Bruno Woltzenlogel Paleo

Automating Gödel's ontological proof of God's existence with higher-order automated theorem provers

Kurt Godels ontological argument for Gods existence has been formalized and automated on a computer with higher-order automated theorem provers. From Godels premises, the computer proved: necessarily, there exists God. On the other hand, the theorem provers have also confirmed prominent criticism on Godels ontological argument, and they found some new results about it. The background theory of the work presented here offers a novel perspective towards a computational theoretical philosophy.Attempts to prove the existence (or non-existence) of God by means of abstract ontological arguments are an old tradition in philosophy and theology. Gödel’s proof [12,13] is a modern culmination of this tradition, following particularly the footsteps of Leibniz. Gödel defines God as a being who possesses all positive properties. He does not extensively discuss what positive properties are, but instead he states a few reasonable (but debatable) axioms that they should satisfy. Various slightly different versions of axioms and definitions have been considered by Gödel and by several philosophers who commented on his proof (cf. [19,2,11,1,10]). Dana Scott’s version of Gödel’s proof [18] employs the following axioms (A), definitions (D), corollaries (C) and theorems (T), and it proceeds in the following order:

Herbrand Sequent Extraction

Computer generated proofs of interesting mathematical theorems are usually too large and full of trivial structural information, and hence hard to understand for humans. Techniques to extract specific essential information from these proofs are needed. In this paper we describe an algorithm to extract Herbrand sequents from proofs written in Gentzens sequent calculus LK for classical first-order logic. The extracted Herbrand sequent summarizes the creative information of the formal proof, which lies in the instantiations chosen for the quantifiers, and can be used to facilitate its analysis by humans. Furthermore, we also demonstrate the usage of the algorithm in the analysis of a proof of the equivalence of two different definitions for the mathematical concept of lattice, obtained with the proof transformation system CERES .

Compression of propositional resolution proofs via partial regularization

This paper describes two algorithms for the compression of propositional resolution proofs. The first algorithm, RecyclePivots-WithIntersection, performs partial regularization, removing an inference η when it is redundant in the sense that its pivot literal already occurs as the pivot of another inference located below in the path from η to the root of the proof. The second algorithm, LowerUnits, delays the resolution of (both input and derived) unit clauses, thus removing (some) inferences having the same pivot but possibly occurring also in different branches of the proof.

Exploiting symmetry in SMT problems

Methods exploiting problem symmetries have been very successful in several areas including constraint programming and SAT solving. We here recast a technique to enhance the performance of SMTsolvers by detecting symmetries in the input formulas and use them to prune the search space of the SMT algorithm. This technique is based on the concept of (syntactic) invariance by permutation of constants. An algorithm for solving SMT by taking advantage of such symmetries is presented. The implementation of this algorithm in the SMT-solver veriT is used to illustrate the practical benefits of this approach. It results in a significant improvement of veriTs performances on the SMTLIB benchmarks that places it ahead of the winners of the last editions of the SMT-COMP contest in the QF-UF category.

Atomic cut introduction by resolution: proof structuring and compression

The careful introduction of cut inferences can be used to structure and possibly compress formal sequent calculus proofs. This paper 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 compression is shown to be exponential in the length of proofs.

Compression of Propositional Resolution Proofs by Lowering Subproofs

This paper describes a generalization of the LowerUnits algorithm for the compression of propositional resolution proofs. The generalized algorithm, here called LowerUnivalents, is able to lower not only units but also subproofs of non-unit clauses, provided that they satisfy some additional conditions. This new algorithm is particularly suited to be combined with the RecyclePivotsWithIntersection algorithm. A formal proof that LowerUnivalents always compresses more than LowerUnits is shown, and both algorithms are empirically compared on thousands of proofs produced by the SMT-Solver veriT.

Interacting with Modal Logics in the Coq Proof Assistant

This paper describes an embedding of higher-order modal logics in the Coq proof assistant. Coq’s capabilities are used to implement modal logics in a minimalistic manner, which is nevertheless sufficient for the formalization of significant, non-trivial modal logic proofs. The elegance, flexibility and convenience of this approach, from a user perspective, are illustrated here with the successful formalization of Godel’s ontological argument.

Higher-Order Modal Logics: Automation and Applications

These are the lecture notes of a tutorial on higher-order modal logics held at the 11th Reasoning Web Summer School. After defining the syntax and (possible worlds) semantics of some higher-order modal logics, we show that they can be embedded into classical higher-order logic by systematically lifting the types of propositions, making them depend on a new atomic type for possible worlds. This approach allows several well-established automated and interactive reasoning tools for classical higher-order logic to be applied also to modal higher-order logic problems. Moreover, also meta reasoning about the embedded modal logics becomes possible. Finally, we illustrate how our approach can be useful for reasoning with web logics and expressive ontologies, and we also sketch a possible solution for handling inconsistent data.

PROOFTOOL: a GUI for the GAPT Framework

This paper introduces PROOFTOOL, the graphical user interface for the General Architecture for Proof Theory (GAPT) framework. Its features are described with a focus not only on the visualization but also on the analysis and transformation of proofs and related tree-like structures, and its implementation is explained. Finally, PROOFTOOL is compared with three other graphical interfaces for proofs.This paper introduces PROOFTOOL, the graphical user interface for the General Architecture for Proof Theory (GAPT) framework. Its features are described with a focus not only on the visualization but also on the analysis and transformation of proofs and related tree-like structures, and its implementation is explained. Finally, PROOFTOOL is compared with three other graphical interfaces for proofs.

Skeptik: A Proof Compression System

This paper introduces Skeptik: a system for checking, compressing and improving proofs obtained by SAT- and SMT-solvers.