Sharif Digital Repository

Automatic theorem proving has been attractive in artificial intelligence (AI). The aim of the usual automatic theorem proving methods is applying of the mathematical logic rules to decide intelligently, and reasoning automatically. In contrast the aim of our research is to apply computers abilities and artificial intelligence methods to simulate human abilities to prove mathematical theorems. For this purpose, we invent a typed formal language (TFL) to describe mathematical theories. The language must be clearly computable and should cover the First Order Logic (FOL) and the usual mathematical sentences. Then some algorithms and inference rules to compare sentences conceptually are... 

The thesis opens with a discussion of the distinction between some of the classical and the constructive notions in mathematical analysis. There then follows a description of the three main varieties of modern constructive mathematics: Bishop’s constructive mathematics, the recursive constructive mathematics of the Russian School, and Brouwer’s intuitionistic mathematics. The main purpose of the thesis is to prove the independence, relative to Bishop’s constructive mathematics, of each of the following statements:There exists a bounded, point-wise continuous map of [0, 1] into R that is not Lebesgue measurable.If µis a positive measure on a locally compact space, then every real-valued map... 

In this thesis, we provide a comprehensive overview of the efforts made towards achieving the goals of Hilbert’s program, and evaluate how successful They are. We begin by discussing Gödel’s incompleteness theorems, and then as a first attempt we present a brief presentation of Gentzen’s consistency theorem. Subsequently, various research fields that branched out from these initial efforts are explored, encompassing the application of the Omega-Rule, reverse mathematics, and self-verifying systems. Furthermore, we present the concept of “Almost Finitistic Consistency” as an alternative approach within Hilbert’s program. In conclusion, a thorough assessment of the success of Hilbert’s program... 

In much of the financial economics literature, banks are modeled as financial intermediaries that transfer household deposits to loan applicants. In these models, deposits are obtained from the accumulation of real amounts (household savings). In addition to this, another approach describes banks not as intermediaries of loanable funds, but as money-creators who do not need to accumulate deposit resources to lend. In this study, by modeling the money-creation behavior of the banking network, we try to answer the question that if the bank can create credit and purchasing power for the firm based on its future income, how will the general variables of the economy, including consumption and... 

The importance of a quick and authentic response in today's hectic business environment shows us the necessity of being alert. By understanding the value of alertness, we seek to find factors affecting it. In this article, we try to learn about new aspects of this structure. We expect that personality attributes and the way we look at the environment are influential in this matter. The amount of "curiosity" that is institutionalized in our personality and our "attitude" towards entrepreneurship has often been examined along with other concepts like mindfulness. However, the exact relationship between them has not been specifically analyzed. For this purpose, we started this research by... 

In 1933, Gödel’s paper “An interpretation of the intuitionistic propositional calculus”[12] introduced a proof interpretation of the intuitionistic propositional logic to the modal system, S4, which interprets the logical constant as a provability predicate. Fortunately, this interpretation has the soundness-completeness property and therefore we can prove the constructivist’s viewpoint which states that the intuitionistic truth and provability are identical. However, there is a mathematical trick in his work. In fact, Gödel used a proof theory of the concept of proof instead of an explicit interpretation. Furtheremore, he proved the failure of any interpretation of this implicit proof as a... 

In this thesis, we study recursion theory in the context of Bishop’s constructive mathematics.For this goal, at first, we redefine primitive recursive functions and then based on works of Fred Richman in [Church’s Thesis Without Tears, The Journal of Symbolic Logic, Vol. 48, No. 3 (Sep., 1983), pp. 797-803], give an interpretation of partial recursive functions. Finally we study some important theorems in recursion theory,for example, universal computation predicate, recursion theorem, s-m-n theorem and
Rice’s theorem  

This dissertation consists of five parts. In the fist part we draw a scheme of Logical Paradoxes: What is a paradox, what are logical paradoxes, different versions of logical paradoxes, etc. The three next parts are respectively about: Tarski’s Hierarchy Approach, Kripke’s Fixed Point Theory and Herzberger and Gupta’s Revision Theory of Truth. In the end of each of these three parts, we consider some critics. In the last part, we introduce our new approach. This is based on a new concept called Pseudo-Contradiction. Relative to a (consistent) set S of axioms, a sentence φ is a Pseudo-contradiction if and only if both φ and ¬φ are inconsistent with S. We try to show the Liar sentence can be... 

In this thesis, we study monadic second order logic and the languages of graphs. We introduce the notion of ∆- regular graph automaton. Then we designed this finite automaton for processing on labeled graphs. Definability of graph languages which are recognizable by this automaton is shown. To do this, we first briefly introduced monadic second order logic, and then we studied definability of regular string and tree languages in monadic second order logic. In the last section of thesis, ∆- regular graph automaton is introduced  

According to increasing computer systems, needs for verification of such systems with respect to desirable properties is critical. Model checking is one of the best methods of verification. Different methods have been proposed for model checking. The most efficient of these methods is automata-theoretic approach. In this approach, formal specification of desirable property, specified by formula in temporal logics, is translated to corresponding automaton. If the system model is expressed as automaton, the problem of model checking is then reduced to a problem of automata-theory. The question is the following. Are all the computations of the corresponding automaton accepted by the automaton... 

Immensity and profundity of mathematical logic have provided an opportunity for its interconnection with other branches of science, one of which is computer science where theory of dependent types and lambda calculus lounges amongst the area of their overlap. The former has been engendered for the sake of deepening the connection between mathematical logic and programming. On the other side, lambda calculus which is a formal system in mathematical logic and computer science, developed as a tool for formal computations, could also be counted as a programming discipline. It is the objective of this thsis to study the two aforementioned fields with emphasis on unification of higher-order...