Martin Suda

A Unifying Principle for Clause Elimination in First-Order Logic

Preprocessing techniques for formulas in conjunctive normal form play an important role in first-order theorem proving. To speed up the proving process, these techniques simplify a formula without affecting its satisfiability or unsatisfiability. In this paper, we introduce the principle of implication modulo resolution, which allows us to lift several preprocessing techniques—in particular, several clause-elimination techniques—from the SAT-solving world to first-order logic. We analyze confluence properties of these new techniques and show how implication modulo resolution yields short soundness proofs for the existing first-order techniques of predicate elimination and blocked-clause elimination.

Testing a Saturation-Based Theorem Prover: Experiences and Challenges

This paper attempts to address the question of how best to assure the correctness of saturation-based automated theorem provers using our experience developing the theorem prover Vampire. We describe the techniques we currently employ to ensure that Vampire is correct and use this to motivate future challenges that need to be addressed to make this process more straightforward and to achieve better correctness guarantees.

Unification with Abstraction and Theory Instantiation in Saturation-Based Reasoning

We make a new contribution to the field by providing a new method of using SMT solvers in saturation-based reasoning. We do this by introducing two new inference rules for reasoning with non-ground clauses. The first rule utilises theory constraint solving (an SMT solver) to perform reasoning within a clause to find an instance where we can remove one or more theory literals. This utilises the power of SMT solvers for theory reasoning with non-ground clauses, reasoning which is currently achieved by the addition of often prolific theory axioms. The second rule is unification with abstraction where the notion of unification is extended to introduce constraints where theory terms may not otherwise unify. This abstraction is performed lazily, as needed, to allow the superposition theorem prover to make as much progress as possible without the search space growing too quickly. Additionally, the first rule can be used to discharge the constraints introduced by the second. These rules were implemented within the Vampire theorem prover and experimental results show that they are useful for solving a considerable number of previously unsolved problems. The current implementation focuses on complete theories, in particular various versions of arithmetic.

Local proofs and AVATAR

With first-order interpolation as the application in mind, we study the problem of generating local proofs in theorem provers employing the AVATAR architecture. The theory is complemented by experimental results based on our implementation of the techniques in theorem prover Vampire.

Reinterpreting Dependency Schemes: Soundness Meets Incompleteness in DQBF

Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes have been treated separately in the literature, even though both treatments extend QBF by replacing the linear order of the quantifier prefix with a partial order. We propose to merge the two, by reinterpreting a dependency scheme as a mapping from QBF into DQBF. Our approach offers a fresh insight on the nature of soundness in proof systems for QBF with dependency schemes, in which a natural property called ‘full exhibition’ is central. We apply our approach to QBF proof systems from two distinct paradigms, termed ‘universal reduction’ and ‘universal expansion’. We show that full exhibition is sufficient (but not necessary) for soundness in universal reduction systems for QBF with dependency schemes, whereas for expansion systems the same property characterises soundness exactly. We prove our results by investigating DQBF proof systems, and then employing our reinterpretation of dependency schemes. Finally, we show that the reflexive resolution path dependency scheme is fully exhibited, thereby proving a conjecture of Slivovsky.

Splitting proofs for interpolation

We study interpolant extraction from local first-order refutations. We present a new theoretical perspective on interpolation based on clearly separating the condition on logical strength of the formula from the requirement on the common signature. This allows us to highlight the space of all interpolants that can be extracted from a refutation as a space of simple choices on how to split the refutation into two parts. We use this new insight to develop an algorithm for extracting interpolants which are linear in the size of the input refutation and can be further optimized using metrics such as number of non-logical symbols or quantifiers. We implemented the new algorithm in first-order theorem prover Vampire and evaluated it on a large number of examples coming from the first-order proving community. Our experiments give practical evidence that our work improves the state-of-the-art in first-order interpolation.

Simulation of the human arterial system-static and dynamic

This paper introduces and compares two new models for the blood flow in the human arterial system. The first one is a hydrostatic model for the average flow and pressure values within the network. It is capable of being identified individually for each patient with a minimum amount of work. The second one is a hydrodynamic model for the pulse wave propagation within the arterial network. Due to the complexity of this one, it is a difficult task to obtain individual patient data, but it is designed for principal investigations of pathological pulse forms.