Discrete mathematics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Discrete_mathematics

Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithms.

 

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus or Euclidean geometry. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.

The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.

Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.

Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.

In university curricula, "Discrete Mathematics" appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction with efforts by ACM and MAA into a course that is basically intended to develop mathematical maturity in first-year students; therefore it is nowadays a prerequisite for mathematics majors in some universities as well. Some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics is sometimes seen as a preparatory course, not unlike precalculus in this respect.

The Fulkerson Prize is awarded for outstanding papers in discrete mathematics.

Grand challenges, past and present

Much research in graph theory was motivated by attempts to prove that all maps, like this one, can be colored using only four colors so that no areas of the same color share an edge. Kenneth Appel and Wolfgang Haken proved this in 1976.

 

The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance).

In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent. Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself. Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done.

The need to break German codes in World War II led to advances in cryptography and theoretical computer science, with the first programmable digital electronic computer being developed at England's Bletchley Park with the guidance of Alan Turing and his seminal work, On Computable Numbers. At the same time, military requirements motivated advances in operations research. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades. Operations research remained important as a tool in business and project management, with the critical path method being developed in the 1950s. The telecommunication industry has also motivated advances in discrete mathematics, particularly in graph theory and information theory. Formal verification of statements in logic has been necessary for software development of safety-critical systems, and advances in automated theorem proving have been driven by this need.

Computational geometry has been an important part of the computer graphics incorporated into modern video games and computer-aided design tools.

Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and combinatorics, are important in addressing the challenging bioinformatics problems associated with understanding the tree of life.

Currently, one of the most famous open problems in theoretical computer science is the P = NP problem, which involves the relationship between the complexity classes P and NP. The Clay Mathematics Institute has offered a $1 million USD prize for the first correct proof, along with prizes for six other mathematical problems.

Topics in discrete mathematics

Theoretical computer science

Complexity studies the time taken by algorithms, such as this sorting routine.

Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and mathematical logic. Included within theoretical computer science is the study of algorithms and data structures. Computability studies what can be computed in principle, and has close ties to logic, while complexity studies the time, space, and other resources taken by computations. Automata theory and formal language theory are closely related to computability. Petri nets and process algebras are used to model computer systems, and methods from discrete mathematics are used in analyzing VLSI electronic circuits. Computational geometry applies algorithms to geometrical problems, while computer image analysis applies them to representations of images. Theoretical computer science also includes the study of various continuous computational topics.

Information theory

The ASCII codes for the word "Wikipedia", given here in binary, provide a way of representing the word in information theory, as well as for information-processing algorithms.

Information theory involves the quantification of information. Closely related is coding theory which is used to design efficient and reliable data transmission and storage methods. Information theory also includes continuous topics such as: analog signals, analog coding, analog encryption.

Logic

Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, and completeness. For example, in most systems of logic (but not in intuitionistic logic) Peirce's law (((P→Q)→P)→P) is a theorem. For classical logic, it can be easily verified with a truth table. The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.

Logical formulas are discrete structures, as are proofs, which form finite trees or, more generally, directed acyclic graph structures (with each inference step combining one or more premise branches to give a single conclusion). The truth values of logical formulas usually form a finite set, generally restricted to two values: true and false, but logic can also be continuous-valued, e.g., fuzzy logic. Concepts such as infinite proof trees or infinite derivation trees have also been studied,^[17] e.g. infinitary logic.

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas. 

In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infinite set, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outside the scope of discrete mathematics. Indeed, contemporary work in descriptive set theory makes extensive use of traditional continuous mathematics.

Combinatorics

Combinatorics studies the way in which discrete structures can be combined or arranged. Enumerative combinatorics concentrates on counting the number of certain combinatorial objects - e.g. the twelvefold way provides a unified framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration (i.e., determining the number) of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties. Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, partition theory is now considered a part of combinatorics or an independent field. Order theory is the study of partially ordered sets, both finite and infinite.

Graph theory

Graph theory has close links to group theory. This truncated tetrahedron graph is related to the alternating group A₄.

 

Graph theory, the study of graphs and networks, is often considered part of combinatorics, but has grown large enough and distinct enough, with its own kind of problems, to be regarded as a subject in its own right. Graphs are one of the prime objects of study in discrete mathematics. They are among the most ubiquitous models of both natural and human-made structures. They can model many types of relations and process dynamics in physical, biological and social systems. In computer science, they can represent networks of communication, data organization, computational devices, the flow of computation, etc. In mathematics, they are useful in geometry and certain parts of topology, e.g. knot theory. Algebraic graph theory has close links with group theory. There are also continuous graphs; however, for the most part, research in graph theory falls within the domain of discrete mathematics.

Probability

Discrete probability theory deals with events that occur in countable sample spaces. For example, count observations such as the numbers of birds in flocks comprise only natural number values {0, 1, 2, ...}. On the other hand, continuous observations such as the weights of birds comprise real number values and would typically be modeled by a continuous probability distribution such as the normal. Discrete probability distributions can be used to approximate continuous ones and vice versa. For highly constrained situations such as throwing dice or experiments with decks of cards, calculating the probability of events is basically enumerative combinatorics.

Number theory

The Ulam spiral of numbers, with black pixels showing prime numbers. This diagram hints at patterns in the distribution of prime numbers.

Number theory is concerned with the properties of numbers in general, particularly integers. It has applications to cryptography and cryptanalysis, particularly with regard to modular arithmetic, diophantine equations, linear and quadratic congruences, prime numbers and primality testing. Other discrete aspects of number theory include geometry of numbers. In analytic number theory, techniques from continuous mathematics are also used. Topics that go beyond discrete objects include transcendental numbers, diophantine approximation, p-adic analysis and function fields. 

