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124-759: It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and the logician William Alvin Howard . It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer , Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation ) and Stephen Kleene (see Realizability ). The relationship has been extended to include category theory as

248-741: A monad is a structure that combines program fragments ( functions ) and wraps their return values in a type with additional computation. In addition to defining a wrapping monadic type , monads define two operators : one to wrap a value in the monad type, and another to compose together functions that output values of the monad type (these are known as monadic functions ). General-purpose languages use monads to reduce boilerplate code needed for common operations (such as dealing with undefined values or fallible functions, or encapsulating bookkeeping code). Functional languages use monads to turn complicated sequences of functions into succinct pipelines that abstract away control flow , and side-effects . Both

372-428: A proof system and as a typed functional programming language . This includes Martin-Löf 's intuitionistic type theory and Coquand 's Calculus of Constructions , two calculi in which proofs are regular objects of the discourse and in which one can state properties of proofs the same way as of any program. This field of research is usually referred to as modern type theory . Such typed lambda calculi derived from

496-417: A "product" type (this may be called a tuple, a struct, a list, or some other term depending on the language), disjunction as a sum type (this type may be called a union), the false formula as the empty type and the true formula as the unit type (whose sole member is the null object). Quantifiers correspond to dependent function space or products (as appropriate). This is summarized in the following table: At

620-557: A "strongly normalizing" cartesian closed category. To clarify this distinction, the underlying syntactic structure of cartesian closed categories is rephrased below. Objects (propositions/types) include Morphisms (deductions/terms) include Equivalently to the annotations above, well-defined morphisms (typed terms) in any cartesian-closed category can be constructed according to the following typing rules . The usual categorical morphism notation f : α → β {\displaystyle f:\alpha \to \beta }

744-460: A "wrapping" type, and this is where its connection to monads comes in. In languages with some form of the Maybe type, there are functions that aid in their use such as composing monadic functions with each other and testing if a Maybe contains a value. In the following hard-coded example, a Maybe type is used as a result of functions that may fail, in this case the type returns nothing if there

868-504: A Hilbert-style logic are given below . If one restricts to the implicational intuitionistic fragment, a simple way to formalize logic in Hilbert's style is as follows. Let Γ be a finite collection of formulas, considered as hypotheses. Then δ is derivable from Γ, denoted Γ ⊢ δ, in the following cases: This can be formalized using inference rules , as in the left column of the following table. Typed combinatory logic can be formulated using

992-421: A cartesian-closed category can be interpreted as propositions (types), and morphisms as deductions mapping a set of assumptions ( typing context ) to a valid consequent (well-typed term). Lambek's correspondence is a correspondence of equational theories, abstracting away from dynamics of computation such as beta reduction and term normalization, and is not the expression of a syntactic identity of structures as it

1116-463: A category (called the Kleisli category ) and a monoid in the category of functors (from values to computations), with monadic composition as a binary operator in the monoid and unit as identity in the monoid. The value of the monad pattern goes beyond merely condensing code and providing a link to mathematical reasoning. Whatever language or default programming paradigm a developer uses, following

1240-416: A child may spontaneously engage in comparison and learn an abstract relationship, without the need for prompts. Comparison is more likely when the objects to be compared are close together in space and/or time, are highly similar (although not so similar that they match, which interfere with identifying relationships), or share common labels. In law , analogy is a method of resolving issues on which there

1364-515: A circuit. In a circuit, the electricity is carried through wires and the current, or rate of flow of electricity, is determined by the voltage, or electrical pressure. Given the similarity in structure, or structural alignment, between these domains, structure mapping theory would predict that relationships from one of these domains, would be inferred in the other using analogy. Children do not always need prompting to make comparisons in order to learn abstract relationships. Eventually, children undergo

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1488-419: A conclusion about the analogy and comparing the new material with the already learned material. Typically this method is used to learn topics in science. In 1989, teacher Kerry Ruef began a program titled The Private Eye Project . It is a method of teaching that revolves around using analogies in the classroom to better explain topics. She thought of the idea to use analogies as a part of curriculum because she

1612-457: A distinct monad. If a language does not support monads by default, it is still possible to implement the pattern, often without much difficulty. When translated from category-theory to programming terms, the monad structure is a generic concept and can be defined directly in any language that supports an equivalent feature for bounded polymorphism . A concept's ability to remain agnostic about operational details while working on underlying types

1736-432: A domain is viewed as consisting of objects, their properties, and the relationships that characterise their interactions. The process of analogy then involves: In general, it has been found that people prefer analogies where the two systems correspond highly to each other (e.g. have similar relationships across the domains as opposed to just having similar objects across domains) when these people try to compare and contrast

1860-457: A few applications that have monads at the heart of their designs: The term "monad" in programming dates to the APL and J programming languages, which do tend toward being purely functional. However, in those languages, "monad" is only shorthand for a function taking one parameter (a function with two parameters being a "dyad", and so on). The mathematician Roger Godement was the first to formulate

1984-490: A fully-featured monad. In most languages, the Maybe monad is also known as an option type , which is just a type that marks whether or not it contains a value. Typically they are expressed as some kind of enumerated type . In this Rust example we will call it Maybe<T> and variants of this type can either be a value of generic type T , or the empty variant: Nothing . Maybe<T> can also be understood as

