Permissioned or Permissionless blockchain? Which one is PoS more adapted for? Since in proof-of-stake (PoS) there is no mining process, according to functionality of permissioned and permissionless blockchains, which type is PoS more adapted for? 

In the other words, is a PoS-based public permissionless blockchain as secure a PoW-based public permissionless blockchain ? Or Do we need to restrict the permission for the transactions validation to a limited permitted and known validators (i.e. private permissioned blockchain) ?


The term "permissioneless or public blockchain" is referred to as a blockchain platform in which a user does not have any permission for using (sending transaction) or participating in validating transactions process. For example, Bitcoin and Ethereum are public permissionless blockchain, such that they are open to everybody for both sending transaction and participating in validation of transactions (i.e. PoW process). 
On the other hands, "permissioned blockchain" allows the network to appoint a group of participants in the network who are given the authority to provide the validation of blocks and transactions, meaning that there is an an access control layer to allow certain actions to be performed only by certain identifiable participants. Since access to this type of blockchains is not open to everybody, unlike Bitcoin or Ethereum, they are usually called also private blockchains. For example, Ripple and Hyperledger Fabric are permissioned blockchain.


If you need additional explanation for more clarification, kindly let me know to extend the question.
Frankly, I don't think your definitions are useful. But by your definitions, PoS makes a network permissioned by definition. In a proof-of-stake network, you can only participate in the validation of transactions if you possess the token, which you can only get from someone else who has the token and chooses to give it you. By your definitions, that makes the system permissioned.

Again, I don't think your definitions are useful.

To propose more helpful definitions:

A system is permissionless or public if it permits open participation in both the submission of transactions and the choice of transactions to make forward progress on. There must be no special permissions required to submit and execute transactions beyond the possession of some way to pay transaction fees, and this must (in actual practice) be widely and easily available to anyone who makes a reasonable effort to acquire it. The method in which the ledger makes forward progress must justify a reasonable expectation that hard censorship will not occur -- that is, anyone who submits validly signed transactions to the network should be able to reasonably expect them to execute without having to worry that some particular clique or group can decide to disallow their transactions specifically.

A system is permissioned or private if it does not permit open participation in either the submission of transactions or the choice of transactions to make forward progress on. That is, submitting a transaction requires some permission beyond mere possession of a widely-available, anti-spam asset or participants cannot reasonably expect the system to resist censorship, that is, not all participants have reasonable assurance that their transactions won't be discriminated against in a way that significantly affects their ability to use the system and get its benefits.