Algebraic structures

Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: boolean algebra used in logic gates and programming; relational algebra used in databases; discrete and finite versions of groups, rings and fields are important in algebraic coding theory; discrete semigroups and monoids appear in the theory of formal languages.

Calculus of finite differences, discrete calculus or discrete analysis

A function defined on an interval of the integers is usually called a sequence. A sequence could be a finite sequence from a data source or an infinite sequence from a discrete dynamical system. Such a discrete function could be defined explicitly by a list (if its domain is finite), or by a formula for its general term, or it could be given implicitly by a recurrence relation or difference equation. Difference equations are similar to a differential equations, but replace differentiation by taking the difference between adjacent terms; they can be used to approximate differential equations or (more often) studied in their own right. Many questions and methods concerning differential equations have counterparts for difference equations. For instance, where there are integral transforms in harmonic analysis for studying continuous functions or analogue signals, there are discrete transforms for discrete functions or digital signals. As well as the discrete metric there are more general discrete or finite metric spaces and finite topological spaces.

Geometry

Computational geometry applies computer algorithms to representations of geometrical objects.

Discrete geometry and combinatorial geometry are about combinatorial properties of discrete collections of geometrical objects. A long-standing topic in discrete geometry is tiling of the plane. Computational geometry applies algorithms to geometrical problems.

Topology

Although topology is the field of mathematics that formalizes and generalizes the intuitive notion of "continuous deformation" of objects, it gives rise to many discrete topics; this can be attributed in part to the focus on topological invariants, which themselves usually take discrete values. See combinatorial topology, topological graph theory, topological combinatorics, computational topology, discrete topological space, finite topological space, topology (chemistry).

Operations research

PERT charts like this provide a project management technique based on graph theory.

 

Operations research provides techniques for solving practical problems in engineering, business, and other fields — problems such as allocating resources to maximize profit, and scheduling project activities to minimize risk. Operations research techniques include linear programming and other areas of optimization, queuing theory, scheduling theory, and network theory. Operations research also includes continuous topics such as continuous-time Markov process, continuous-time martingales, process optimization, and continuous and hybrid control theory.

Game theory, decision theory, utility theory, social choice theory

	Cooperate	Defect
Cooperate	−1, −1	−10, 0
Defect	0, −10	−5, −5
Payoff matrix for the Prisoner's dilemma, a common example in game theory. One player chooses a row, the other a column; the resulting pair gives their payoffs

Decision theory is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision. 

Utility theory is about measures of the relative economic satisfaction from, or desirability of, consumption of various goods and services. 

Social choice theory is about voting. A more puzzle-based approach to voting is ballot theory.

Game theory deals with situations where success depends on the choices of others, which makes choosing the best course of action more complex. There are even continuous games, see differential game. Topics include auction theory and fair division.

Discretization

Discretization concerns the process of transferring continuous models and equations into discrete counterparts, often for the purposes of making calculations easier by using approximations. Numerical analysis provides an important example.

Discrete analogues of continuous mathematics

There are many concepts in continuous mathematics which have discrete versions, such as discrete calculus, discrete probability distributions, discrete Fourier transforms, discrete geometry, discrete logarithms, discrete differential geometry, discrete exterior calculus, discrete Morse theory, difference equations, discrete dynamical systems, and discrete vector measures.

In applied mathematics, discrete modelling is the discrete analogue of continuous modelling. In discrete modelling, discrete formulae are fit to data. A common method in this form of modelling is to use recurrence relation.

In algebraic geometry, the concept of a curve can be extended to discrete geometries by taking the spectra of polynomial rings over finite fields to be models of the affine spaces over that field, and letting subvarieties or spectra of other rings provide the curves that lie in that space. Although the space in which the curves appear has a finite number of points, the curves are not so much sets of points as analogues of curves in continuous settings. For example, every point of the form ${\displaystyle V(x-c)\subset \operatorname {Spec} K[x]=\mathbb {A} ^{1}}$ for ${\displaystyle K}$ a field can be studied either as ${\displaystyle \operatorname {Spec} K[x]/(x-c)\cong \operatorname {Spec} K}$, a point, or as the spectrum ${\displaystyle \operatorname {Spec} K[x]_{(x-c)}}$ of the local ring at (x-c), a point together with a neighborhood around it. Algebraic varieties also have a well-defined notion of tangent space called the Zariski tangent space, making many features of calculus applicable even in finite settings.

Hybrid discrete and continuous mathematics

The time scale calculus is a unification of the theory of difference equations with that of differential equations, which has applications to fields requiring simultaneous modelling of discrete and continuous data. Another way of modeling such a situation is the notion of hybrid dynamical systems.

Code of Hammurabi

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Code_of_Hammurabi 

Code of Hammurabi
A side view of the stele "fingertip" at the Louvre Museum
Created	c. 1754 BC
Author(s)	Hammurabi

The Code of Hammurabi is a well-preserved Babylonian code of law of ancient Mesopotamia, dated to about 1754 BC (Middle Chronology). It is one of the oldest deciphered writings of significant length in the world. The sixth Babylonian king, Hammurabi, enacted the code. A partial copy exists on a 2.25-metre-tall (7.5 ft) stone stele. It consists of 282 laws, with scaled punishments, adjusting "an eye for an eye, a tooth for a tooth" (lex talionis) as graded based on social stratification depending on social status and gender, of slave versus free, man versus woman.