2108-504: A functor f from C to D can be thought of as an analogy between C and D, because f has to map objects of C to objects of D and arrows of C to arrows of D in such a way that the structure of their respective parts is preserved. This is similar to the structure mapping theory of analogy of Dedre Gentner, because it formalises the idea of analogy as a function which makes certain conditions true. A computer algorithm has achieved human-level performance on multiple-choice analogy questions from

2232-594: A given program execution so that this evaluation context can be later on reinstalled. The basic Curry–Howard-style correspondence for classical logic is given below. Note the correspondence between the double-negation translation used to map classical proofs to intuitionistic logic and the continuation-passing-style translation used to map lambda terms involving control to pure lambda terms. More particularly, call-by-name continuation-passing-style translations relates to Kolmogorov 's double negation translation and call-by-value continuation-passing-style translations relates to

2356-447: A kind of computation, they can also be used to implement convenient language features. Some languages, such as Haskell , even offer pre-built definitions in their core libraries for the general monad structure and common instances. "For a monad m , a value of type m a represents having access to a value of type a within the context of the monad." —C. A. McCann More exactly, a monad can be used where unrestricted access to

2480-446: A kind of double-negation translation due to Kuroda. A finer Curry–Howard correspondence exists for classical logic if one defines classical logic not by adding an axiom such as Peirce's law , but by allowing several conclusions in sequents. In the case of classical natural deduction, there exists a proofs-as-programs correspondence with the typed programs of Parigot's λμ-calculus . A proofs-as-programs correspondence can be settled for

2604-488: A logical point of view is still an actively debated research question, but one popular approach is based on using monads to segregate provably terminating from potentially non-terminating code (an approach that also generalizes to much richer models of computation, and is itself related to modal logic by a natural extension of the Curry–Howard isomorphism). A more radical approach, advocated by total functional programming ,

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2728-412: A monad and a basic functor. In the applicative context, unit is sometimes referred to as pure but is still the same function. What does differ in this construction is the law unit must satisfy; as bind is not defined, the constraint is given in terms of map instead: The final leap from applicative functor to monad comes with the second transformation, the join function (in category theory this

2852-478: A monad from a simpler functor can come in handy. In many languages, a list structure comes pre-defined along with some basic features, so a List type constructor and append operator (represented with ++ for infix notation) are assumed as already given here. Embedding a plain value in a list is also trivial in most languages: From here, applying a function iteratively with a list comprehension may seem like an easy choice for bind and converting lists to

2976-506: A monad is derived from a pre-existing functor, whereupon the monad inherits map automatically. (For historical reasons, this map is instead called fmap in Haskell.) A monad's first transformation is actually the same unit from the Kleisli triple, but following the hierarchy of structures closely, it turns out unit characterizes an applicative functor , an intermediate structure between

3100-428: A more elegant form similar to function composition . With >>= available, chainable_division can be expressed much more succinctly with the help of anonymous functions (i.e. lambdas). Notice in the expression below how the two nested lambdas each operate on the wrapped value in the passed Maybe monad using the bind operator. What has been shown so far is basically a monad, but to be more concise,

3224-409: A non-exhaustive list: At the time of Curry, and also at the time of Howard, the proofs-as-programs correspondence concerned only intuitionistic logic , i.e. a logic in which, in particular, Peirce's law was not deducible. The extension of the correspondence to Peirce's law and hence to classical logic became clear from the work of Griffin on typing operators that capture the evaluation context of

3348-444: A particular order. One example of a monad is the Maybe type. Undefined null results are one particular pain point that many procedural languages don't provide specific tools for dealing with, requiring use of the null object pattern or checks to test for invalid values at each operation to handle undefined values. This causes bugs and makes it harder to build robust software that gracefully handles errors. The Maybe type forces

3472-420: A program that, given values with the types listed in Γ, manufactures an object of type α . An axiom/hypothesis corresponds to the introduction of a new variable with a new, unconstrained type, the →  I rule corresponds to function abstraction and the →  E rule corresponds to function application. Observe that the correspondence is not exact if the context Γ is taken to be a set of formulas as, e.g.,

3596-497: A relational shift, after which they begin seeing similar relations across different situations instead of merely looking at matching objects. This is critical in their cognitive development as continuing to focus on specific objects would reduce children's ability to learn abstract patterns and reason analogically. Interestingly, some researchers have proposed that children's basic brain functions (i.e., working memory and inhibitory control) do not drive this relational shift. Instead, it

3720-548: A renewed interest in analogy, most notably in cognitive science . Cajetan named several kinds of analogy that had been used but previously unnamed, particularly: In ancient Greek the word αναλογια ( analogia ) originally meant proportionality , in the mathematical sense, and it was indeed sometimes translated to Latin as proportio . Analogy was understood as identity of relation between any two ordered pairs , whether of mathematical nature or not. Analogy and abstraction are different cognitive processes, and analogy

3844-518: A sequence, which has led some to describe monads as "programmable semicolons", a reference to how many imperative languages use semicolons to separate statements . However, monads do not actually order computations; even in languages that use them as central features, simpler function composition can arrange steps within a program. A monad's general utility rather lies in simplifying a program's structure and improving separation of concerns through abstraction. The monad structure can also be seen as

