Certifying circuits in Type Theory

Abstract

We investigate how to take advantage of the particular features of the calculus of inductive constructions in the framework of hardware verification. First, we emphasize in a short case study the use of dependent types and of the constructive aspect of the logic for specifying and synthesizing combinatorial circuits. Then, co-inductive types are introduced to model the temporal aspects of sequential synchronous devices. Moore and Mealy automata are co-inductively axiomatized and are used to represent uniformly both the structures and the behaviors of the circuits. This leads to clear, general and elegant proof processes as is illustrated on the example of a realistic circuit: the ATM Switch Fabric. All the proofs are carried out using Coq.

“Backward” coinduction, Nash equilibrium and the rationality of escalation

“…If one of them gives up, the other receives the object and both pay the amount of their bid. 11 For instance, if agent Alice stops immediately, she pays nothing and agent Bob, who acquires the object, has a payoff v. In the general turn of the auction, if Alice abandons, she looses the auction and has a payoff −n and Bob who has already bid −n has a payoff v − n. At the next turn after Alice decides to continue, bids $1 for this and acquires the object due to Bob stopping, Alice has a payoff v − (n + 1) and Bob has a payoff −n. In our formalization we have considered the dollar auction up to infinity.…”

“…Alice → −(v + n), Bob → −n 11 In a variant, each bidder, when he bids, puts a dollar bill in a hat or in a piggy bank and their is no return at the end of the auction. The last bidder gets the object.…”

“…Another strongly connected work is the one of Coupet-Grimal [10] on temporal logic. Other applications are on representation of real numbers by infinite sequences [3,18] and implementation of streams (infinite lists) in electronic circuits [11]. An ancestor of our description of infinite games and infinite strategy profiles is the constructive description of finite games, finite strategy profiles, and equilibria by Vestergaard [32].…”

“…Interested readers may have a look at Coupet-Grimal [10], Coupet-Grimal and Jakubiec [11], Lescanne [20], Bertot [2,3] and especially Bertot and Castéran [4, chap. 13] for other examples of cofix reasoning.…”

Formally certified stable marriages

“…As an example of the former, the correctness of three sorting algorithms was formalized in [6]. The Coq proof assistant has also been used for the verification of hardware circuits [4] and a square root algorithm [2]. Other developments involve more complex data structures, such as a union-find implementation [3], memory allocation library [19], and even verification of semantics preservation for a complete compiler, CompCert [13].…”