded using only bits.
- The required bits can be reliably conveyed over the noisy channels at any rate below Shannon capacity.
5. Achieving Shannon Capacity with Two States
- Alice sends Bob her latest (odd) time index in the block for which , (i.e., her latest constant composite function), along with value of . If such an index does not exist she sends zero to Bob. Bob then repeats the same process with the appropriate alterations. We use to denote the maximum of the indices, which therefore represents the location of the last constant composite function in the block. We now set if and if . This process requires exchanging bits between Alice and Bob.
- We now note, that since is the index of the latest constant composite function in the block, then for all , for some . The final state in the block, , can therefore be calculated byWe finally note, that and are single bits that can be calculated by their respective parties and then exchanged, leaving the total number of required exchanged bits for the algorithm .
- Before the simulation begins, both parties communicate the locations of the first (rather than the last) constant composite function in the block: the smallest value for which , for some . This process requires exchanging bits.
- The parties exchange the identities of their transmission functions (i.e., ) before the location of the first constant composite function in the block, using a single bit per time index. In the sequel we show that there are indeed only two relevant functions to describe, so their description requires only a single bit. At the end of this process, the parties can independently simulate the transcripts for both initial states until the location of the first constant composite function.
- For time indices after the location of the first constant composite function, the transcripts associated with both initial states coincide, so they can both be simulated using a single bit per time index.
6. Failure of the Coding Scheme for Three States
7. Achieving Shannon Capacity with More than Two States
8. Concluding Remarks
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Proof of Lemma 1
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|Block #||Initial State||Transcript|
|vertical block #||1||2||⋯||m|
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