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3968-729: A significant role in human thought processes. It has been argued that analogy lies at "the core of cognition". The English word analogy derives from the Latin analogia , itself derived from the Greek ἀναλογία , "proportion", from ana- "upon, according to" [also "again", "anew"] + logos "ratio" [also "word, speech, reckoning"]. Analogy plays a significant role in problem solving , as well as decision making , argumentation , perception , generalization , memory , creativity , invention , prediction, emotion , explanation , conceptualization and communication . It lies behind basic tasks such as

4092-506: A similar syntax: let Γ be a finite collection of variables, annotated with their types. A term T (also annotated with its type) will depend on these variables [Γ ⊢ T: δ ] when: The generation rules defined here are given in the right-column below. Curry's remark simply states that both columns are in one-to-one correspondence. The restriction of the correspondence to intuitionistic logic means that some classical tautologies , such as Peirce's law (( α → β ) → α ) → α , are excluded from

4216-461: A special case of induction . In their view analogy is an inductive inference from common known attributes to another probable common attribute, which is known about only in the source of the analogy, in the following form: Contemporary cognitive scientists use a wide notion of analogy, extensionally close to that of Plato and Aristotle, but framed by Gentner's (1983) structure-mapping theory . The same idea of mapping between source and target

4340-419: A teacher may refer to other concepts, items or phenomena that pupils are more familiar with. It may help to create or clarify one theory (or theoretical model) via the workings of another theory (or theoretical model). Thus an analogy, as used in teaching, would be comparing a topic that students are already familiar with, with a new topic that is being introduced, so that students can get a better understanding of

4464-416: A third element that they are considered to share. In logic, it is an inference or an argument from one particular to another particular, as opposed to deduction , induction , and abduction . It is also used of where at least one of the premises , or the conclusion, is general rather than particular in nature. It has the general form A is to B as C is to D . In a broader sense, analogical reasoning

4588-576: A tool for aspect-oriented programming . One other noteworthy use for monads is isolating side-effects, like input/output or mutable state , in otherwise purely functional code. Even purely functional languages can still implement these "impure" computations without monads, via an intricate mix of function composition and continuation-passing style (CPS) in particular. With monads though, much of this scaffolding can be abstracted away, essentially by taking each recurring pattern in CPS code and bundling it into

4712-423: A uniquely mathematical and compile time variation on the decorator pattern . Some monads can pass along extra data that is inaccessible to functions, and some even exert finer control over execution, for example only calling a function under certain conditions. Because they let application programmers implement domain logic while offloading boilerplate code onto pre-developed modules, monads can even be considered

4836-456: A value is inappropriate for reasons specific to the scenario. In the case of the Maybe monad, it is because the value may not exist. In the case of the IO monad, it is because the value may not be known yet, such as when the monad represents user input that will only be provided after a prompt is displayed. In all cases the scenarios in which access makes sense are captured by the bind operation defined for

4960-485: A very important part in morality . This may be because morality is supposed to be impartial and fair. If it is wrong to do something in a situation A, and situation B corresponds to A in all related features, then it is also wrong to perform that action in situation B. Moral particularism accepts such reasoning, instead of deduction and induction, since only the first can be used regardless of any moral principles. Structure mapping, originally proposed by Dedre Gentner ,

5084-407: A way to canonically transform proofs into other proofs of the same statement. One can also transfer the notion of normal terms to a notion of normal proofs, expressing that the hypotheses of the axioms never need to be all detached (since otherwise a simplification can happen). Conversely, the non provability in intuitionistic logic of Peirce's law can be transferred back to combinatory logic: there

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5208-590: A wider notion of analogy. They saw analogy as a shared abstraction. Analogous objects did not share necessarily a relation, but also an idea, a pattern, a regularity, an attribute, an effect or a philosophy. These authors also accepted that comparisons, metaphors and "images" (allegories) could be used as arguments , and sometimes they called them analogies . Analogies should also make those abstractions easier to understand and give confidence to those who use them. James Francis Ross in Portraying Analogy (1982),

5332-420: Is a cognitive process of transferring some information or meaning of a particular subject (the analog, or source) onto another (the target); and also the linguistic expression corresponding to such a process. The term analogy can also refer to the relation between the source and the target themselves, which is often (though not always) a similarity , as in the biological notion of analogy . Analogy plays

5456-426: Is a divide-by-zero . One such way to test whether or not a Maybe contains a value is to use if statements. Other languages may have pattern matching Monads can compose functions that return Maybe , putting them together. A concrete example might have one function take in several Maybe parameters, and return a single Maybe whose value is Nothing when any of the parameters are Nothing , as in

5580-425: Is a natural transformation usually called μ ), which "flattens" nested applications of the monad: As the characteristic function, join must also satisfy three variations on the monad laws: Regardless of whether a developer defines a direct monad or a Kleisli triple, the underlying structure will be the same, and the forms can be derived from each other easily: The List monad naturally demonstrates how deriving

5704-462: Is a theory in psychology that describes the psychological processes involved in reasoning through, and learning from, analogies. More specifically, this theory aims to describe how familiar knowledge, or knowledge about a base domain, can be used to inform an individual's understanding of a less familiar idea, or a target domain. According to this theory, individuals view their knowledge of ideas, or domains, as interconnected structures. In other words,