Nearly half of the code deals with matters of contract, establishing the wages to be paid to an ox driver or a surgeon for example. Other provisions set the terms of a transaction, the liability of a builder for a house that collapses, or property that is damaged while left in the care of another. A third of the code addresses issues concerning household and family relationships such as inheritance, divorce, paternity, and reproductive behavior. Only one provision appears to impose obligations on a government official; this provision establishes that a judge who alters his decision after it is written down is to be fined and removed from the bench permanently. A few provisions address issues related to military service.

The code was discovered by modern archaeologists in 1901, and its editio princeps translation published in 1902 by Jean-Vincent Scheil. This nearly complete example of the code is carved into a diorite stele in the shape of a huge index finger, 2.25 m (7.4 ft) tall. The code is inscribed in the Akkadian language, using cuneiform script carved into the stele. The material was imported into Sumeria from Magan - today the area covered by the United Arab Emirates and Oman.

It is currently on display in the Louvre, with replicas in numerous institutions, including the Oriental Institute at the University of Chicago, the Northwestern Pritzker School of Law in Chicago, the Clendening History of Medicine Library & Museum at the University of Kansas Medical Center, the library of the Theological University of the Reformed Churches in the Netherlands, the Pergamon Museum of Berlin, the Arts Faculty of the University of Leuven in Belgium, the National Museum of Iran in Tehran, the Department of Anthropology, National Museum of Natural History, Smithsonian Institution, the University Museum at the University of Pennsylvania, the Pushkin State Museum of Fine Arts in Russia, the Prewitt-Allen Archaeological Museum at Corban University, Garrett-Evangelical Theological Seminary, and Museum of the Bible in Washington, DC.

History

The code on clay tablets

 

The code on a diorite stele

 

Hammurabi ruled from 1792 to 1750 BC (according to the middle chronology). At the head of the stone slab is Hammurabi receiving the law from Shamash, and in the preface, he states, "Anu and Bel called by name me, Hammurabi, the exalted prince, who feared God, to bring about the rule of righteousness in the land, to destroy the wicked and the evil-doers; so that the strong should not harm the weak; so that I should rule over the black-headed people like Shamash, and enlighten the land, to further the well-being of mankind." The laws were arranged in 44 columns and 28 paragraphs; some follow along the rules of "an eye for an eye".

It was taken as plunder by the Elamite king Shutruk-Nahhunte in the 12th century BC and was taken to Susa in Elam (located in the present-day Khuzestan Province of Iran), where it was no longer available to the Babylonian people. However, when Cyrus the Great brought both Babylon and Susa under the rule of his Persian Empire and placed copies of the document in the Library of Sippar, the text became available for all the peoples of the vast Persian Empire to view.

In 1901, Egyptologist Gustave Jéquier, a member of an expedition headed by Jacques de Morgan, found the stele containing the Code of Hammurabi during archaeological excavations at the ancient site of Susa in Khuzestan.

The stele unearthed in 1901 had many laws scraped off by Shutruk-Naknunte. Early estimates pegged the number of missing laws at 34, however the exact number is still not determined and only 30 have been discovered so far. The common belief is that the code contained 282 laws in total. 

Laws of Hammurabi's Code

Detail of paragraph 165.

The Code of Hammurabi was one of the only sets of laws in the ancient Near East and also one of the first forms of law. The code of laws was arranged in orderly groups, so that all who read the laws would know what was required of them. Earlier collections of laws include the Code of Ur-Nammu, king of Ur (c.  2050 BC), the Laws of Eshnunna (c. 1930 BC) and the codex of Lipit-Ishtar of Isin (c. 1870 BC), while later ones include the Hittite laws, the Assyrian laws, and Mosaic Law. These codes come from similar cultures in a relatively small geographical area, and they have passages that resemble each other.

Figures at the top of the stele "fingernail", above Hammurabi's code of laws.

 

The Code of Hammurabi is the longest surviving text from the Old Babylonian period. The code has been seen as an early example of a fundamental law, regulating a government – i.e., a primitive constitution. The code is also one of the earliest examples of the idea of presumption of innocence, and it also suggests that both the accused and accuser have the opportunity to provide evidence. The occasional nature of many provisions suggests that the code may be better understood as a codification of Hammurabi's supplementary judicial decisions, and that, by memorializing his wisdom and justice, its purpose may have been the self-glorification of Hammurabi rather than a modern legal code or constitution. However, its copying in subsequent generations indicates that it was used as a model of legal and judicial reasoning.

While the Code of Hammurabi was trying to achieve equality, biases still existed against those categorized in the lower end of the social spectrum and some of the punishments and justice could be gruesome. The magnitude of criminal penalties often was based on the identity and gender of both the person committing the crime and the victim. The Code issues justice following the three classes of Babylonian society: property owners, freed men, and slaves.

Punishments for someone assaulting someone from a lower class were far lighter than if they had assaulted someone of equal or higher status. For example, if a doctor killed a rich patient, he would have his hands cut off, but if he killed a slave, only financial restitution was required. Women could also receive punishments that their male counterparts would not, as men were permitted to have affairs with their servants and slaves, whereas married women would be harshly punished for committing adultery.

Other copies

The Hammurabi stele at the American Museum of Natural History, New York.

 

A version of the code at the Istanbul Archaeological Museums.

