Logic and Algebraic Structures in Quantum Computing

Logic and Algebraic Structures in Quantum Computing
Logic and Algebraic Structures in Quantum Computing

Arising from a special session held at the 2010 North American Annual Meeting of the Association for Symbolic Logic, this volume is an international cross-disciplinary collaboration with contributions from leading experts exploring connections across their respective fields. Themes range from philosophical examination of the foundations of physics and quantum logic, to exploitations of the methods and structures of operator theory, category theory, and knot theory in an effort to gain insight into the fundamental questions in quantum theory and logic. The book will appeal to researchers and students working in related fields, including logicians, mathematicians, computer scientists, and physicists. A brief introduction provides essential background on quantum mechanics and category theory, which, together with a thematic selection of articles, may also serve as the basic material for a graduate course or seminar.


Al-Farabi also considered the theories of conditional syllogisms and analogical inference, which were part of the Stoic tradition of logic rather than the Aristotelian. But means that there is some particular boy whom every girl kissed. This period overlaps with the work of what is known as the "mathematical school", which included Dedekind, Pasch, Peano, Hilbert, Zermelo, Huntington, Veblen and Heyting. More specifically, Boole agreed with what Aristotle said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. A consequence is a hypothetical, conditional proposition: two propositions joined by the terms 'if. His purpose is to show the rational structure of the "Absolute"-indeed of rationality itself. The most significant innovation, however, was his explanation of the quantifier in terms of mathematical functions. In particular, one of the schools that grew out of Mohism, the Logicians, are credited by some scholars for their early investigation of formal logic. The field offers a wide variety of application areas, ranging from particle simulations for the study of protein folding to mesh calculations in climate change prediction. Since the third and fourth columns match, we would conclude that  +  =  +  is a universal law. This idea occurred to Boole in his teenage years, working as an usher in a private school in Lincoln, Lincolnshire. Both Thales and Pythagoras of the Pre-Socratic philosophers seem aware of geometry's methods. Logicians formalized some common mistakes, such as the temptation to conclude that if implies , and if holds, then must hold also. Thus only singular propositions are of subject-predicate form, and they are irreducibly singular, i.e. Along with learning applications of these tools, you’ll examine probability and statistics as fully developed areas of mainstream mathematics. nor." and equally well "not both. Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize rather than. Logic and Algebraic Structures in Quantum Computing. Traditional logic generally means the textbook tradition that begins with Antoine Arnauld's and Pierre Nicole's Logic, or the Art of Thinking, better known as the. I followed the rules above and it did work, plus it was pretty quick, but it was a bit like doing a magic trick. Matilal remarks that Dignāga's analysis is much like John Stuart Mill's Joint Method of Agreement and Difference, which is inductive. Synthetic Division I don't like this method so I don't really want to mention it here, but for completeness I suppose I should. Each row will represent a combination of values for the first two variables' values, and each column will represent a combination of the last two variables' values. Without this device, the project of logicism would have been doubtful or impossible. , an analysis of simple categorical propositions into simple terms, negation, and signs of quantity. We call this technique of building an expression from a truth table the sum of products technique. Although Hegel's has had little impact on mainstream logical studies, its influence can be seen elsewhere: The work of the British Idealists, such as F.H. An expression in which elective symbols are used is called an elective function, and an equation of which the members are elective functions, is an elective equation. This model of analogy has been used in the recent work of John F. Boethius Commentary on the Perihermenias, Secunda Editio, ed. The proofs of Euclid of Alexandria are a paradigm of Greek geometry. The algebraic period from Boole's Analysis to Schröder's Vorlesungen. The economic, political, and philosophical studies of Karl Marx, and in the various schools of Marxism. But then the second terms would have been   , which has one more variable in it than we used previously. The ancient Egyptians discovered geometry, including the formula for the volume of a truncated pyramid.Ancient Babylon was also skilled in mathematics. Later in the decade, Gödel developed the concept of set-theoretic constructibility, as part of his proof that the axiom of choice and the continuum hypothesis are consistent with Zermelo–Fraenkel set theory. Logic and Algebraic Structures in Quantum Computing. In this track you’ll focus on analysing the large-scale systems that are central to various fields of science and many real-world applications. His technique, which was simplified and