5828-442: Is actually based on a Kleisli triple ⟨T, η, μ⟩ rather than category theory's standard definition. The two constructs turn out to be mathematically equivalent, however, so either definition will yield a valid monad. Given any well-defined basic types T and U , a monad consists of three parts: To fully qualify as a monad though, these three parts must also respect a few laws: Algebraically, this means any monad both gives rise to

5952-480: Is called a heuristic function of analogical reasoning. Analogical arguments can also be probative, meaning that they serve as a means of proving the rightness of particular theses and theories. This application of analogical reasoning in science is debatable. Analogy can help prove important theories, especially in those kinds of science in which logical or empirical proof is not possible such as theology , philosophy or cosmology when it relates to those areas of

6076-509: Is close to the one of some abstract machines . The informal correspondence is as follows: N. G. de Bruijn used the lambda notation for representing proofs of the theorem checker Automath , and represented propositions as "categories" of their proofs. It was in the late 1960s at the same period of time Howard wrote his manuscript; de Bruijn was likely unaware of Howard's work, and stated the correspondence independently (Sørensen & Urzyczyn [1998] 2006, pp 98–99). Some researchers tend to use

6200-409: Is driven by their relational knowledge, such as having labels for the objects that make the relationships clearer(see previous section). However, there is not enough evidence to determine whether the relational shift is actually because basic brain functions become better or relational knowledge becomes deeper. Additionally, research has identified several factors that may increase the likelihood that

6324-417: Is high-level perception. Forbus et al. (1998) claim that this is only a metaphor. It has been argued (Morrison and Dietrich 1995) that Hofstadter's and Gentner's groups do not defend opposite views, but are instead dealing with different aspects of analogy. In anatomy , two anatomical structures are considered to be analogous when they serve similar functions but are not evolutionarily related, such as

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6448-457: Is no clear line between perception , including high-level perception, and analogical thinking. In fact, analogy occurs not only after, but also before and at the same time as high-level perception. In high-level perception, humans make representations by selecting relevant information from low-level stimuli . Perception is necessary for analogy, but analogy is also necessary for high-level perception. Chalmers et al. concludes that analogy actually

6572-433: Is no previous authority. The legal use of analogy is distinguished by the need to use a legally relevant basis for drawing an analogy between two situations. It may be applied to various forms of legal authority , including statutory law and case law . In the civil law tradition, analogy is most typically used for filling gaps in a statutory scheme. In the common law tradition, it is most typically used for extending

6696-443: Is no typed term of combinatory logic that is typable with type Results on the completeness of some sets of combinators or axioms can also be transferred. For instance, the fact that the combinator X constitutes a one-point basis of (extensional) combinatory logic implies that the single axiom scheme which is the principal type of X , is an adequate replacement to the combination of the axiom schemes After Curry emphasized

6820-401: Is not apparent in some lexical definitions of palm and sole , where the former is defined as the inner surface of the hand , and the latter as the underside of the foot . Kant's Critique of Judgment held to this notion of analogy, arguing that there can be exactly the same relation between two completely different objects. Greek philosophers such as Plato and Aristotle used

6944-402: Is now used by thousands of schools around the country. The Fourth Lateran Council of 1215 taught: For between creator and creature there can be noted no similarity so great that a greater dissimilarity cannot be seen between them. The theological exploration of this subject is called the analogia entis . The consequence of this theory is that all true statements concerning God (excluding

7068-481: Is often an easier one. This analogy is not comparing all the properties between a hand and a foot, but rather comparing the relationship between a hand and its palm to a foot and its sole. While a hand and a foot have many dissimilarities, the analogy focuses on their similarity in having an inner surface. The same notion of analogy was used in the US -based SAT college admission tests, that included "analogy questions" in

7192-409: Is powerful, but the unique features and stringent behavior of monads set them apart from other concepts. Discussions of specific monads will typically focus on solving a narrow implementation problem since a given monad represents a specific computational form. In some situations though, an application can even meet its high-level goals by using appropriate monads within its core logic. Here are just

7316-462: Is replaced with typing context notation α ⊢ f : β {\displaystyle \alpha \vdash f:\beta } . Identity: Composition: Unit type ( terminal object ): Cartesian product: Left and right projection: Currying : Application : Finally, the equations of the category are Analogy Analogy is a comparison or correspondence between two things (or two groups of things) because of

7440-573: Is the analogue ear based on electrical, electronic or mechanical devices. Some types of analogies can have a precise mathematical formulation through the concept of isomorphism . In detail, this means that if two mathematical structures are of the same type, an analogy between them can be thought of as a bijection which preserves some or all of the relevant structure. For example, R 2 {\displaystyle \mathbb {R} ^{2}} and C {\displaystyle \mathbb {C} } are isomorphic as vector spaces, but

7564-400: Is the case for each of Curry's and Howard's correspondences: i.e. the structure of a well-defined morphism in a cartesian-closed category is not comparable to the structure of a proof of the corresponding judgment in either Hilbert-style logic or natural deduction. For example, it is not possible to state or prove that a morphism is normalizing, establish a Church-Rosser type theorem, or speak of

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7688-422: Is to and as when representing the analogous relationship between two pairs of expressions, for example, "Smile is to mouth, as wink is to eye." In the field of mathematics and logic, this can be formalized with colon notation to represent the relationships, using single colon for ratio, and double colon for equality. In the field of testing, the colon notation of ratios and equality is often borrowed, so that

