scholarly journals Learning from Łukasiewicz and Meredith: Investigations into Proof Structures

2021 ◽  
pp. 58-75
Author(s):  
Christoph Wernhard ◽  
Wolfgang Bibel
Keyword(s):  
Automated Deduction ◽  
Wide Spread ◽  
Global Features ◽  
First Order ◽  
Proof Search ◽  
Substantial Progress ◽  
Lemma Generation

AbstractThe material presented in this paper contributes to establishing a basis deemed essential for substantial progress in Automated Deduction. It identifies and studies global features in selected problems and their proofs which offer the potential of guiding proof search in a more direct way. The studied problems are of the wide-spread form of “axiom(s) and rule(s) imply goal(s)”. The features include the well-known concept of lemmas. For their elaboration both human and automated proofs of selected theorems are taken into a close comparative consideration. The study at the same time accounts for a coherent and comprehensive formal reconstruction of historical work by Łukasiewicz, Meredith and others. First experiments resulting from the study indicate novel ways of lemma generation to supplement automated first-order provers of various families, strengthening in particular their ability to find short proofs.

scholarly journals HOL-λσ: an intentional first-order expression of higher-order logic

2001 ◽  
Vol 11 (1) ◽  
pp. 21-45 ◽  
Author(s):  
GILLES DOWEK ◽  
THERESE HARDIN ◽  
CLAUDE KIRCHNER
Keyword(s):  
Order Theory ◽  
Higher Order ◽  
Search Method ◽  
Order Logic ◽  
Automated Deduction ◽  
Higher Order Logic ◽  
First Order ◽  
Proof Search ◽  
First Order Theory ◽  
Explicit Substitutions

We give a first-order presentation of higher-order logic based on explicit substitutions. This presentation is intentionally equivalent to the usual presentation of higher-order logic based on λ-calculus, that is, a proposition can be proved without the extensionality axioms in one theory if and only if it can be in the other. We show that the Extended Narrowing and Resolution first-order proof-search method can be applied to this theory. In this way we get a step-by-step simulation of higher-order resolution. Hence, expressing higher-order logic as a first-order theory and applying a first-order proof search method is a relevant alternative to a direct implementation. In particular, the well-studied improvements of proof search for first-order logic could be reused at no cost for higher-order automated deduction. Moreover, as we stay in a first-order setting, extensions, such as equational higher-order resolution, may be easier to handle.

scholarly journals CSE_E 1.0: An Integrated Automated Theorem Prover for First-Order Logic

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1142
Author(s):  
Feng Cao ◽  
Yang Xu ◽  
Jun Liu ◽  
Shuwei Chen ◽  
Xinran Ning
Keyword(s):  
Order Logic ◽  
Theorem Prover ◽  
First Order Logic ◽  
Inference Mechanism ◽  
First Order ◽  
Proof Search ◽  
Interdisciplinary Field ◽  
Heuristic Strategies ◽  
Automated Theorem Prover ◽  
Hard Problems

First-order logic is an important part of mathematical logic, and automated theorem proving is an interdisciplinary field of mathematics and computer science. The paper presents an automated theorem prover for first-order logic, called C S E _ E 1.0, which is a combination of two provers contradiction separation extension (CSE) and E, where CSE is based on the recently-introduced multi-clause standard contradiction separation (S-CS) calculus for first-order logic and E is the well-known equational theorem prover for first-order logic based on superposition and rewriting. The motivation of the combined prover C S E _ E 1.0 is to (1) evaluate the capability, applicability and generality of C S E _ E , and (2) take advantage of novel multi-clause S-CS dynamic deduction of CSE and mature equality handling of E to solve more and harder problems. In contrast to other improvements of E, C S E _ E 1.0 optimizes E mainly from the inference mechanism aspect. The focus of the present work is given to the description of C S E _ E including its S-CS rule, heuristic strategies, and the S-CS dynamic deduction algorithm for implementation. In terms of combination, in order not to lose the capability of E and use C S E _ E to solve some hard problems which are unsolved by E, C S E _ E 1.0 schedules the running of the two provers in time. It runs plain E first, and if E does not find a proof, it runs plain CSE, then if it does not find a proof, some clauses inferred in the CSE run as lemmas are added to the original clause set and the combined clause set handed back to E for further proof search. C S E _ E 1.0 is evaluated through benchmarks, e.g., CASC-26 (2017) and CASC-J9 (2018) competition problems (FOFdivision). Experimental results show that C S E _ E 1.0 indeed enhances the performance of E to a certain extent.

