Synthesizing multiple boolean functions using interpolation on a single proof

Abstract

Abstract-It is often difficult to correctly implement a Boolean controller for a complex system, especially when concurrency is involved. Yet, it may be easy to formally specify a controller. For instance, for a pipelined processor it suffices to state that the visible behavior of the pipelined system should be identical to a nonpipelined reference system (Burch-Dill paradigm). We present a novel procedure to efficiently synthesize multiple Boolean control signals from a specification given as a quantified first-order formula (with a specific quantifier structure). Our approach uses uninterpreted functions to abstract details of the design. We construct an unsatisfiable SMT formula from the given specification. Then, from just one proof of unsatisfiability, we use a variant of Craig interpolation to compute multiple coordinated interpolants that implement the Boolean control signals. Our method avoids iterative learning and back-substitution of the control functions. We applied our approach to synthesize a controller for a simple two-stage pipelined processor, and present first experimental results.

Interpolation-Based Semantic Gate Extraction and Its Applications to QBF Preprocessing

“…Hofferek et al use interpolation to synthesize multiple functional implementations from a single proof and thus avoid the increase in formula size incurred by repeated substitution [24]. This has an analogue in strategy extraction for QBF, which allows for implementations of all (existential or universal) variables to be obtained from a proof [3].…”

Program Synthesis for Program Analysis

“…Algorithmic approaches to the problem have frequently been connected to automated theorem proving [61,67]. Recent developments include an application of Craig interpolation to synthesis by [55].…”

Greedy pebbling for proof space compression

“…The parsimony of the resolution calculus also makes checking a resolution proof conceptually simpler and implementable in fewer lines of trusted code. The extra information contained in resolution proofs is essential for further manipulation and compression of the proof [3,13,21,29,57] and for applications that rely, for instance, on interpolants extracted from proofs [23,38,51]. And finally, although alternative detailed proof systems for conflict-driven clause learning, mixing resolution and natural deduction have recently been proposed [60], resolution remains the primary format chosen by major SMT-solvers [4,5,14] and conversion from DRUP to resolution is possible for SAT-solvers that are not able to output resolution proofs directly [37].…”

“…In an ongoing project for interpolant-based controller synthesis [38], for example, extracting an interpolant from an SMT-proof took hours and reached the limit of memory (256 GB) available in a single node of the computer cluster used in the project. The example shows one serious issue with proofs: They tend to be huge, easily filling up all available memory, and therefore are hard to process independently.…”

“…An example of an application where certified correctness is crucial is the verification and synthesis of software [10,38]. In an ongoing project for interpolant-based controller synthesis [38], for example, extracting an interpolant from an SMT-proof took hours and reached the limit of memory (256 GB) available in a single node of the computer cluster used in the project.…”