7812-444: Is to eliminate unrestricted recursion (and forgo Turing completeness , although still retaining high computational complexity), using more controlled corecursion wherever non-terminating behavior is actually desired. In its more general formulation, the Curry–Howard correspondence is a correspondence between formal proof calculi and type systems for models of computation . In particular, it splits into two correspondences. One at

7936-507: Is unclear, even in the case of natural deduction. Joachim Lambek showed in the early 1970s that the proofs of intuitionistic propositional logic and the combinators of typed combinatory logic share a common equational theory, the theory of cartesian closed categories . The expression Curry–Howard–Lambek correspondence is now used by some people to refer to the relationships between intuitionistic logic, typed lambda calculus and cartesian closed categories. Under this correspondence, objects of

8060-400: Is used by conceptual metaphor and conceptual blending theorists. Structure mapping theory concerns both psychology and computer science . According to this view, analogy depends on the mapping or alignment of the elements of source and target. The mapping takes place not only between objects, but also between relations of objects and between relations of relations. The whole mapping yields

8184-449: The MONIAC (an analogue computer ) used the flow of water in its pipes as an analogue to the flow of money in an economy. Where two or more biological or physical participants meet, they communicate and the stresses produced describe internal models of the participants. Pask in his conversation theory asserts an analogy that describes both similarities and differences between any pair of

8308-669: The SAT test. The algorithm measures the similarity of relations between pairs of words (e.g., the similarity between the pairs HAND:PALM and FOOT:SOLE) by statistically analysing a large collection of text. It answers SAT questions by selecting the choice with the highest relational similarity. The analogical reasoning in the human mind is free of the false inferences plaguing conventional artificial intelligence models, (called systematicity ). Steven Phillips and William H. Wilson use category theory to mathematically demonstrate how such reasoning could arise naturally by using relationships between

8432-430: The complex numbers , C {\displaystyle \mathbb {C} } , have more structure than R 2 {\displaystyle \mathbb {R} ^{2}} does: C {\displaystyle \mathbb {C} } is a field as well as a vector space . Category theory takes the idea of mathematical analogy much further with the concept of functors . Given two categories C and D,

8556-466: The lambda calculus . Moggi's key insight was that a real-world program is not just a function from values to other values, but rather a transformation that forms computations on those values. When formalized in category-theoretic terms, this leads to the conclusion that monads are the structure to represent these computations. Several others popularized and built on this idea, including Philip Wadler and Simon Peyton Jones , both of whom were involved in

8680-450: The legs of vertebrates and the legs of insects . Analogous structures are the result of independent evolution and should be contrasted with structures which shared an evolutionary line. Often a physical prototype is built to model and represent some other physical object. For example, wind tunnels are used to test scale models of wings and aircraft which are analogous to (correspond to) full-size wings and aircraft. For example,

8804-519: The " coherence " of an analogy depends on structural consistency, semantic similarity and purpose. Structural consistency is the highest when the analogy is an isomorphism , although lower levels can be used as well. Similarity demands that the mapping connects similar elements and relationships between source and target, at any level of abstraction. It is the highest when there are identical relations and when connected elements have many identical attributes. An analogy achieves its purpose if it helps solve

8928-478: The 1980s, a vague notion of the monad pattern began to surface in the computer science community. According to programming language researcher Philip Wadler , computer scientist John C. Reynolds anticipated several facets of it in the 1970s and early 1980s, when he discussed the value of continuation-passing style , of category theory as a rich source for formal semantics, and of the type distinction between values and computations. The research language Opal , which

9052-437: The Curry–Howard isomorphism as it allows the deduction rules to be stated more cleanly) with implicit weakening and the right-hand side shows the typing rules of lambda calculus . In the left-hand side, Γ, Γ 1 and Γ 2 denote ordered sequences of formulas while in the right-hand side, they denote sequences of named (i.e., typed) formulas with all names different. To paraphrase the correspondence, proving Γ ⊢ α means having

9176-442: The Curry–Howard paradigm led to software like Coq in which proofs seen as programs can be formalized, checked, and run. A converse direction is to use a program to extract a proof , given its correctness —an area of research closely related to proof-carrying code . This is only feasible if the programming language the program is written for is very richly typed: the development of such type systems has been partly motivated by

9300-454: The University of Georgia, developed a theory on teaching with analogies and developed steps to explain the process of teaching with this method. The steps for teaching with analogies are as follows: Step one is introducing the new topic that is about to be taught and giving some general knowledge on the subject. Step two is reviewing the concept that the students already know to ensure they have

9424-457: The assignment of a predicate or a relation to the target. Structure mapping theory has been applied and has found considerable confirmation in psychology . It has had reasonable success in computer science and artificial intelligence (see below). Some studies extended the approach to specific subjects, such as metaphor and similarity. Logicians analyze how analogical reasoning is used in arguments from analogy . An analogy can be stated using

9548-400: The base analogue is selected and mapping from base to target occurs in series. Empirical evidence shows that humans are better at using and creating analogies when the information is presented in an order where an item and its analogue are placed together.. Eqaan Doug and his team challenged the shared structure theory and mostly its applications in computer science. They argue that there

9672-411: The composition of computations. Not only can the monad laws be used to check an instance's validity, but features from related structures (like functors) can be used through subtyping . Returning to the Maybe example, its components were declared to make up a monad, but no proof was given that it satisfies the monad laws. This can be rectified by plugging the specifics of Maybe into one side of

