Alexander Bentkamp

Numerical and symbolic methods for optimization are used extensively in engineering, industry, and finance. Various methods are used to reduce problems of interest to ones that are amenable to solution by these methods. We develop a framework for designing and applying such reductions, using the Lean programming language and interactive proof assistant. Formal verification makes the process more reliable, and the availability of an interactive framework and ambient mathematical library provides a robust environment for constructing the reductions and reasoning about them.

Numerical and symbolic methods for optimization are used extensively in engineering, industry, and finance. Various methods are used to reduce problems of interest to ones that are amenable to solution by such software. We develop a framework for designing and applying such reductions, using the Lean programming language and interactive proof assistant. Formal verification makes the process more reliable, and the availability of an interactive framework and ambient mathematical library provides a robust environment for constructing the reductions and reasoning about them.

In 1982, Raymond Smullyan published an article, “The Asylum of Doctor Tarr and Professor Fether,” that consists of a series of puzzles. These were later reprinted in the anthology The Lady or The Tiger? and Other Logic Puzzles. The last puzzle, which describes the asylum alluded to in the title, was designed to be especially difficult. With the help of automated reasoning, we show that the puzzle’s hypotheses are, in fact, inconsistent, which is to say, no such asylum can possibly exist.

We present a tool for verification of hybrid systems expressed in the sequential fragment of HCSP (Hybrid Communicating Sequential Processes). The tool permits annotating HCSP programs with pre- and postconditions, invariants, and proof rules for reasoning about ordinary differential equations. Verification conditions are generated from the annotations following the rules of hybrid Hoare logic. We designed labeling and highlighting mechanisms to distinguish and visualize different verification conditions. The tool is implemented in Python and has a web-based user interface. We evaluated the effectiveness of the tool on translations of Simulink/Stateflow models and on KeYmaera X benchmarks.

We developed a procedure to enumerate complete sets of higher-order unifiers based on work by Jensen and Pietrzykowski. Our procedure removes many redundant unifiers by carefully restricting the search space and tightly integrating decision procedures for fragments that admit a finite complete set of unifiers. We identify a new such fragment and describe a procedure for computing its unifiers. Our unification procedure, together with new higher-order term indexing data structures, is implemented in the Zipperposition theorem prover. Experimental evaluation shows a clear advantage over Jensen and Pietrzykowski's procedure.

Superposition is among the most successful calculi for first-order logic. Its extension to higher-order logic introduces new challenges such as infinitely branching inference rules, new possibilities such as reasoning about formulas, and the need to curb the explosion of specific higher-order rules. We describe techniques that address these issues and extensively evaluate their implementation in the Zipperposition theorem prover. Largely thanks to their use, Zipperposition won the higher-order division of the CASC-J10 competition.