 

Various copies of portions of the Code of Hammurabi have been found on baked clay tablets, some possibly older than the celebrated basalt stele now in the Louvre. The Prologue of the Code of Hammurabi (the first 305 inscribed squares on the stele) is on such a tablet, also at the Louvre (Inv #AO 10237). Some gaps in the list of benefits bestowed on cities recently annexed by Hammurabi may imply that it is older than the famous stele (currently dated to the early 18th century BC). Likewise, the Museum of the Ancient Orient, part of the Istanbul Archaeology Museums, also has a "Code of Hammurabi" clay tablet, dated to 1790 BC (in Room 5, Inv # Ni 2358).

In July 2010, archaeologists reported that a fragmentary Akkadian cuneiform tablet was discovered at Tel Hazor, Israel, containing a c. 1700 BC text that was said to be partly parallel to portions of the Hammurabi code. The Hazor law code fragments are currently being prepared for publication by a team from the Hebrew University of Jerusalem.

Laws covered

Today, approximately 275 laws from Hammurabi’s Code are known. Each law is written in two parts: A specific situation or case is outlined, then a corresponding decision is given.

One of the best known laws from Hammurabi's code was:

Ex. Law #196: "If a man destroy the eye of another man, they shall destroy his eye. If one break a man's bone, they shall break his bone. If one destroy the eye of a freeman or break the bone of a freeman he shall pay one gold mina. If one destroy the eye of a man's slave or break a bone of a man's slave he shall pay one-half his price."

Hammurabi had many other punishments, as well. If a son strikes his father, his hands shall be hewn off. Translations vary.

The laws covered such subjects as:

Slander

Ex. Law #127: "If any one 'point the finger' at a sister of a god or the wife of any one, and can not prove it, this man shall be taken before the judges and his brow shall be marked (by cutting the skin, or perhaps hair)."

 

Fraud

Ex. Law #265: "If a herdsman, to whose care cattle or sheep have been entrusted, be guilty of fraud and make false returns of the natural increase, or sell them for money, then shall he be convicted and pay the owner ten times the loss."

 

Slavery and status of slaves as property

Ex. Law #15: "If any one take a male or female slave of the court, or a male or female slave of a freed man, outside the city gates, he shall be put to death."

 

The duties of workers

Ex. Law #42: "If any one take over a field to till it, and obtain no harvest therefrom, it must be proved that he did no work on the field, and he must deliver grain, just as his neighbor raised, to the owner of the field."

 

Theft

Ex. Law #22: "If any one is committing a robbery and is caught, then he shall be put to death."

 

Trade

Ex. Law #104: "If a merchant give an agent grain, wool, oil, or any other goods to transport, the agent shall give a receipt for the amount, and compensate the merchant therefore, he shall obtain a receipt from the merchant for the money that he gives the merchant."

 

Liability

Ex. Law #53: "If any one be too apathetic to keep his dam in primly condition, and does not so keep it; if then the dam break and all the fields be flooded, then shall he in whose dam the break occurred be sold for money, and the money shall replace the crops which he has caused to be ruined."

 

Divorce

Ex. Law #142: "If a woman quarrel with her husband, and say: "You are not congenial to me," the reasons for her prejudice must be presented. If she is guiltless, and there is no fault on her part, but he leaves and neglects her, then no guilt attaches to this woman, she shall take her dowry and go back to her father's house."

 

Adultery

Ex. Law #129: "If the wife of a man has been caught lying with another man, they shall bind them and throw them into the waters. If the owner of the wife would save his wife then in turn the king could save his servant."

Perjury

Ex. Law #3: "If a man has borne false witness in a trial, or has not established the statement that he has made, if that case be a capital trial, that man shall be put to death."

Crime

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Crime 

In ordinary language, a crime is an unlawful act punishable by a state or other authority. The term crime does not, in modern criminal law, have any simple and universally accepted definition, though statutory definitions have been provided for certain purposes. The most popular view is that crime is a category created by law; in other words, something is a crime if declared as such by the relevant and applicable law. One proposed definition is that a crime or offence (or criminal offence) is an act harmful not only to some individual but also to a community, society, or the state ("a public wrong"). Such acts are forbidden and punishable by law.

The notion that acts such as murder, rape, and theft are to be prohibited exists worldwide. What precisely is a criminal offence is defined by criminal law of each country. While many have a catalogue of crimes called the criminal code, in some common law countries no such comprehensive statute exists.

The state (government) has the power to severely restrict one's liberty for committing a crime. In modern societies, there are procedures to which investigations and trials must adhere. If found guilty, an offender may be sentenced to a form of reparation such as a community sentence, or, depending on the nature of their offence, to undergo imprisonment, life imprisonment or, in some jurisdictions, execution.

Usually, to be classified as a crime, the "act of doing something criminal" (actus reus) must – with certain exceptions – be accompanied by the "intention to do something criminal" (mens rea).

While every crime violates the law, not every violation of the law counts as a crime. Breaches of private law (torts and breaches of contract) are not automatically punished by the state, but can be enforced through civil procedure.

Overview

When informal relationships prove insufficient to establish and maintain a desired social order, a government or a state may impose more formalized or stricter systems of social control. With institutional and legal machinery at their disposal, agents of the state can compel populations to conform to codes and can opt to punish or attempt to reform those who do not conform.

Authorities employ various mechanisms to regulate (encouraging or discouraging) certain behaviors in general. Governing or administering agencies may for example codify rules into laws, police citizens and visitors to ensure that they comply with those laws, and implement other policies and practices that legislators or administrators have prescribed with the aim of discouraging or preventing crime. In addition, authorities provide remedies and sanctions, and collectively these constitute a criminal justice system. Legal sanctions vary widely in their severity; they may include (for example) incarceration of temporary character aimed at reforming the convict. Some jurisdictions have penal codes written to inflict permanent harsh punishments: legal mutilation, capital punishment, or life without parole.