9796-424: The concept of a monad (dubbing it a "standard construction") in the late 1950s, though the term "monad" that came to dominate was popularized by category-theorist Saunders Mac Lane . The form defined above using bind , however, was originally described in 1965 by mathematician Heinrich Kleisli in order to prove that any monad could be characterized as an adjunction between two (covariant) functors. Starting in

9920-527: The concept of a monad and the term originally come from category theory , where a monad is defined as a functor with additional structure. Research beginning in the late 1980s and early 1990s established that monads could bring seemingly disparate computer-science problems under a unified, functional model. Category theory also provides a few formal requirements, known as the monad laws , which should be satisfied by any monad and can be used to verify monadic code. Since monads make semantics explicit for

10044-495: The concrete details of Jesus' earthly life) are rough analogies, without implying any falsehood. Such analogical and true statements would include God is , God is Love , God is a consuming fire , God is near to all who call him , or God as Trinity, where being , love , fire , distance , number must be classed as analogies that allow human cognition of what is infinitely beyond positive or negative language. Monad (functional programming) In functional programming ,

10168-508: The correspondence. Seen at a more abstract level, the correspondence can be restated as shown in the following table. Especially, the deduction theorem specific to Hilbert-style logic matches the process of abstraction elimination of combinatory logic. Thanks to the correspondence, results from combinatory logic can be transferred to Hilbert-style logic and vice versa. For instance, the notion of reduction of terms in combinatory logic can be transferred to Hilbert-style logic and it provides

10292-617: The cosmos (the universe) that are beyond any data-based observation and knowledge about them stems from the human insight and thinking outside the senses. Analogy can be used in theoretical and applied sciences in the form of models or simulations which can be considered as strong indications of probable correctness. Other, much weaker, analogies may also assist in understanding and describing nuanced or key functional behaviours of systems that are otherwise difficult to grasp or prove. For instance, an analogy used in physics textbooks compares electrical circuits to hydraulic circuits. Another example

10416-418: The duality between the two evaluation strategies known as call-by-name and call-by-value. Because of the possibility of writing non-terminating programs, Turing-complete models of computation (such as languages with arbitrary recursive functions ) must be interpreted with care, as naive application of the correspondence leads to an inconsistent logic. The best way of dealing with arbitrary computation from

10540-466: The example above might be rendered, "Smile : mouth :: wink : eye" and pronounced the same way. Analogy is also a term used in the Neogrammarian school of thought as a catch-all to describe any morphological change in a language that cannot be explained merely sound change or borrowing. Analogies are mainly used as a means of creating new ideas and hypotheses, or testing them, which

10664-406: The first substantive examination of the topic since Cajetan's De Nominum Analogia , demonstrated that analogy is a systematic and universal feature of natural languages, with identifiable and law-like characteristics which explain how the meanings of words in a sentence are interdependent. On the contrary, Ibn Taymiyya , Francis Bacon and later John Stuart Mill argued that analogy is simply

10788-585: The following correspondences: It was at the beginning a simple remark in Curry and Feys's 1958 book on combinatory logic: the simplest types for the basic combinators K and S of combinatory logic surprisingly corresponded to the respective axiom schemes α → ( β → α ) and ( α → ( β → γ )) → (( α → β ) → ( α → γ )) used in Hilbert-style deduction systems . For this reason, these schemes are now often called axioms K and S. Examples of programs seen as proofs in

10912-405: The following generalization arises: a proof is a program, and the formula it proves is the type for the program . More informally, this can be seen as an analogy that states that the return type of a function (i.e., the type of values returned by a function) is analogous to a logical theorem, subject to hypotheses corresponding to the types of the argument values passed to the function; and that

11036-403: The following is a strict list of qualities necessary for a monad as defined by the following section. These are the 3 things necessary to form a monad. Other monads may embody different logical processes, and some may have additional properties, but all of them will have these three similar components. The more common definition for a monad in functional programming, used in the above example,

11160-426: The following: Instead of repeating Just expressions, we can use something called a bind operator. (also known as "map", "flatmap", or "shove" ). This operation takes a monad and a function that returns a monad and runs the function on the inner value of the passed monad, returning the monad from the function. In Haskell, there is an operator bind , or ( >>= ) that allows for this monadic composition in

11284-537: The form "A is to B as C is to what ?" For example, "Hand is to palm as foot is to ____?" These questions were usually given in the Aristotelian format: HAND : PALM : : FOOT : ____ While most competent English speakers will immediately give the right answer to the analogy question ( sole ), it is more difficult to identify and describe the exact relation that holds both between pairs such as hand and palm , and between foot and sole . This relation

11408-405: The formalism known as Gentzen 's sequent calculus but it is not a correspondence with a well-defined pre-existing model of computation as it was for Hilbert-style and natural deductions. Sequent calculus is characterized by the presence of left introduction rules, right introduction rule and a cut rule that can be eliminated. The structure of sequent calculus relates to a calculus whose structure

11532-415: The function received as a parameter. Between each pair of composed function calls, the bind operator can inject into the monadic value m a some additional information that is not accessible within the function f , and pass it along down the pipeline. It can also exert finer control of the flow of execution, for example by calling the function only under some conditions, or executing the function calls in

