scholarly journals Generalisation of Induction Formulae based on Proving by Symbolic Execution

10.29007/72nn ◽  
2018 ◽  
Author(s):  
Angela Wallenburg
Keyword(s):  
Symbolic Execution ◽  
Theorem Prover ◽  
Constraint Solving ◽  
Powerful Method ◽  
First Order ◽  
Total Correctness ◽  
The Right

Induction is a powerful method that can be used to prove the total correctness of program loops. Unfortunately the induction proving process in an interactive theorem prover is often very cumbersome. In particular it can be difficult to find the right induction formula. We describe a method for generalising induction formulae by analysing a symbolic proof attempt in a semi-interactive first-order theorem prover. Based on the proof attempt we introduce universally quantified variables, meta-variables and sets of constraints on these. The constraints describe the conditions for a successful proof. By the help of examples, we outline some classes of problems and their associated constraint solutions, and possible ways to automate the constraint solving.

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 Global Schauder Estimates for the $$\mathbf {p}$$-Laplace System

2021 ◽  
Author(s):  
D. Breit ◽  
A. Cianchi ◽  
L. Diening ◽  
S. Schwarzacher
Keyword(s):  
Global Regularity ◽  
Regularity Theory ◽  
Divergence Form ◽  
Dirichlet Problems ◽  
Schauder Estimates ◽  
First Order ◽  
Right Hand ◽  
Regularity Results ◽  
The Right ◽  
Linear Setting

AbstractAn optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the p-Laplace equation and system, with the right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder, $$\text {BMO}$$ BMO and $${{\,\mathrm{VMO}\,}}$$ VMO spaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when $$p=2$$ p = 2 , and hence the differential operator is the plain Laplacian. Yet in this classical linear setting, our contribution completes and augments the celebrated Schauder theory in Hölder spaces. A distinctive trait of our results is their sharpness, which is demonstrated by a family of apropos examples.

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%.

DIFFERENTIAL EQUATIONS IN A HYPERCOMPLEX SYSTEM

2020 ◽  
Vol 69 (1) ◽  
pp. 7-11
Author(s):  
A.K. Abirov ◽  
◽  
N.K. Shazhdekeeva ◽  
T.N. Akhmurzina ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Inhomogeneous System ◽  
Homogeneous Solutions ◽  
First Order ◽  
Zero Divisors ◽  
Hypercomplex System ◽  
Independent Variable ◽  
The Right

The article considers the problem of solving an inhomogeneous first-order differential equation with a variable with a constant coefficient in a hypercomplex system. The structure of the solution in different cases of the right-hand side of the differential equation is determined. The structure of solving the equation in the case of the appearance of zero divisors is shown. It turns out that when the component of a hypercomplex function is a polynomial of an independent variable, the differential equation turns into an inhomogeneous system of real variables from n equations and its solution is determined by certain methods of the theory of differential equations. Thus, obtaining analytically homogeneous solutions of inhomogeneous differential equations in a hypercomplex system leads to an increase in the efficiency of modeling processes in various fields of science and technology.

An axiomatization of the logic with the rough quantifier

1991 ◽  
Vol 56 (2) ◽  
pp. 608-617 ◽  
Author(s):  
Michał Krynicki ◽  
Hans-Peter Tuschik
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Order Structure ◽  
Partition Model ◽  
Weak Model ◽  
First Order ◽  
Basic Properties ◽  
Generalized Quantifier ◽  
Order Language ◽  
The Right

We consider the language L(Q), where L is a countable first-order language and Q is an additional generalized quantifier. A weak model for L(Q) is a pair 〈, q〉 where is a first-order structure for L and q is a family of subsets of its universe. In case that q is the set of classes of some equivalence relation the weak model 〈, q〉 is called a partition model. The interpretation of Q in partition models was studied by Szczerba [3], who was inspired by Pawlak's paper [2]. The corresponding set of tautologies in L(Q) is called rough logic. In the following we will give a set of axioms of rough logic and prove its completeness. Rough logic is designed for creating partition models.The partition models are the weak models arising from equivalence relations. For the basic properties of the logic of weak models the reader is referred to Keisler's paper [1]. In a weak model 〈, q〉 the formulas of L(Q) are interpreted as usual with the additional clause for the quantifier Q: 〈, q〉 ⊨ Qx φ(x) iff there is some X ∊ q such that 〈, q〉 ⊨ φ(a) for all a ∊ X.In case X satisfies the right side of the above equivalence we say that X is contained in φ(x) or, equivalently, φ(x) contains X.

scholarly journals SPASS-AR: A First-Order Theorem Prover Based on Approximation-Refinement into the Monadic Shallow Linear Fragment

2020 ◽  
Vol 64 (3) ◽  
pp. 611-640
Author(s):  
Andreas Teucke ◽  
Christoph Weidenbach
Keyword(s):  
Theorem Prover ◽  
First Order ◽  
Linear Fragment
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