Usually, a natural person perpetrates a crime, but legal persons may also commit crimes. Historically, several premodern societies believed that non-human animals were capable of committing crimes, and prosecuted and punished them accordingly.

The sociologist Richard Quinney has written about the relationship between society and crime. When Quinney states "crime is a social phenomenon" he envisages both how individuals conceive crime and how populations perceive it, based on societal norms.

Etymology

The word crime is derived from the Latin root cernō, meaning "I decide, I give judgment". Originally the Latin word crīmen meant "charge" or "cry of distress." The Ancient Greek word κρίμα, krima, from which the Latin cognate derives, typically referred to an intellectual mistake or an offense against the community, rather than a private or moral wrong.

In 13th century English crime meant "sinfulness", according to the Online Etymology Dictionary. It was probably brought to England as Old French crimne (12th century form of Modern French crime), from Latin crimen (in the genitive case: criminis). In Latin, crimen could have signified any one of the following: "charge, indictment, accusation; crime, fault, offense".

The word may derive from the Latin cernere – "to decide, to sift" (see crisis, mapped on Kairos and Chronos). But Ernest Klein (citing Karl Brugmann) rejects this and suggests *cri-men, which originally would have meant "cry of distress". Thomas G. Tucker suggests a root in "cry" words and refers to English plaint, plaintiff, and so on. The meaning "offense punishable by law" dates from the late 14th century. The Latin word is glossed in Old English by facen, also "deceit, fraud, treachery", [cf. fake]. Crime wave is first attested in 1893 in American English.

Definition

England and Wales

Whether a given act or omission constitutes a crime does not depend on the nature of that act or omission. It depends on the nature of the legal consequences that may follow it. An act or omission is a crime if it is capable of being followed by what are called criminal proceedings.

History
The following definition of crime was provided by the Prevention of Crimes Act 1871, and applied for the purposes of section 10 of the Prevention of Crime Act 1908:

The expression "crime" means, in England and Ireland, any felony or the offence of uttering false or counterfeit coin, or of possessing counterfeit gold or silver coin, or the offence of obtaining goods or money by false pretences, or the offence of conspiracy to defraud, or any misdemeanour under the fifty-eighth section of the Larceny Act, 1861.

Scotland

For the purpose of section 243 of the Trade Union and Labour Relations (Consolidation) Act 1992, a crime means an offence punishable on indictment, or an offence punishable on summary conviction, and for the commission of which the offender is liable under the statute making the offence punishable to be imprisoned either absolutely or at the discretion of the court as an alternative for some other punishment.

Sociology

A normative definition views crime as deviant behavior that violates prevailing norms – cultural standards prescribing how humans ought to behave normally. This approach considers the complex realities surrounding the concept of crime and seeks to understand how changing social, political, psychological, and economic conditions may affect changing definitions of crime and the form of the legal, law-enforcement, and penal responses made by society.

These structural realities remain fluid and often contentious. For example: as cultures change and the political environment shifts, societies may criminalise or decriminalise certain behaviours, which directly affects the statistical crime rates, influence the allocation of resources for the enforcement of laws, and (re-)influence the general public opinion.

Similarly, changes in the collection and/or calculation of data on crime may affect the public perceptions of the extent of any given "crime problem". All such adjustments to crime statistics, allied with the experience of people in their everyday lives, shape attitudes on the extent to which the state should use law or social engineering to enforce or encourage any particular social norm. Behaviour can be controlled and influenced by a society in many ways without having to resort to the criminal justice system.

Indeed, in those cases where no clear consensus exists on a given norm, the drafting of criminal law by the group in power to prohibit the behaviour of another group may seem to some observers an improper limitation of the second group's freedom, and the ordinary members of society have less respect for the law or laws in general – whether the authorities actually enforce the disputed law or not.

Other definitions

Legislatures can pass laws (called mala prohibita) that define crimes against social norms. These laws vary from time to time and from place to place: note variations in gambling laws, for example, and the prohibition or encouragement of duelling in history. Other crimes, called mala in se, count as outlawed in almost all societies, (murder, theft and rape, for example).

English criminal law and the related criminal law of Commonwealth countries can define offences that the courts alone have developed over the years, without any actual legislation: common law offences. The courts used the concept of malum in se to develop various common law offences.

Criminalization

The spiked heads of executed criminals once adorned the gatehouse of the medieval London Bridge.

 

One can view criminalization as a procedure deployed by society as a preemptive harm-reduction device, using the threat of punishment as a deterrent to anyone proposing to engage in the behavior causing harm. The state becomes involved because governing entities can become convinced that the costs of not criminalizing (through allowing the harms to continue unabated) outweigh the costs of criminalizing it (restricting individual liberty, for example, to minimize harm to others).

States control the process of criminalization because:

• Even if victims recognize their own role as victims, they may not have the resources to investigate and seek legal redress for the injuries suffered: the enforcers formally appointed by the state often have better access to expertise and resources.
• The victims may only want compensation for the injuries suffered, while remaining indifferent to a possible desire for deterrence.
• Fear of retaliation may deter victims or witnesses of crimes from taking any action. Even in policed societies, fear may inhibit from reporting incidents or from co-operating in a trial.
• Victims, on their own, may lack the economies of scale that could allow them to administer a penal system, let alone to collect any fines levied by a court. Garoupa and Klerman (2002) warn that a rent-seeking government has as its primary motivation to maximize revenue and so, if offenders have sufficient wealth, a rent-seeking government will act more aggressively than a social-welfare-maximizing government in enforcing laws against minor crimes (usually with a fixed penalty such as parking and routine traffic violations), but more laxly in enforcing laws against major crimes.
• As a result of the crime, victims may die or become incapacitated.