11656-429: The general laws, then algebraically building a chain of equalities to reach the other side: Though rarer in computer science, one can use category theory directly, which defines a monad as a functor with two additional natural transformations . So to begin, a structure requires a higher-order function (or "functional") named map to qualify as a functor: This is not always a major issue, however, especially when

11780-422: The identification of places, objects and people, for example, in face perception and facial recognition systems . Hofstadter has argued that analogy is "the core of cognition". An analogy is not a figure of speech but a kind of thought. Specific analogical language uses exemplification , comparisons , metaphors , similes , allegories , and parables , but not metonymy . Phrases like and so on , and

11904-449: The internal arrows that keep the internal structures of the categories rather than the mere relationships between the objects (called "representational states"). Thus, the mind, and more intelligent AIs, may use analogies between domains whose internal structures transform naturally and reject those that do not. Keith Holyoak and Paul Thagard (1997) developed their multiconstraint theory within structure mapping theory. They defend that

12028-414: The level of formulas and types that is independent of which particular proof system or model of computation is considered, and one at the level of proofs and programs which, this time, is specific to the particular choice of proof system and model of computation considered. At the level of formulas and types, the correspondence says that implication behaves the same as a function type, conjunction as

12152-423: The level of proof systems and models of computations, the correspondence mainly shows the identity of structure, first, between some particular formulations of systems known as Hilbert-style deduction system and combinatory logic , and, secondly, between some particular formulations of systems known as natural deduction and lambda calculus . Between the natural deduction system and the lambda calculus there are

12276-590: The like , as if , and the very word like also rely on an analogical understanding by the receiver of a message including them. Analogy is important not only in ordinary language and common sense (where proverbs and idioms give many examples of its application) but also in science , philosophy , law and the humanities . The concepts of association , comparison, correspondence, mathematical and morphological homology , homomorphism , iconicity , isomorphism , metaphor, resemblance, and similarity are closely related to analogy. In cognitive linguistics ,

12400-640: The monad pattern brings many of the benefits of purely functional programming . By reifying a specific kind of computation, a monad not only encapsulates the tedious details of that computational pattern, but it does so in a declarative way, improving the code's clarity. As monadic values explicitly represent not only computed values, but computed effects , a monadic expression can be replaced with its value in referentially transparent positions , much like pure expressions can be, allowing for many techniques and optimizations based on rewriting . Typically, programmers will use bind to chain monadic functions into

12524-460: The monad; for the Maybe monad a value is bound only if it exists, and for the IO monad a value is bound only after the previous operations in the sequence have been performed. A monad can be created by defining a type constructor M and two operations: (An alternative but equivalent construct using the join function instead of the bind operator can be found in the later section § Derivation from functors .) With these elements,

12648-413: The new topic by relating back to existing knowledge. This can be particularly helpful when the analogy serves across different disciplines: indeed, there are various teaching innovations now emerging that use sight-based analogies for teaching and research across subjects such as science and the humanities. Shawn Glynn, a professor in the department of educational psychology and instructional technology at

12772-449: The notion of conceptual metaphor may be equivalent to that of analogy. Analogy is also a basis for any comparative arguments as well as experiments whose results are transmitted to objects that have been not under examination (e.g., experiments on rats when results are applied to humans). Analogy has been studied and discussed since classical antiquity by philosophers, scientists, theologists and lawyers . The last few decades have shown

12896-400: The notion of normal forms in lambda calculus matches Prawitz 's notion of normal deduction in natural deduction , from which it follows that the algorithms for the type inhabitation problem can be turned into algorithms for deciding intuitionistic provability. Howard's correspondence naturally extends to other extensions of natural deduction and simply typed lambda calculus . Here is

13020-403: The pair of a recursive function and of a proof of a formula expressing that the recursive function "realizes", i.e. correctly instantiates the disjunctions and existential quantifiers of the initial formula so that the formula gets true. Kreisel 's modified realizability applies to intuitionistic higher-order predicate logic and shows that the simply typed lambda term inductively extracted from

13144-430: The participants' internal models or concepts exists. In historical science, comparative historical analysis often uses the concept of analogy and analogical reasoning. Recent methods involving calculation operate on large document archives, allowing for analogical or corresponding terms from the past to be found as a response to random questions by users (e.g., Myanmar - Burma) and explained. Analogical reasoning plays

13268-509: The problem at hand. The multiconstraint theory faces some difficulties when there are multiple sources, but these can be overcome. Hummel and Holyoak (2005) recast the multiconstraint theory within a neural network architecture. A problem for the multiconstraint theory arises from its concept of similarity, which, in this respect, is not obviously different from analogy itself. Computer applications demand that there are some identical attributes or relations at some level of abstraction. The model

13392-430: The program to compute that function is analogous to a proof of that theorem. This sets a form of logic programming on a rigorous foundation: proofs can be represented as programs, and especially as lambda terms , or proofs can be run . The correspondence has been the starting point of a large spectrum of new research after its discovery, leading in particular to a new class of formal systems designed to act both as

13516-439: The programmer composes a sequence of function calls (a "pipeline") with several bind operators chained together in an expression. Each function call transforms its input plain-type value, and the bind operator handles the returned monadic value, which is fed into the next step in the sequence. Typically, the bind operator >>= may contain code unique to the monad that performs additional computation steps not available in

