scholarly journals Topical Neural Theorem Prover that Induces Rules

10.29007/wscr ◽  
2020 ◽  
Author(s):  
Shuang Xia ◽  
Krysia Broda ◽  
Alessandra Russo
Keyword(s):  
Knowledge Base ◽  
Predictive Accuracy ◽  
Theorem Prover ◽  
Computational Time ◽  
Backward Chaining ◽  
First Order ◽  
Computation Tree ◽  
Computation Trees ◽  
Reasoning Mechanism ◽  
The Given

Various sub-symbolic approaches for reasoning and learning have been proposed in the literature. Among these approaches, the neural theorem prover (NTP) approach uses a backward chaining reasoning mechanism to guide a machine learning architecture to learn vector embedding representations of predicates and to induce first-order clauses from a given knowledge base. NTP is however known for being not scalable, as the computation trees generated by the backward chaining process can grow exponentially with the size of the given knowledge base. In this paper we address this limitation by extending the NTP approach with a topic-based method for controlling the induction of first-order clauses. Our proposed approach, called TNTP for Topical NTP, identifies topic-based clusters over a large knowledge-base and uses these clusters to control the soft unification of predicates during the learning process with the effect of reducing the size of the computation tree needed to induce first-order clauses. Our TNTP framework is capable of learning a diverse set of induced rules that have improved predictive accuracy, whilst reducing the computational time by several orders of magnitude. We demonstrated this by evaluating our approach on three different datasets (UMLS, Kinship and Nations) and comparing our results with that of the NTP method, chosen here as our baseline.

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%.

On two pursuit differential game problems with state and geometric constraints in a Hilbert space

2021 ◽  
Vol 65 (3) ◽  
pp. 5-16
Author(s):  
Abbas Ja’afaru Badakaya ◽  
Keyword(s):  
Differential Game ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Nonempty Closed Convex Subset ◽  
Geometric Constraints ◽  
First Order ◽  
Control Functions ◽  
Closed Convex Set ◽  
The Given ◽  
First Order Differential Equations

This paper concerns with the study of two pursuit differential game problems of many pursuers and many evaders on a nonempty closed convex subset of R^n. Throughout the period of the games, players must stay within the given closed convex set. Players’ laws of motion are defined by certain first order differential equations. Control functions of the pursuers and evaders are subject to geometric constraints. Pursuit is said to be completed if the geometric position of each of the evader coincides with that of a pursuer. We proved two theorems each of which is solution to a problem. Sufficient conditions for the completion of pursuit are provided in each of the theorems. Moreover, we constructed strategies of the pursuers that ensure completion of pursuit.

Big Data Mining Based on Computational Intelligence and Fuzzy Clustering

2015 ◽  
pp. 130-148 ◽  
Author(s):  
Usman Akhtar ◽  
Mehdi Hassan
Keyword(s):  
Big Data ◽  
Computational Intelligence ◽  
Clustering Algorithms ◽  
Heterogeneous Data ◽  
Computational Time ◽  
Statistical Features ◽  
Feature Sets ◽  
Clustering Quality ◽  
The Given ◽  
Different Sources

The availability of a huge amount of heterogeneous data from different sources to the Internet has been termed as the problem of Big Data. Clustering is widely used as a knowledge discovery tool that separate the data into manageable parts. There is a need of clustering algorithms that scale on big databases. In this chapter we have explored various schemes that have been used to tackle the big databases. Statistical features have been extracted and most important and relevant features have been extracted from the given dataset. Reduce and irrelevant features have been eliminated and most important features have been selected by genetic algorithms (GA).Clustering with reduced feature sets requires lower computational time and resources. Experiments have been performed at standard datasets and results indicate that the proposed scheme based clustering offers high clustering accuracy. To check the clustering quality various quality measures have been computed and it has been observed that the proposed methodology results improved significantly. It has been observed that the proposed technique offers high quality clustering.

Big Data Mining Based on Computational Intelligence and Fuzzy Clustering

Web Services ◽  
2019 ◽  
pp. 413-430
Author(s):  
Usman Akhtar ◽  
Mehdi Hassan
Keyword(s):  
Big Data ◽  
Computational Intelligence ◽  
Clustering Algorithms ◽  
Heterogeneous Data ◽  
Computational Time ◽  
Statistical Features ◽  
Feature Sets ◽  
Clustering Quality ◽  
The Given ◽  
Different Sources

The availability of a huge amount of heterogeneous data from different sources to the Internet has been termed as the problem of Big Data. Clustering is widely used as a knowledge discovery tool that separate the data into manageable parts. There is a need of clustering algorithms that scale on big databases. In this chapter we have explored various schemes that have been used to tackle the big databases. Statistical features have been extracted and most important and relevant features have been extracted from the given dataset. Reduce and irrelevant features have been eliminated and most important features have been selected by genetic algorithms (GA). Clustering with reduced feature sets requires lower computational time and resources. Experiments have been performed at standard datasets and results indicate that the proposed scheme based clustering offers high clustering accuracy. To check the clustering quality various quality measures have been computed and it has been observed that the proposed methodology results improved significantly. It has been observed that the proposed technique offers high quality clustering.

Two variable first-order logic over ordered domains

2001 ◽  
Vol 66 (2) ◽  
pp. 685-702 ◽  
Author(s):  
Martin Otto
Keyword(s):  
Order Logic ◽  
Decision Problems ◽  
First Order Logic ◽  
Satisfiability Problem ◽  
Exponential Time ◽  
Infinite Domains ◽  
First Order ◽  
Computation Tree ◽  
Common Extension ◽  
Undecidability Result

AbstractThe satisfiability problem for the two-variable fragment of first-order logic is investigated over finite and infinite linearly ordered, respectively wellordered domains, as well as over finite and infinite domains in which one or several designated binary predicates are interpreted as arbitrary wellfounded relations.It is shown that FO2 over ordered, respectively wellordered. domains or in the presence of one well-founded relation, is decidable for satisfiability as well as for finite satisfiability. Actually the complexity of these decision problems is essentially the same as for plain unconstrained FO2. namely non-deterministic exponential time.In contrast FO2 becomes undecidable for satisfiability and for finite satisfiability, if a sufficiently large number of predicates are required to be interpreted as orderings. wellorderings. or as arbitrary wellfounded relations. This undecidability result also entails the undecidability of the natural common extension of FO2 and computation tree logic CTL.

scholarly journals Spectrum of a Differential Operator with Periodic Generalized Potential

2007 ◽  
Vol 2007 ◽  
pp. 1-8
Author(s):  
Mehmet Sahin ◽  
Manaf Dzh. Manafov
Keyword(s):  
Differential Operator ◽  
Infinite Number ◽  
Periodic Potential ◽  
Generalized Functions ◽  
Order Differential Operator ◽  
Nonzero Constant ◽  
Classic Case ◽  
First Order ◽  
Generalized Potential ◽  
The Given

We study some spectral problems for a second-order differential operator with periodic potential. Notice that the given potential is a sum of zero- and first-order generalized functions. It is shown that the spectrum of the investigated operator consists of infinite number of gaps whose length limit unlike the classic case tends to nonzero constant in some place and to infinity in other place.

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