extended soon after its introduction, has since been applied to many other problems in all areas of mathematical logic. The major logicists were Frege, Russell, and the early Wittgenstein. Thales was said to have had a sacrifice in celebration of discovering Thales' theorem just as Pythagoras had the Pythagorean theorem. Since designers want circuits to work as quickly as possible, they work to minimize the circuit depth, which is the maximum distance from any input through the circuit to an output. In proof theory, the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the realizability method invented by Georg Kreisel and Gödel's interpretation. Though it is difficult to determine the dates, the probable order of writing of Aristotle's logical works is: , a study of the ten kinds of primitive term. Somewhat more surprising is that the NAND gate alone is universal - that is, any truth table can be implemented by a circuit that includes only NAND gates. Furthermore, the demonstration of the connection between the meanings of the words if, and, not, or, there is, some, all, and so forth, deserves attention". The rich geometric structure of Lie groups will provide you with the opportunity to develop theories that are both amazingly general and surprisingly concrete. In modern notation, this would be expressed as In English, "for all x, if Ax then Bx". A truth table contains a row for every possible combination of input values and each row tells what the value of the circuit's output would be for that combination of inputs. They appear in many situations in mathematics and physics, where continuous symmetries play a role. He was examining the field of logic, created for thinking about the validity of philosophical arguments. Most notable was Hilbert's Program, which sought to ground all of mathematics to a finite set of axioms, proving its consistency by "finitistic" means and providing a procedure which would decide the truth or falsity of any mathematical statement. The aim of the "logicist school" was to incorporate the logic of all mathematical and scientific discourse in a single unified system which, taking as a fundamental principle that all mathematical truths are logical, did not accept any non-logical terminology. A more complex Karnaugh map Another example will illustrate some additional features of a Karnaugh map. Given such a truth table defining a function, we'll build up a Boolean expression representing the function. This method is known as inductive reasoning, a method which starts from empirical observation and proceeds to lower axioms or propositions; from these lower axioms, more general ones can be induced. He noticed that logical functions could be built from the AND, OR, and NOT operations and that this observation leads one to be able to reason about logic in a mathematical system. then"; it is the definition used in modern logic. A warning: Students new to Boolean expressions frequently try to abbreviate   as - that is, they draw a single line over the whole expression, rather than two separate lines over the two individual pieces. In contrast to Heraclitus, Parmenides held that all is one and nothing changes. The title translates as "new instrument". The Curry-Howard correspondence emerged as a deep analogy between logic and computation, including a correspondence between systems of natural deduction and typed lambda calculi used in computer science. It culminates with the , an important work which includes a thorough examination and attempted solution of the antinomies which had been an obstacle to earlier progress. We could draw a truth table comparing the results for these two expressions. Syncategoremata are terms which are necessary for logic, but which, unlike categorematic terms, do not signify on their own behalf, but 'co-signify' with other words.. This psychological approach to logic was rejected by Gottlob Frege. What exists can in no way not exist. You can see that some wires intersect in a small, solid circle: This circle indicates that the wires are connected, and so values coming into the circle continue down all the wires connected to the circle. Reprinted: South Bend, IN: St. The logic of Aristotle, and particularly his theory of the syllogism, has had an enormous influence in Western thought. The theory of syncategoremata. We can use parentheses when this order of operations isn't what we want. Model theory applies the methods of mathematical logic to study models of particular mathematical theories. The subject of your Master’s thesis will depend on your interests in the field. He was the first to deal with the principles of contradiction and excluded middle in a systematic way. Boole's system for writing down logical expressions is called Boolean algebra. Peirce contrasted this with the disputation and uncertainty surrounding traditional logic, and especially reasoning in metaphysics. and give a homogeneous presentation of the lot. The fields of constructive analysis and computable analysis were developed to study the effective content of classical mathematical theorems; these in turn inspired the program of reverse mathematics. Courses in History of Science and in Concrete Geometry are an advantage. We know these as the familiar curves of straight lines, conics, and higher degree classical curves. This holds always but humans always prove unable to understand it, both before hearing it and when they have first heard it.

