Abstract Lagrangian-based duality, traditionally applied in optimization, has recently been generalized to serve as the basis for a unifying framework for primal-dual search algorithms in the context of program verification and automated reasoning. In this paper, we analyze Quantified Satisfiability Modulo Theories (QSMT) algorithms using this framework. Interestingly, our Lagrangian-based analysis reveals that three recently proposed algorithms for quantified linear real arithmetic (LRA) share a common structure, and that their main differences lie in the approach to a certain problem—model-based projection for $$\exists \forall $$ ∃ ∀ -formulas. Moreover, in the course of this Lagrangian-based analysis, we identify an issue with the progress property of one of the algorithms, propose a way to fix the issue, and experimentally demonstrate that the proposed fix improves performance.

We propose a compositional approach to combine and scale automated reasoning in the static analysis of decentralized system security, such as blockchains. Our focus lies in the game-theoretic security analysis of such systems, allowing us to examine economic incentives behind user actions. In this context, it is particularly important to certify that deviating from the intended, honest behavior of the decentralized protocol is not beneficial: as long as users follow the protocol, they cannot be financially harmed, regardless of how others behave. Such an economic analysis of blockchain protocols can be encoded as an automated reasoning problem in the first-order theory of real arithmetic, reducing game-theoretic reasoning to satisfiability modulo theories (SMT). However, analyzing an entire game-theoretic model (called a game) as a single SMT instance does not scale to protocols with millions of interactions. We address this challenge and propose a divide-and-conquer security analysis based on compositional reasoning over games. Our compositional analysis is incremental: we divide games into subgames such that changes to one subgame do not necessitate re-analyzing the entire game, but only the ancestor nodes. Our approach is sound, complete, and effective: combining the security properties of subgames yields security of the entire game. Experimental results show that compositional reasoning discovers intra-game properties and errors while scaling to games with millions of nodes, enabling security analysis of large protocols.