Labelling theory

The label of "crime" and the accompanying social stigma normally confine their scope to those activities seen as injurious to the general population or to the state, including some that cause serious loss or damage to individuals. Those who apply the labels of "crime" or "criminal" intend to assert the hegemony of a dominant population, or to reflect a consensus of condemnation for the identified behavior and to justify any punishments prescribed by the state (in the event that standard processing tries and convicts an accused person of a crime).

Natural-law theory

Justifying the state's use of force to coerce compliance with its laws has proven a consistent theoretical problem. One of the earliest justifications involved the theory of natural law. This posits that the nature of the world or of human beings underlies the standards of morality or constructs them. Thomas Aquinas wrote in the 13th century: "the rule and measure of human acts is the reason, which is the first principle of human acts". He regarded people as by nature rational beings, concluding that it becomes morally appropriate that they should behave in a way that conforms to their rational nature. Thus, to be valid, any law must conform to natural law and coercing people to conform to that law is morally acceptable. In the 1760s, William Blackstone described the thesis:

"This law of nature, being co-eval with mankind and dictated by God himself, is of course superior in obligation to any other. It is binding over all the globe, in all countries, and at all times: no human laws are of any validity, if contrary to this; and such of them as are valid derive all their force, and all their authority, mediately or immediately, from this original."

But John Austin (1790–1859), an early positivist, applied utilitarianism in accepting the calculating nature of human beings and the existence of an objective morality. He denied that the legal validity of a norm depends on whether its content conforms to morality. Thus in Austinian terms, a moral code can objectively determine what people ought to do, the law can embody whatever norms the legislature decrees to achieve social utility, but every individual remains free to choose what to do. Similarly, H.L.A. Hart saw the law as an aspect of sovereignty, with lawmakers able to adopt any law as a means to a moral end.

Thus the necessary and sufficient conditions for the truth of a proposition of law simply involved internal logic and consistency, and that the state's agents used state power with responsibility. Ronald Dworkin rejects Hart's theory and proposes that all individuals should expect the equal respect and concern of those who govern them as a fundamental political right. He offers a theory of compliance overlaid by a theory of deference (the citizen's duty to obey the law) and a theory of enforcement, which identifies the legitimate goals of enforcement and punishment. Legislation must conform to a theory of legitimacy, which describes the circumstances under which a particular person or group is entitled to make law, and a theory of legislative justice, which describes the law they are entitled or obliged to make.

There are natural-law theorists who have accepted the idea of enforcing the prevailing morality as a primary function of the law. This view entails the problem that it makes any moral criticism of the law impossible: if conformity with natural law forms a necessary condition for legal validity, all valid law must, by definition, count as morally just. Thus, on this line of reasoning, the legal validity of a norm necessarily entails its moral justice.

One can solve this problem by granting some degree of moral relativism and accepting that norms may evolve over time and, therefore, one can criticize the continued enforcement of old laws in the light of the current norms. People may find such law acceptable, but the use of state power to coerce citizens to comply with that law lacks moral justification. More recent conceptions of the theory characterise crime as the violation of individual rights. 

Since society considers so many rights as natural (hence the term right) rather than man-made, what constitutes a crime also counts as natural, in contrast to laws (seen as man-made). Adam Smith illustrates this view, saying that a smuggler would be an excellent citizen, "...had not the laws of his country made that a crime which nature never meant to be so." 

Natural-law theory therefore distinguishes between "criminality" (which derives from human nature) and "illegality" (which originates with the interests of those in power). Lawyers sometimes express the two concepts with the phrases malum in se and malum prohibitum respectively. They regard a "crime malum in se" as inherently criminal; whereas a "crime malum prohibitum" (the argument goes) counts as criminal only because the law has decreed it so.

It follows from this view that one can perform an illegal act without committing a crime, while a criminal act could be perfectly legal. Many Enlightenment thinkers (such as Adam Smith and the American Founding Fathers) subscribed to this view to some extent, and it remains influential among so-called classical liberals and libertarians.

History

Some religious communities regard sin as a crime; some may even highlight the crime of sin very early in legendary or mythological accounts of origins – note the tale of Adam and Eve and the theory of original sin. What one group considers a crime may cause or ignite war or conflict. However, the earliest known civilizations had codes of law, containing both civil and penal rules mixed together, though not always in recorded form.

Ancient Near East

The Sumerians produced the earliest surviving written codes. Urukagina (reigned c. 2380 BC – c. 2360 BC, short chronology) had an early code that has not survived; a later king, Ur-Nammu, left the earliest extant written law system, the Code of Ur-Nammu (c. 2100 – c. 2050 BC), which prescribed a formal system of penalties for specific cases in 57 articles. The Sumerians later issued other codes, including the "code of Lipit-Ishtar". This code, from the 20th century BCE, contains some fifty articles, and scholars have reconstructed it by comparing several sources.

The Sumerian was deeply conscious of his personal rights and resented any encroachment on them, whether by his King, his superior, or his equal. No wonder that the Sumerians were the first to compile laws and law codes.