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. Other logic gates and universality Until now, we've dealt only with AND, OR, and NOT gates. Diodorus used the plausibility of the first two to prove that nothing is possible if it neither is nor will be true. Valid reasoning has been employed in all periods of human history. First, though, we'll take a necessary detour through the study of Boolean expressions. His direct influence was small, but his influence through pupils such as John of Salisbury was great, and his method of applying rigorous logical analysis to theology shaped the way that theological criticism developed in the period that followed.

Major physical theories such as classical mechanics and general relativity acquire their most natural and insightful formulation in such terms.


. First, in the realm of foundations, Boole reduced the four propositional forms of Aristotelian logic to formulas in the form of equations - by itself a revolutionary idea. Your studies in this track will allow you to evaluate a manifold in terms of differentiable functions, vector fields, and other related tools. Ockham's Theory of Propositions: Part II of the Summa Logicae, translated by Alfred J. Let no one ignorant of geometry enter here. For more information about this track, please contact  Track title: Pure AnalysisTrack description: The Pure Analysis programme provides you with the tools to perform analysis on manifolds, with a strong emphasis on geometric concepts. Avicenna wrote on the hypothetical syllogism and on the propositional calculus, which were both part of the Stoic logical tradition. Ноутбук Lenovo IdeaPad 320-17AST 80XW0002RK (AMD A6-9220 2.5 GHz/4096Mb/1000Gb/DVD-RW/AMD Radeon R520M 2048Mb/Wi-Fi/Bluetooth/Cam/17.3/1600x900/Windows 10 64-bit). What underlies every definition is a Platonic Form, the common nature present in different particular things. Traditional logic regards the sentence "Caesar is a man" as of fundamentally the same form as "all men are mortal." Sentences with a proper name subject were regarded as universal in character, interpretable as "every Caesar is a man". If two wires intersect with no circle, this means that one wire goes over the other, like an Interstate overpass, and a value on one wire has no influence on the other. Track title: Algebraic Geometry and Number TheoryTrack description: These areas have their roots in classical Greek mathematics, and thus belong to the oldest branches of mathematics. In practice, the subject of a M.Sc. The Prior Analytics contains his exposition of the "syllogism", where three important principles are applied for the first time in history: the use of variables, a purely formal treatment, and the use of an axiomatic system. While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative proof. Husserl argued forcefully that grounding logic in psychological observations implied that all logical truths remained unproven, and that skepticism and relativism were unavoidable consequences. Tarski also produced important work on the methodology of deductive systems, and on fundamental principles such as completeness, decidability, consistency and definability. The animation below shows the equivalence of long division and synthetic division. Perhaps next year I'll try an alternative. In this period, there were more practitioners, and a greater continuity of development. This free service is available to anyone who has published and whose publication is in Scopus. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory. Such a table is called a truth table. The geometric part, or algebraic geometry, is an essential tool in modern mathematical physics. His pupils and successors were called "Megarians", or "Eristics", and later the "Dialecticians". The same idea is found in the work of Leibniz, who had read both Llull and Hobbes, and who argued that logic can be represented through a combinatorial process or calculus. As a result, research into this class of formal systems began to address both logical and computational aspects; this area of research came to be known as modern type theory. The Stoics adopted the Megarian logic and systemized it. Diodorus Cronus defined the possible as that which either is or will be, the impossible as what will not be true, and the contingent as that which either is already, or will be false. You might even ask for points with coordinates that are integers modulo a prime.  All these viewpoints open up completely different directions in the field, referred to as arithmetic-algebraic geometry. Since Gentzen's work, natural deduction and sequent calculi have been widely applied in the fields of proof theory, mathematical logic and computer science. This work inspired the contemporary area of proof mining. As Boole was working in the nineteenth century, of course, he wasn't thinking about logic circuits. The period between the fourteenth century and the beginning of the nineteenth century had been largely one of decline and neglect, and is generally regarded as barren by historians of logic. thesis should have a substantial mathematical content in the area of the specialisation. Source Books in the History of the Sciences. The “sum of products technique” that we saw for converting a Boolean function into a circuit isn't too bad using these criteria. Mathematics in Aristotle, Oxford University Press. For example, the commutative law applies to both OR and AND. But, like Llull and Hobbes, he failed to develop a detailed or comprehensive system, and his work on this topic was not published until long after his death. The revival of logic occurred in the mid-nineteenth century, at the beginning of a revolutionary period where the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in mathematics. This diagram consists of some peculiar shapes connected with some lines. According to Aristotle, the Megarians of his day claimed there was no distinction between potentiality and actuality. Frege's objective was the program of Logicism, i.e. In some cases, students are able to specialise in subjects that overlap with our programmes in Applied Analysis, Differential Geometry and Topology, and Mathematical Physics. For Boolean functions with four or fewer inputs, the Karnaugh map is a particularly convenient way to find the smallest possible sum-of-products expression. However, logic studies the of valid reasoning, inference and demonstration. When the study of logic resumed after the Dark Ages, the main source was the work of the Christian philosopher Boethius, who was familiar with some of Aristotle's logic, but almost none of the work of the Stoics. We'll now turn to investigating a technique for building circuits from a truth table, which results in smaller circuits without making any compromises in depth. James Tanton calls this the Galley Method - his Curriculum Essay about how it works includes exercises and interesting questions. Taking the specialisation in Pure Analysis will give you the opportunity to engage in research in these and additional fields of interest. The first logicians to debate conditional statements were Diodorus and his pupil Philo of Megara. The second question is a result of Plato's theory of Forms. The Development of Modern Logic Oxford University Press. Universal and particular propositions, by contrast, are not of simple subject-predicate form at all. The former attempts to model logical reasoning as it 'naturally' occurs in practice and is most easily applied to intuitionistic logic, while the latter was devised to clarify the derivation of logical proofs in any formal system. To understand how computers work, we will want to understand the fundamentals of digital circuits. SJR uses a similar algorithm as the Google page rank; it provides a quantitative and a qualitative measure of the journal’s impact. In the previous section, we saw how logic circuits work