Formalization, Mechanization and Automation of G\"odel's Proof of God's Existence

G\"odel's ontological proof has been analysed for the first-time with an unprecedent degree of detail and formality with the help of higher-order theorem provers. The following has been done (and in this order): A detailed natural deduction proof. A formalization of the axioms, definitions and theorems in the TPTP THF syntax. Automatic verification of the consistency of the axioms and definitions with Nitpick. Automatic demonstration of the theorems with the provers LEO-II and Satallax. A step-by-step formalization using the Coq proof assistant. A formalization using the Isabelle proof assistant, where the theorems (and some additional lemmata) have been automated with Sledgehammer and Metis.Comment: 2 page

Atomic Cut Introduction by Resolution: Proof Structuring and Compression

The original publication is available at www.springerlink.comInternational audienceThe careful introduction of cut inferences can be used to structure and possibly compress formal sequent calculus proofs. This pa- per presents CIRes, an algorithm for the introduction of atomic cuts based on various modifications and improvements of the CERes method, which was originally devised for efficient cut-elimination. It is also demonstrated that CIRes is capable of compressing proofs, and the amount of compres- sion is shown to be exponential in the length of proofs

Physics and Proof Theory

Axiomatization of Physics (and Science in general) has many drawbacks that are correctly criticized by opposing philosophical views of Science. This paper shows that, by giving formal proofs a more promi- nent role in the formalization, many of the drawbacks can be solved and many of the opposing views are naturally conciliated. Moreover, this ap- proach allows, by means of Proof Theory, to open new conceptual bridges between the disciplines of Physics and Computer Science

Exploring and Exploiting Algebraic and Graphical Properties of Resolution

International audienceIntegrating an SMT solver in a certified environment such as an LF-style proof assistant requires the solver to output proofs. Unfortunately, those proofs may be quite large, and the overhead of rechecking the proof may account for a significant fraction of the proof time. In this paper we explore techniques for reducing the sizes of propositional proofs, which are at the core of SMT proofs. Our techniques are justified in an algebra of resolution and rely on a graph-theoretical representation of proofs that allows us to detect the potential for reordering and combining resolution inferences

NP-completeness of small conflict set generation for congruence closure

International audienceThe efficiency of Satisfiability Modulo Theories (SMT) solvers is dependent on the capability of theory reasoners to provide small conflict sets, i.e. small unsatisfiable subsets from unsatisfiable sets of literals. Decision procedures for uninterpreted symbols (i.e. congruence closure algorithms) date back from the very early days of SMT. Nevertheless, to the best of our knowledge , the complexity of generating smallest conflict sets for sets of literals with uninterpreted symbols and equalities had not yet been determined, although the corresponding decision problem was believed to be NP-complete. We provide here an NP-completeness proof, using a simple reduction from SAT

Tinker, Tailor, Solver, Proof

We introduce Tinker, a tool for designing and evaluating proof strategies based on proof-strategy graphs, a formalism previously introduced by the authors. We represent proof strategies as open-graphs, which are directed graphs with additional input/output edges. Tactics appear as nodes in a graph, and can be `piped' together by adding edges between them. Goals are added to the input edges of such a graph, and flow through the graph as the strategy is evaluated. Properties of the edges ensure that only the right `type' of goals are accepted. In this paper, we detail the Tinker tool and show how it can be integrated with two different theorem provers: Isabelle and ProofPower.Comment: In Proceedings UITP 2014, arXiv:1410.785