— Kramer

Successive legal codes in Babylon, including the code of Hammurabi (c. 1790 BC), reflected Mesopotamian society's belief that law derived from the will of the gods (see Babylonian law). Many states at this time functioned as theocracies, with codes of conduct largely religious in origin or reference. In the Sanskrit texts of Dharmaśāstra (c. 1250 BC), issues such as legal and religious duties, code of conduct, penalties and remedies, etc. have been discussed and forms one of the elaborate and earliest source of legal code.

Sir Henry Maine studied the ancient codes available in his day, and failed to find any criminal law in the "modern" sense of the word. While modern systems distinguish between offences against the "state" or "community", and offences against the "individual", the so-called penal law of ancient communities did not deal with "crimes" (Latin: crimina), but with "wrongs" (Latin: delicta). Thus the Hellenic laws treated all forms of theft, assault, rape, and murder as private wrongs, and left action for enforcement up to the victims or their survivors. The earliest systems seem to have lacked formal courts.

Rome and its Legacy in Europe

The Romans systematized law and applied their system across the Roman Empire. Again, the initial rules of Roman law regarded assaults as a matter of private compensation. The most significant Roman law concept involved dominion. The pater familias owned all the family and its property (including slaves); the pater enforced matters involving interference with any property. The Commentaries of Gaius (written between 130 and 180 AD) on the Twelve Tables treated furtum (in modern parlance: "theft") as a tort. 

Similarly, assault and violent robbery involved trespass as to the pater's property (so, for example, the rape of a slave could become the subject of compensation to the pater as having trespassed on his "property"), and breach of such laws created a vinculum juris (an obligation of law) that only the payment of monetary compensation (modern "damages") could discharge. Similarly, the consolidated Teutonic laws of the Germanic tribes, included a complex system of monetary compensations for what courts would now consider the complete range of criminal offences against the person, from murder down.

Even though Rome abandoned its Britannic provinces around 400 AD, the Germanic mercenaries – who had largely become instrumental in enforcing Roman rule in Britannia – acquired ownership of land there and continued to use a mixture of Roman and Teutonic Law, with much written down under the early Anglo-Saxon kings. But only when a more centralized English monarchy emerged following the Norman invasion, and when the kings of England attempted to assert power over the land and its peoples, did the modern concept emerge, namely of a crime not only as an offence against the "individual", but also as a wrong against the "state".

This idea came from common law, and the earliest conception of a criminal act involved events of such major significance that the "state" had to usurp the usual functions of the civil tribunals, and direct a special law or privilegium against the perpetrator. All the earliest English criminal trials involved wholly extraordinary and arbitrary courts without any settled law to apply, whereas the civil (delictual) law operated in a highly developed and consistent manner (except where a king wanted to raise money by selling a new form of writ). The development of the idea that the "state" dispenses justice in a court only emerges in parallel with or after the emergence of the concept of sovereignty.

In continental Europe, Roman law persisted, but with a stronger influence from the Christian Church. Coupled with the more diffuse political structure based on smaller feudal units, various legal traditions emerged, remaining more strongly rooted in Roman jurisprudence, but modified to meet the prevailing political climate.

In Scandinavia the effect of Roman law did not become apparent until the 17th century, and the courts grew out of the things – the assemblies of the people. The people decided the cases (usually with largest freeholders dominating). This system later gradually developed into a system with a royal judge nominating a number of the most esteemed men of the parish as his board, fulfilling the function of "the people" of yore. 

From the Hellenic system onwards, the policy rationale for requiring the payment of monetary compensation for wrongs committed has involved the avoidance of feuding between clans and families. If compensation could mollify families' feelings, this would help to keep the peace. On the other hand, the institution of oaths also played down the threat of feudal warfare. Both in archaic Greece and in medieval Scandinavia, an accused person walked free if he could get a sufficient number of male relatives to swear him not guilty. (Compare the United Nations Security Council, in which the veto power of the permanent members ensures that the organization does not become involved in crises where it could not enforce its decisions.)

These means of restraining private feuds did not always work, and sometimes prevented the fulfillment of justice. But in the earliest times the "state" did not always provide an independent policing force. Thus criminal law grew out of what 21st-century lawyers would call torts; and, in real terms, many acts and omissions classified as crimes actually overlap with civil-law concepts.

The development of sociological thought from the 19th century onwards prompted some fresh views on crime and criminality, and fostered the beginnings of criminology as a study of crime in society. Nietzsche noted a link between crime and creativity – in The Birth of Tragedy he asserted: "The best and brightest that man can acquire he must obtain by crime". In the 20th century, Michel Foucault in Discipline and Punish made a study of criminalization as a coercive method of state control.

Classification and categorisation

Categorisation by type

The following classes of offences are used, or have been used, as legal terms:

• Offence against the person
• Violent offence
• Sexual offence
• Offence against property

Researchers and commentators have classified crimes into the following categories, in addition to those above:

• Forgery, personation and cheating
• Firearms and offensive weapons
• Offences against the state/offences against the Crown and Government, or political offences
• Harmful or dangerous drugs
• Offences against religion and public worship
• Offences against public justice, or offences against the administration of public justice
• Public order offence
• Commerce, financial markets and insolvency
• Offences against public morals and public policy
• Motor vehicle offences
• Conspiracy, incitement and attempt to commit crime
• Inchoate offence
• Juvenile delinquency
• Victimless crime

Categorisation by penalty

One can categorise crimes depending on the related punishment, with sentencing tariffs prescribed in line with the perceived seriousness of the offence. Thus fines and noncustodial sentences may address the crimes seen as least serious, with lengthy imprisonment or (in some jurisdictions) capital punishment reserved for the most serious.