scholarly journals Deep Network Guided Proof Search

10.29007/8mwc ◽  
2018 ◽  
Author(s):  
Sarah Loos ◽  
Geoffrey Irving ◽  
Christian Szegedy ◽  
Cezary Kaliszyk
Keyword(s):  
Deep Learning ◽  
Program Analysis ◽  
Network Models ◽  
Theorem Prover ◽  
Neural Network Models ◽  
Two Phase ◽  
First Order ◽  
Proof Search ◽  
System Verification ◽  
Theory Reasoning

Deep learning techniques lie at the heart of several significant AI advances in recent years including object recognition and detection, image captioning, machine translation, speech recognition and synthesis, and playing the game of Go.Automated first-order theorem provers can aid in the formalization and verification of mathematical theorems and play a crucial role in program analysis, theory reasoning, security, interpolation, and system verification.Here we suggest deep learning based guidance in the proof search of the theorem prover E. We train and compare several deep neural network models on the traces of existing ATP proofs of Mizar statements and use them to select processed clauses during proof search. We give experimental evidence that with a hybrid, two-phase approach, deep learning based guidance can significantly reduce the average number of proof search steps while increasing the number of theorems proved.Using a few proof guidance strategies that leverage deep neural networks, we have found first-order proofs of 7.36% of the first-order logic translations of the Mizar Mathematical Library theorems that did not previously have ATP generated proofs. This increases the ratio of statements in the corpus with ATP generated proofs from 56% to 59%.

Lifting propositional proof compression algorithms to first-order logic

2020 ◽  
Author(s):  
Jan Gorzny ◽  
Ezequiel Postan ◽  
Bruno Woltzenlogel Paleo
Keyword(s):  
Propositional Logic ◽  
Empirical Evaluation ◽  
Order Logic ◽  
Automated Deduction ◽  
First Order Logic ◽  
Compression Algorithms ◽  
International Conference ◽  
First Order ◽  
Theorem Provers ◽  
Resolution Proofs

Abstract Proofs are a key feature of modern propositional and first-order theorem provers. Proofs generated by such tools serve as explanations for unsatisfiability of statements. However, these explanations are complicated by proofs which are not necessarily as concise as possible. There are a wide variety of compression techniques for propositional resolution proofs but fewer compression techniques for first-order resolution proofs generated by automated theorem provers. This paper describes an approach to compressing first-order logic proofs based on lifting proof compression ideas used in propositional logic to first-order logic. The first approach lifted from propositional logic delays resolution with unit clauses, which are clauses that have a single literal. The second approach is partial regularization, which removes an inference $\eta $ when it is redundant in the sense that its pivot literal already occurs as the pivot of another inference in every path from $\eta $ to the root of the proof. This paper describes the generalization of the algorithms LowerUnits and RecyclePivotsWithIntersection (Fontaine et al.. Compression of propositional resolution proofs via partial regularization. In Automated Deduction—CADE-23—23rd International Conference on Automated Deduction, Wroclaw, Poland, July 31–August 5, 2011, p. 237--251. Springer, 2011) from propositional logic to first-order logic. The generalized algorithms compresses resolution proofs containing resolution and factoring inferences with unification. An empirical evaluation of these approaches is included.

scholarly journals Proof systems for Geometric theories (PROGEO)

2020 ◽  
Author(s):  
Elaine Pimentel
Keyword(s):  
Projective Geometry ◽  
Automated Deduction ◽  
First Order ◽  
Systematic Procedure ◽  
Proof Systems ◽  
Conservative Extensions

