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Logical deduction and abstraction from detail are fundamental, yet distinct aspects of reasoning about programs. This dissertation shows that the combination of logic and abstract interpretation enables a unified and simple treatment of several theoretical and practical topics which encompass the model theory of temporal logics, the analysis of satisfiability solvers, and the construction of Craig interpolants. In each case, the combination of logic and abstract interpretation leads to more general results, simpler proofs, and a unification of ideas from seemingly disparate fields. The first contribution of this dissertation is a framework for combining temporal logics and abstraction. Chapter 3 introduces trace algebras, a new lattice-based semantics for linear and branching time logics. A new representation theorem shows that trace algebras precisely capture the standard trace-based semantics of temporal logics. We prove additional representation theorems to show how structures that have been independently discovered in static program analysis, model checking, and algebraic modal logic, can be derived from trace algebras by abstract interpretation. The second contribution of this dissertation is a framework for proving when two lattice-based algebras satisfy the same logical properties. Chapter 5 introduces functions called subsumption and bisubsumption and shows that these functions characterise logical equivalence of two algebras. We also characterise subsumption and bisubsumption using fixed points and finitary logics. We prove a representation theorem and apply it to derive the transition system analogues of subsumption and bisubsumption. These analogues strictly generalise the well studied notions of simulation and bisimulation. Our fixed point characterisations also provide a technique to construct property preserving abstractions. The third contribution of this dissertation is abstract satisfaction, an abstract interpretation framework for the design and analysis of satisfiability procedures. We show that formula satisfiability has several different fixed point characterisations, and that satisfiability procedures can be understood as abstract interpreters. Our main result is that the propagation routine in modern sat solvers is a greatest fixed point computation involving abstract transformers, and that clause learning is an abstract transformer for a form of negation. The final contribution of this dissertation is an abstract interpretation based analysis of algorithms for constructing Craig interpolants. We identify and analyse a lattice of interpolant constructions. Our main result is that existing algorithms are two of three optimal abstractions of this lattice. A second new result we derive in this framework is that the lattice of interpolation algorithms can be ordered by logical strength, so that there is a strongest and a weakest possible construction. |