Common law

Under the common law of England, crimes were classified as either treason, felony or misdemeanour, with treason sometimes being included with the felonies. This system was based on the perceived seriousness of the offence. It is still used in the United States but the distinction between felony and misdemeanour is abolished in England, Wales and Northern Ireland.

Classification by mode of trial

The following classes of offence are based on mode of trial:

• Indictable-only offence
• Indictable offence
• Hybrid offence, a.k.a. either-way offence in England and Wales
• Summary offence, a.k.a. infraction in the US

Classification by origin

In common law countries, crimes may be categorised into common law offences and statutory offences. In the US, Australia and Canada (in particular), they are divided into federal crimes and under state crimes.

Other classifications

• Arrestable offence

U.S. classification

Felony Sentences in State Courts, study by the United States Department of Justice.

In the United States since 1930, the FBI has tabulated Uniform Crime Reports (UCR) annually from crime data submitted by law enforcement agencies across the United States. Officials compile this data at the city, county, and state levels into the UCR. They classify violations of laws based on common law as Part I (index) crimes in UCR data. These are further categorized as violent or property crimes. Part I violent crimes include murder and criminal homicide (voluntary manslaughter), forcible rape, aggravated assault, and robbery; while Part I property crimes include burglary, arson, larceny/theft, and motor-vehicle theft. All other crimes count come under Part II.

For convenience, such lists usually include infractions although, in the U.S., they may come into the sphere not of the criminal law, but rather of the civil law. Compare tortfeasance.

Booking arrests require detention for a time-frame ranging 1 to 24 hours.

Reports, studies and organizations

There are several national and International organizations offering studies and statistics about global and local crime activity, such as United Nations Office on Drugs and Crime, the United States of America Overseas Security Advisory Council (OSAC) safety report or national reports generated by the law-enforcement authorities of EU state member reported to the Europol.

Offence in common law jurisdictions

In England and Wales, as well as in Hong Kong, the term offence means the same thing as, and is interchangeable with, the term crime, They are further split into:

• Summary offences
• Indictable offences

Causes and correlates of crime

Many different causes and correlates of crime have been proposed with varying degree of empirical support. They include socioeconomic, psychological, biological, and behavioral factors. Controversial topics include media violence research and effects of gun politics. 

Emotional state (both chronic and current) have a tremendous impact on individual thought processes and, as a result, can be linked to criminal activities. The positive psychology concept of Broaden and Build posits that cognitive functioning expands when an individual is in a good-feeling emotional state and contracts as emotional state declines. In positive emotional states an individual is able to consider more possible solutions to problems, but in lower emotional states fewer solutions can be ascertained. The narrowed thought-action repertoires can result in the only paths perceptible to an individual being ones they would never use if they saw an alternative, but if they can't conceive of the alternatives that carry less risk they will choose one that they can see. Criminals who commit even the most horrendous of crimes, such as mass murders, did not see another solution.

Crimes in international law

Kang Kek Iew before the Cambodian Genocide Tribunal on July 20, 2009

 

Crimes defined by treaty as crimes against international law include:

• Crimes against peace
• Crimes of apartheid
• Forced disappearance
• Genocide
• Incitement to genocide
• Piracy
• Sexual slavery
• Slavery
• Waging a war of aggression
• War crimes

From the point of view of state-centric law, extraordinary procedures (international courts or national courts operating with universal jurisdiction) may prosecute such crimes. Note the role of the International Criminal Court at The Hague in the Netherlands.

Religion and crime

Religious sentiment often becomes a contributory factor of crime. In the 1819 anti-Jewish riots in Frankfurt, rioters attacked Jewish businesses and destroyed property.

 

Different religious traditions may promote distinct norms of behaviour, and these in turn may clash or harmonise with the perceived interests of a state. Socially accepted or imposed religious morality has influenced secular jurisdictions on issues that may otherwise concern only an individual's conscience. Activities sometimes criminalized on religious grounds include (for example) alcohol consumption (prohibition), abortion and stem-cell research. In various historical and present-day societies, institutionalized religions have established systems of earthly justice that punish crimes against the divine will and against specific devotional, organizational and other rules under specific codes, such as Roman Catholic canon law.

Military jurisdictions and states of emergency

In the military sphere, authorities can prosecute both regular crimes and specific acts (such as mutiny or desertion) under martial-law codes that either supplant or extend civil codes in times of (for example) war.

Many constitutions contain provisions to curtail freedoms and criminalize otherwise tolerated behaviors under a state of emergency in the event of war, natural disaster or civil unrest. Undesired activities at such times may include assembly in the streets, violation of curfew, or possession of firearms.

Employee crime

Two common types of employee crime exist: embezzlement and wage theft.

The complexity and anonymity of computer systems may help criminal employees camouflage their operations. The victims of the most costly scams include banks, brokerage houses, insurance companies, and other large financial institutions.

In the United States, it is estimated that workers are not paid at least $19 billion every year in overtime and that in total $40 billion to $60 billion are lost annually due to all forms of wage theft. This compares to national annual losses of $340 million due to robbery, $4.1 billion due to burglary, $5.3 billion due to larceny, and $3.8 billion due to auto theft in 2012. In Singapore, as in the United States, wage theft was found to be widespread and severe. In a 2014 survey it was found that as many as one-third of low wage male foreign workers in Singapore, or about 130,000, were affected by wage theft from partial to full denial of pay.