Solving the Unsolvable Using an Oracle Eli Gafni UCLA [email protected] ABSTRACT Oracle accepts it. If not, it tells the system that Sol1 is not valid so- In Distributed Computing Theory the relation between a Model lution of T . Based on this, the system deliberates again and propose and a Task is Binary: Either the Model solves the Task, or it does another solution Sol2, etc. not. But if it does not, there is no measure of how “close the Model But the system is distributed. What do we mean the system is to solve the Task.” proposes a solution? We propose such a directed closeness measure: Processors can Processors interface with the Oracle. First by announcing to the coordinate among themselves by the power they possess and each Oracle their participation. Then they coordinate through the given proposes an output to an Oracle. The Oracle, based on the set of synchronization power and consequently, a processor proposes a processors participating, and outputs it already approved so far, pair ¹processor; outputº by sending it to the Oracle. The Oracle disapproves or approves the next proposal it examines. If disap- judges whether a proposal can live with the processors’ output it proved, a processor returns to consult its peers to come up with a approved (by answering “yes”) already and the current participating next proposal. set. If the up-to now approved outputs, together with the new We take the number of Oracle queries as a measure to how proposal can be completed to a full valid output, it responds to the close the Model is to the Task. The less the number of queries the processor with a “yes.”Else the Oracle responds with a “no,”in which closer the model is to solve the task, culminating with a traditional case the processor returns to coordinate with other processors to solvability that incurs zero cost. come up with a new output proposal. The complexity is how many Oracle queries in the worst case CCS CONCEPTS a processor may go through. Notice, that a processor also may decide an output without approval from the Oracle. For instance, • Theory of computation ! Distributed computing models. this is the case for implementation. Nevertheless, we will see mixed KEYWORDS example: Queries are required for getting a solution, nevertheless, some processors may output without requiring approval from the message adversary, synchronization, model of distributed comput- Oracle, as they are certain their next proposal will necessarily be ing compatible with the other outputs. ACM Reference Format: Thus, the closeness is min-max: For each algorithm assign the Eli Gafni. 2020. Solving the Unsolvable Using an Oracle. In Proceedings of maximum number of queries a processor made, and then take min- 22nd International Conference on Distributed Computing and Networking imum over the algorithms. (ICDCN 2021). ACM, New York, NY, USA, 5 pages. https://doi.org/10.1145/ We consider three variations on the details of the Oracle interac- nnnnnnn.nnnnnnn tion with the system, and conjecture they are all equivalent as far 1 INTRODUCTION as the complexity of Oracle-implementing a given task in a given model. When considering a one-shot distributed problemT called task [12], The three variations are as follows: and a system S, we relate them by a binary notion. The task T is (1) FfA (Free for All) The pair ¹p ;valueº for any specific p either solvable by the system S, or not. But if S does not solve T , j j and any specific value can be submitted many times by dif- the more coordination power it has, makes an intuitive sense that ferent processors, by that learning whether the proposal first the closer it may be to solving T . submitted was approved or not. We propose to quantify this closeness. When we can model a (2) Private The pair ¹p ;valueº for any specific p and any spe- system as an iterated task [11], this measure induces a directed j j cific value can be submitted at most a single time. distance between tasks. (3) Random When a proposal can obtain a “yes” value, but The general idea to do something impossible is to call on extra- nevertheless with all “yes” completions, it is also compatible ordinary powers i.e. an Oracle. The system deliberates and then with a “no” answer, then the Oracle can choose randomly propose to the Oracle a solution Sol1, for T . If Sol1 is a solution, the between the two. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed There are two interesting potential counter-example to the conjec- for profit or commercial advantage and that copies bear this notice and the full citation ture that the complexity of a pair Model/Task is invariant over the on the first page. Copyrights for components of this work owned by others than ACM three Oracles. must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a In both cases we refute the counter example and show that the fee. Request permissions from [email protected]. complexity coincide. Technically, it was challenging and constitutes ICDCN 2021, Jan 05-08, 2021, Nara, Japan the main technical contribution of the paper. © 2020 Association for Computing Machinery. ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 A potential counter-example to separate FfA and Private is the https://doi.org/10.1145/nnnnnnn.nnnnnnn pair read-write wait-free (rw-wf)/Total-Order (TO). ICDCN 2021, Jan 05-08, 2021, Nara, Japan Eli Gafni Total-Order is repeated consensus processor goes though offer- The algorithm is then straight forward: Processors return IS. ing its id to the consensus (election). It departs with the sequence Processor pi that returns ISi will propose to the Oracle it will of ids once its id was accepted as a consensus value. output the integer jISi j, where jISi j is the size of the snapshot ISi . We cannot automatically derive Total-Order from Election as It will either return from the Oracle with “yes” in which case it the communication with the Oracle is pretty restricted. It becomes returns jISi j. Else, if it returns with a “no” from the Oracle, it outputs easy to emulate sequence of election as a result of the flexibility jISi j − 1 (no need to query the Oracle since “yes” is guaranteed.) of offering outputs for others. And indeed the complexity ofrw- The paper is organized as follows: wf/Election is O(n), in both FfA and Private is O¹nº, it is easy to In the next section, the Model section, we just introduce commonly see it is O¹n2º in FfA. known notions making the “language” self contained. We then It is non-trivial to get O¹n2º for Private, but eventually we found pick up 3 hierarchies of task: The test-and-set hierarchy, the set an algorithm. In hindsight, it is pretty simple, consensus hierarchy, and the total order hierarchy. In each we The pair that could potentially separate Private and Random analyse the complexity of going up a step up the ladder, as well is the pair of read-write wait-free model with access to k + 1- Fetch- as jumping directly from the bottom (rw-wf) to the top. We then and-Add (k + 1-F&A) and the task k-F&A. Notice that for instance conclude. the model above is equivalent to repeated k + 1-F&A a la [11]. The complexity in Private is O¹1º. The algorithm relies heavily 2 MODEL on the determinism of the Oracle. To get O¹1º for Random is not A task is the analogue of a function in distributed computing. At a small feat even after it was achieved. its fundamental theoretical basis it is what distributed computing To make the presentation less abstract, an example of the Oracle does. Processors do not compute, processors coordinate. in action will be instructive. A task, like its analogue the function, is a mathematical object. Given a set P, an input domain I of values, and an output domain O, it is a binary relation between a subset of processors P ⊆ P each 1.1 Motivating Example with input from I, to the same subset of processors, each with an output from O. Consider the task of TightRenaming [2]. Thus, the task of consensus between 2 processors p and p , and The specification of this inputless task, is that for participating 0 1 input and output domain f0; 1g, is the relation: set P of cardinality jP j each processor returns a unique positive integer in the range 1; 2;:::; jP j. (1) i 2 f0; 1g;m 2 f0; 1g; ¹pi ;mº ! ¹pi ;mº 8 It is easy to see that the task can implement 2-processors consen- (2) m 2 f0; 1g; ¹¹p0;mº; ¹¹p1;mºº ! ¹¹p0;mº; ¹¹p1;mºº 8 sus, and is implementable in a read-write system S that has access (3) m 2 f0; 1g; ¹¹p0;mº; ¹¹p1;m¯ºº ! ¹¹p0;mº; ¹¹p1;mºº; 8 to 2-processors consensus. ¹¹p0;m¯º; ¹¹p1;m¯ºº Now weaken the system that implements TightRenaming, and The task of F&A is as follows: P ⊆ P, Pi 2 P returns a subset consider the system S to have access to 3-processor set-consensus, 8 Si ⊆ P, such that: rather than access to 2-processors consensus.