13640-438: The programmer to deal with these potentially undefined results by explicitly defining the two states of a result: Just ⌑result⌑ , or Nothing . For example the programmer might be constructing a parser, which is to return an intermediate result, or else signal a condition which the parser has detected, and which the programmer must also handle. With just a little extra functional spice on top, this Maybe type transforms into

13764-493: The proof realizes the initial formula. In the case of propositional logic, it coincides with Howard's statement: the extracted lambda term is the proof itself (seen as an untyped lambda term) and the realizability statement is a paraphrase of the fact that the extracted lambda term has the type that the formula means (seen as a type). Gödel 's dialectica interpretation realizes (an extension of) intuitionistic arithmetic with computable functions. The connection with lambda calculus

13888-403: The proper knowledge to assess the similarities between the two concepts. Step three is finding relevant features within the analogy of the two concepts. Step four is finding similarities between the two concepts so students are able to compare and contrast them in order to understand. Step five is indicating where the analogy breaks down between the two concepts. And finally, step six is drawing

14012-434: The scope of precedent . The use of analogy in both traditions is broadly described by the traditional maxim Ubi eadem est ratio, ibi idem ius (where the reason is the same, the law is the same). Analogies as defined in rhetoric are a comparison between words, but an analogy more generally can also be used to illustrate and teach. To enlighten pupils on the relations between or within certain concepts, items or phenomena,

14136-503: The specification of Haskell. In particular, Haskell used a problematic "lazy stream" model up through v1.2 to reconcile I/O with lazy evaluation , until switching over to a more flexible monadic interface. The Haskell community would go on to apply monads to many problems in functional programming, and in the 2010s, researchers working with Haskell eventually recognized that monads are applicative functors ; and that both monads and arrows are monoids . At first, programming with monads

14260-416: The syntactic correspondence between intuitionistic Hilbert-style deduction and typed combinatory logic , Howard made explicit in 1969 a syntactic analogy between the programs of simply typed lambda calculus and the proofs of natural deduction . Below, the left-hand side formalizes intuitionistic implicational natural deduction as a calculus of sequents (the use of sequents is standard in discussions of

14384-476: The systems. This is also known as the systematicity principle. An example that has been used to illustrate structure mapping theory comes from Gentner and Gentner (1983) and uses the base domain of flowing water and the target domain of electricity. In a system of flowing water, the water is carried through pipes and the rate of water flow is determined by the pressure of the water towers or hills. This relationship corresponds to that of electricity flowing through

14508-548: The term Curry–Howard–de Bruijn correspondence in place of Curry–Howard correspondence. The BHK interpretation interprets intuitionistic proofs as functions but it does not specify the class of functions relevant for the interpretation. If one takes lambda calculus for this class of function, then the BHK interpretation tells the same as Howard's correspondence between natural deduction and lambda calculus. Kleene 's recursive realizability splits proofs of intuitionistic arithmetic into

14632-479: The three-way Curry–Howard–Lambek correspondence . The beginnings of the Curry–Howard correspondence lie in several observations: The Curry–Howard correspondence is the observation that there is an isomorphism between the proof systems, and the models of computation. It is the statement that these two families of formalisms can be considered as identical. If one abstracts on the peculiarities of either formalism,

14756-421: The wish to make the Curry–Howard correspondence practically relevant. The Curry–Howard correspondence also raised new questions regarding the computational content of proof concepts that were not covered by the original works of Curry and Howard. In particular, classical logic has been shown to correspond to the ability to manipulate the continuation of programs and the symmetry of sequent calculus to express

14880-419: The λ-terms λ x .λ y . x and λ x .λ y . y of type α → α → α would not be distinguished in the correspondence. Examples are given below . Howard showed that the correspondence extends to other connectives of the logic and other constructions of simply typed lambda calculus. Seen at an abstract level, the correspondence can then be summarized as shown in the following table. Especially, it also shows that

15004-454: Was actively designed up until 1990, also effectively based I/O on a monadic type, but the connection was not realized at the time. The computer scientist Eugenio Moggi was the first to explicitly link the monad of category theory to functional programming, in a conference paper in 1989, followed by a more refined journal submission in 1991. In earlier work, several computer scientists had advanced using category theory to provide semantics for

15128-400: Was extended (Doumas, Hummel, and Sandhofer, 2008) to learn relations from unstructured examples (providing the only current account of how symbolic representations can be learned from examples). Mark Keane and Brayshaw (1988) developed their Incremental Analogy Machine (IAM) to include working memory constraints as well as structural, semantic and pragmatic constraints, so that a subset of

15252-458: Was largely confined to Haskell and its derivatives, but as functional programming has influenced other paradigms, many languages have incorporated a monad pattern (in spirit if not in name). Formulations now exist in Scheme , Perl , Python , Racket , Clojure , Scala , F# , and have also been considered for a new ML standard. One benefit of the monad pattern is bringing mathematical precision on

15376-605: Was observing objects once and she said, "my mind was noting what else each object reminded me of..." This led her to teach with the question, "what does [the subject or topic] remind you of?" The idea of comparing subjects and concepts led to the development of The Private Eye Project as a method of teaching. The program is designed to build critical thinking skills with analogies as one of the main themes revolving around it. While Glynn focuses on using analogies to teach science, The Private Eye Project can be used for any subject including writing, math, art, social studies, and invention. It

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