scholarly journals Formalization of Cost and Utility in Microeconomics

Energies ◽  
2020 ◽  
Vol 13 (3) ◽  
pp. 712
Author(s):  
Asad Ahmed ◽  
Osman Hasan ◽  
Falah Awwad ◽  
Nabil Bastaki
Keyword(s):  
Optimal Power Flow ◽  
Theorem Prover ◽  
Polynomial Functions ◽  
Order Condition ◽  
Accurate Analysis ◽  
Higher Order Logic ◽  
First Order ◽  
Mathematical Properties ◽  
Utility Models ◽  
The Cost

Cost and utility modeling of economics agents based on the differential theory is fundamental to the analysis of the microeconomics models. In particular, the first and second-order derivative tests are used to specify the desired properties of the cost and utility models. Traditionally, paper-and-pencil proof methods and computer-based tools are used to investigate the mathematical properties of these models. However, these techniques do not provide an accurate analysis due to their inability to exhaustively specify and verify the mathematical properties of the cost and utility models. Additionally, these techniques cannot accurately model and analyze pure continuous behaviors of the economic agents due to the utilization of computer arithmetic. On the other hand, an accurate analysis is direly needed in many safety and cost-critical microeconomics applications, such as agriculture and smart grids. To overcome the issues pertaining to the above-mentioned techniques, in this paper, we propose a theorem proving based methodology to formally analyze and specify the mathematical properties of functions used in microeconomics modeling. The proposed methodology is primarily based on a formalization of the derivative tests and root analysis of the polynomial functions, within the sound core of the HOL-Light theorem prover. We also provide a formalization of the first-order condition, which is used to analyze the maximum of the profit function in a higher-order-logic theorem prover. We then present the formal analysis of the utility, cost and first-order condition based on the polynomial functions. To illustrate the usefulness of proposed formalization, the proposed formalization is used to formally analyze and verify the quadratic cost and utility functions, which have been used in an optimal power flow problem and demand response (DR) program, respectively.

scholarly journals Making Higher-Order Superposition Work

2021 ◽  
pp. 415-432
Author(s):  
Petar Vukmirović ◽  
Alexander Bentkamp ◽  
Jasmin Blanchette ◽  
Simon Cruanes ◽  
Visa Nummelin ◽  
...  
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Theorem Prover ◽  
First Order Logic ◽  
Inference Rules ◽  
Higher Order Logic ◽  
First Order ◽  
New Challenges

AbstractSuperposition is among the most successful calculi for first-order logic. Its extension to higher-order logic introduces new challenges such as infinitely branching inference rules, new possibilities such as reasoning about formulas, and the need to curb the explosion of specific higher-order rules. We describe techniques that address these issues and extensively evaluate their implementation in the Zipperposition theorem prover. Largely thanks to their use, Zipperposition won the higher-order division of the CASC-J10 competition.

scholarly journals Integer Induction in Saturation

2021 ◽  
pp. 361-377
Author(s):  
Petra Hozzová ◽  
Laura Kovács ◽  
Andrei Voronkov
Keyword(s):  
Theorem Proving ◽  
Program Analysis ◽  
State Of The Art ◽  
Inductive Reasoning ◽  
Theorem Prover ◽  
Inference Rules ◽  
First Order ◽  
Theorem Provers ◽  
Mathematical Properties

AbstractIntegers are ubiquitous in programming and therefore also in applications of program analysis and verification. Such applications often require some sort of inductive reasoning. In this paper we analyze the challenge of automating inductive reasoning with integers. We introduce inference rules for integer induction within the saturation framework of first-order theorem proving. We implemented these rules in the theorem prover Vampire and evaluated our work against other state-of-the-art theorem provers. Our results demonstrate the strength of our approach by solving new problems coming from program analysis and mathematical properties of integers.

scholarly journals Leo-III Version 1.1 (System description)

10.29007/grmx ◽  
2018 ◽  
Author(s):  
Christoph Benzmüller ◽  
Alexander Steen ◽  
Max Wisniewski
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Theorem Prover ◽  
Higher Order Logic ◽  
System Description ◽  
Agent Based ◽  
First Order ◽  
Theorem Provers ◽  
Automated Theorem Prover ◽  
Ordering Constraints

