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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... 

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... 

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... 

Nowadays, BDI architecture is of the most well known agent’s architectures. BDI architecture or the architecture in which the system is viewed as rational agents based on the attitudes of belief, desire and intention represents an abstraction of human deliberation based on a theory of rational action in the human cognition process. BDI logic introduced by Rao and Georgeff have been widely used as the theoretical basis of specification and implementation of rational agents. BDI logic is limited to deal with crisp assertion, while the assertions encountered in real world are not precise and thus cannot be treated simply by using yes or no. Moreover, In order to obtain more human like agents we... 

Pregroups as a mathematical structure, are replacement for Lambek's type caregorial grammar which much used in Computational Linguistics. Because of computational and logical properties of pregroups, we can use them as strong tool to analyse the sentence structure of many natural languages. This kind of analysis has been done for English, French, German, Polish, Italian, Arabic and Japanese. In case of Persian language, analysis of simple and compound sentences structure with simple tense verbs and explicit subjects and objects has been studied. In this M.Sc. thesis, we will extent analysis of Persian sentence structure to sentences with compound tense verbs and implicit subjects and objects... 

Description logic is a family of knowledge representive languages which represents knowledge via propositional logic (first order logic) propositions and constructors and applies its services for reasoning and consistency checking. Nowadays description logic and its popular reasoner FaCT++ which applies tablue reasoning technique are widely used in applications such as semantic web and onthologies. Model checking is a technique for systems and models verification and to guarantee the accuracy of design. Given a model description and a specification formula, the model checker verifies the model against the specification and decides if the model satisfies the description or not. Main model... 

In logic literature, the phrase “basic logic” can refer to three different logical systems. First, basic propositional logic (BPL) that was introduced by A. Visser in 1981. This logic is a subintuitionistic logic, that can be obtained from intuitionistic logic by a weakening of modus ponens. One decade later, Wim Ruitenberg regarding philosophical critiques of logical connectives, reintroduced BPL and its first order extension, BQC. From then, different aspects of basic logic have been investigated by logicians all over the world. Another logical system with the name “basic logic” is the system that was introduced by G. Sambin and G. Battilotti in 1997. The goal of this system is to provide... 

Deontic logic is used to formalize legal reasoning. To apply this logic in law, we describe tersely some efforts to improve this logic by relativizing its operations with respect to different people and groups of society. Until now, this logic was restricted to formalize “what must be”. We extend this logic to dyadic logic to formalize “what must be done”.
In practice, legal reasoning leads to non-monotonic logics, the most applicable one in law is defeasible logic. So it is necessary to combine deontic and defeasible logics to formalize legal reasoning in a more appropriate way. To do that, we must adjust possible worlds of these two logics. In this way, we find a method for... 

In this dissertation, we study (first-order) arithmetical interpretations for propositional (modal and non-modal) logics. More precisely, the following results are included in this dissertation: an axiomatization for provability logic of Heyting Arithmetic, HA, and its self-completion HA := HA + PrHA(⌜A⌝) ! A for 1-substitutions is provided, and their arithmetical completeness theorems are proved. We also show that they are decidable. The de Jongh property for Basic Arithmetic BA, HA and HA + □ are proved  

In constructive mathematics, one has to construct a mathematical object in order to show that it exists. Consequently, some of classical theorems are not acceptable from a constructive point of view. In particular, the constructive validity of the parts of mathematics that play a role in forming physical theories is of interest. Here, we want to examine the constructive provability of some theorems related to quantum mechanics  

In this thesis, we investigate constructive measure theory in two schools of constructivism; intuitionistic mathematics and Bishop’s constructive mathematics.In this regard, first, we examine some classical propositions on measurability in both schools. Then we give a comparison between measurability in the two schools  

Epistemic algorithms are instructions and rules based on knowledge of agents. These algorithms by individual or group knowledge of agents make decisions about future behavior of systems. Epistemic gossip protocols is one of these algorithms. They are used to for spreading secrets among nodes in a network. According to individual knowledge of each node they decide who calls whom in each step. In security problems epistemic algorithms are used to detect the safety of protocols. Some epistemic notions like ignorance and contingency are used in formalization of security problems. In the thesis we study these epistemic algorithms  

In this thesis, we study proof complexity conjectures and also introduce their mathematical logic equivalents in terms of provability and unprovability in strong enough first-order arithmetical theories. One of the most important conjectures in this theory is the following conjecture. The non-existence of an optimal proof system for propositional tautologies: In general, a proof system is a computable function in polynomial time such that its range is exactly the set of tautologies. We say proof system P, polynomially simulates proof system Q if and only if there exists a polynomial h such that for all tautologies such as A and for all proofs like a, if Qpaq A, then there exists a proof... 

Mathematical logic claims to have a model for various kinds of thinking (mathematical, philosophical, scientific and...) which can provide us with a language at the same time. The relation between art and logic, when art is defined on its own, is somewhat unexplored, as opposed to when it's defined scientifically or philosophically. There are arguments in the literature asserting that art cannot fit into the frame of mathematical logic. The inter-connection between thinking and art has been fairly investigated, even though it's been mainly believed that art is more engaged with emotions rather than rationality; thinking about artistic value, artistic credibility, proof in art, etc... It's... 

In this thesis we construct a bounded translation of intuitionistic predicate logic into Basic predicate logic with restricted use of universal quantifier and without equality inspired by papers [1] and [2]  

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  

IF Logic is an extension of First-Order logic, including syntactic connective ’/’ (independence indicator) which carries the concept of ’independent from’ in language and metalanguage.This logic which has introduced by Hintikka and Sandu in 1989, uses Game-theoretical semantics (GTS) such that every formula is defined as a game and the players are verifier and falsifier of its truth. The formula is True if the verifier has a winning-strategy and it is False where the falsifier has a winning-strategy; and evaluated as indeterminate otherwise. this relates a central concept of logic (truth) to a central concept of game theory(winningstrategy)  

Arising from the intersection of two modern ideas in logic and mathematics, namely Type Theory - specifically Martin-Löf’s intuitionistic type theory, and Homotopy Theory, Homotopy Type Theory has introduced a novel formalism that formalizes every mathematical object - including constructions, definitions, and proofs - in a specific and fundamentally different way from other interpretive frameworks (such as set theory), bringing forth new results - both philosophically and mathematically. The first goal of this research is to understand the meaning of Homotopy Type Theory as a foundation for mathematics through exploring this new formalism. The second goal of this research is to understand... 

The verification of programs is one of the most important subjects in Computer Science. In this thesis, after reviewing preliminary subjects, While-Programming computational model is presented and its semantics is discussed. Next, the axiomatic method of Hoare is presented, which is a deduction system for verification of programs. With the help of Hoare method, the idea of Data-Oriented verification of programs is developed