We plan to study the problem of finding conservative extensions of first order logics. In this project we intend to establish a systematic procedure for adding geometric theories in both intuitionistic and classical logics, as well as to extend this procedure to bipolar axioms, a generalization of the set of geometric axioms. This way, we obtain proof systems for several mathematical theories, such as lattices, algebra and projective geometry, being able to reason about such theories using automated deduction.

scholarly journals Breeding Theorem Proving Heuristics with Genetic Algorithms

10.29007/gms9 ◽  
2018 ◽  
Author(s):  
Simon Schäfer ◽  
Stephan Schulz
Keyword(s):  
Genetic Algorithms ◽  
Large Scale ◽  
First Order ◽  
Proof Search ◽  
Theorem Provers ◽  
Search Heuristics ◽  
Practical Performance ◽  
Set Up ◽  
The Given ◽  
Selection Of

First-order theorem provers have to search for proofs in an infinitespace of possible derivations. Proof search heuristics play a vitalrole for the practical performance of these systems. In the currentgeneration of saturation-based theorem provers like SPASS, E,Vampire or Prover~9, one of the most important decisions is theselection of the next clause to process with the given clausealgorithms. Provers offer a wide variety of basic clause evaluationfunctions, which can often be parameterized and combined in manydifferent ways. Finding good strategies is usually left to the usersor developers, often backed by large-scale experimentalevaluations. We describe a way to automatize this process usinggenetic algorithms, evaluating a population of different strategieson a test set, and applying mutation and crossover operators to goodstrategies to create the next generation. We describe the design andexperimental set-up, and report on first promising results.

scholarly journals V.M. Glushkov and automated theorem proving in Ukraine: evidence algorithm evidence algorithm and SAD systems

2020 ◽  
Vol 4 ◽  
pp. 3-10
Author(s):  
O.V. Lyaletski ◽  
Keyword(s):  
Theorem Proving ◽  
Automated Theorem Proving ◽  
English Language ◽  
Mathematical Reasoning ◽  
Russian Language ◽  
Automated Deduction ◽  
Current Time ◽  
Proof Search ◽  
Online Mode ◽  
Evidence Algorithm

Fifty years ago, in 1970, Academician V.M. Glushkov published a paper, in which he, along with a discussion of some problems of artificial intelligence, formulated a research program called Evidence Algorithm (EA) describing his vision of the problem of a computer support of human activity in looking for a proof of a particular theorem. V.M. Glushkov proposed to focus attention on the construction of an automated theorem-proving system performing simultaneous investigations in: creating formal natural languages for writing mathematical texts in a form accustomed to a human, constructing a procedure for a proof search based on the evolutionary developing of the machine notion of an evidence of a computer-made proof step, using the knowledge gained by the system during its operation and providing a user with the opportunity to assist to the system in its proof search process. Since the inception of EA, two serious attempts have been made to implement this program. The first led to the emergence in 1978 of a Russian-language automated theorem proving and the second led to the appearance in 2002 of its English-language modification named System for Automated Deduction (SAD). And if the development and trial operation of the first system were discontinued in 1992 after the output from service of the ES-line computers, on which it was realized, the SAD system, being placed on the website “nevidal.org”, is now still available in online mode. That is, at the current time, it is possible to carry out different experiments with the SAD system and to solve various problems that require rigorous mathematical reasoning. This work is devoted to a chronological description of studies on the implementation of the EA program for the entire period of its existence and to the highlighting of peculiarities of both the systems, as well as of their common features and distinguishes. Some possible ways of the further development of the SAD system are given.

scholarly journals On structuring proof search for first order linear logic

2006 ◽  
Vol 360 (1-3) ◽  
pp. 42-76 ◽  
Author(s):  
Paola Bruscoli ◽  
Alessio Guglielmi
Keyword(s):  
Linear Logic ◽  
Order Linear ◽  
First Order ◽  
Proof Search
Sign in / Sign up

Export Citation Format

Share Document