Leo-III is an automated theorem prover for (polymorphic) higher-order logic which supports all common TPTP dialects, including THF, TFF and FOF as well as their rank-1 polymorphic derivatives. It is based on a paramodulation calculus with ordering constraints and, in tradition of its predecessor LEO-II, heavily relies on cooperation with external first-order theorem provers.Unlike LEO-II, asynchronous cooperation with typed first-order provers and an agent-based internal cooperation scheme is supported. In this paper, we sketch Leo-III's underlying calculus, survey implementation details and give examples of use.

scholarly journals The Formalization of Discrete Fourier Transform in HOL

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Zhiping Shi ◽  
Yupeng Zhang ◽  
Yong Guan ◽  
Liming Li ◽  
Jie Zhang
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Order Logic ◽  
Theorem Prover ◽  
Formal Definition ◽  
Accurate Analysis ◽  
Higher Order Logic ◽  
Safety Critical ◽  
Fundamental Properties ◽  
Definition Of

Traditionally, Discrete Fourier Transform (DFT) is performed with numerical or symbolic computation, which cannot guarantee 100% accurate analysis which may be necessary for safety-critical applications. Machine theorem proving is one of the formal methods that perform accurate analysis with completeness to some extent. This paper proposes the formalization of DFT in a higher-order logic theorem prover named HOL. We propose the formal definition of DFT and verify the fundamental properties of DFT. Two case studies are presented to illustrate usefulness and correctness of the formalized DFT, including formal verifications of Fast Fourier Transform (FFT) and cosine frequency shift.

scholarly journals Embedding of Quantified Higher-Order Nominal Modal Logic into Classical Higher-Order Logic

10.29007/dzc2 ◽  
2018 ◽  
Author(s):  
Max Wisniewski ◽  
Alexander Steen
Keyword(s):  
Modal Logic ◽  
Higher Order ◽  
Order Logic ◽  
Tense Logic ◽  
Theorem Prover ◽  
Higher Order Logic ◽  
First Order ◽  
Automated Theorem Prover ◽  
Nominal Logic ◽  
The Moment

In this paper, we present an embedding of higher-order nominal modal logicinto classical higher-order logic, and study its automation. There exists no automated theorem prover for first-order or higher-order nominal logic at the moment, hence, this is the first automation for this kind of logic.In our work, we focus on nominal tense logic and have successfully proven some first theorems.

scholarly journals Superposition with First-class Booleans and Inprocessing Clausification

2021 ◽  
pp. 378-395
Author(s):  
Visa Nummelin ◽  
Alexander Bentkamp ◽  
Sophie Tourret ◽  
Petar Vukmirović
Keyword(s):  
State Of The Art ◽  
Higher Order ◽  
The State ◽  
Order Logic ◽  
Theorem Prover ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Superposition Calculus

AbstractWe present a complete superposition calculus for first-order logic with an interpreted Boolean type. Our motivation is to lay the foundation for refutationally complete calculi in more expressive logics with Booleans, such as higher-order logic, and to make superposition work efficiently on problems that would be obfuscated when using clausification as preprocessing. Working directly on formulas, our calculus avoids the costly axiomatic encoding of the theory of Booleans into first-order logic and offers various ways to interleave clausification with other derivation steps. We evaluate our calculus using the Zipperposition theorem prover, and observe that, with no tuning of parameters, our approach is on a par with the state-of-the-art approach.

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 Making Automatic Theorem Provers more Versatile

10.29007/n6j7 ◽  
2018 ◽  
Author(s):  
Simon Cruanes
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Research Directions ◽  
First Order ◽  
Theorem Provers ◽  
Smt Solvers ◽  
Automatic Theorem Provers ◽  
Automatic Theorem

We argue that automatic theorem provers should become more versatile and should be able to tackle problems expressed in richer input formats. Salient research directions include (i) developing tight combinations of SMT solvers and first-order provers; (ii) adding better handling of theories in first-order provers; (iii) adding support for inductive proving; (iv) adding support for user-defined theories and functions; and (v) bringing to the provers some basic abilities to deal with logics beyond first-order, such as higher-order logic.

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%.

Resolution in type theory

1971 ◽  
Vol 36 (3) ◽  
pp. 414-432 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Different Types ◽  
Axiomatic Set Theory ◽  
Order Predicate Calculus

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

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