In programming language theory, semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages. It does so by evaluating the meaning of syntactically valid strings defined by a specific programming language, showing the computation involved. In such a case that the evaluation would be of syntactically invalid strings, the result would be non-computation. Semantics describes the processes a computer follows when executing a program in that specific language. This can be shown by describing the relationship between the input and output of a program, or an explanation of how the program will be executed on a certain platform, hence creating a model of computation.
Programming language theory (PLT) is a branch of computer science that deals with the design, implementation, analysis, characterization, and classification of programming languages and their individual features. It falls within the discipline of computer science, both depending on and affecting mathematics, software engineering, linguistics and even cognitive science. It is a well-recognized branch of computer science, and an active research area, with results published in numerous journals dedicated to PLT, as well as in general computer science and engineering publications.
In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed, or it may be fixed. A string is generally considered as a data type and is often implemented as an array data structure of bytes that stores a sequence of elements, typically characters, using some character encoding. String may also denote more general arrays or other sequence data types and structures.
In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how units of computations, memories, and communications are organized. The computational complexity of an algorithm can be measured given a model of computation. Using a model allows studying the performance of algorithms independently of the variations that are specific to particular implementations and specific technology.
Formal semantics, for instance, helps to write compilers, better understand what a program is doing, and to prove, e.g., that the following if statement
if 1 == 1 then S1 else S2
has the same effect as S1 alone.
The field of formal semantics encompasses all of the following:
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects.
In mathematics, model theory is the study of classes of mathematical structures from the perspective of mathematical logic. The objects of study are models of theories in a formal language. A set of sentences in a formal language is one of the components that form a theory. A model of a theory is a structure that satisfies the sentences of that theory.
It has close links with other areas of computer science such as programming language design, type theory, compilers and interpreters, program verification and model checking.
Computer science is the study of processes that interact with data and that can be represented as data in the form of programs. It enables the use of algorithms to manipulate, store, and communicate digital information. A computer scientist studies the theory of computation and the practice of designing software systems.
In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics. In type theory, every "term" has a "type" and operations are restricted to terms of a certain type.
A compiler is a computer program that translates computer code written in one programming language into another language. The name compiler is primarily used for programs that translate source code from a high-level programming language to a lower level language to create an executable program.
There are many approaches to formal semantics; these belong to three major classes:
In computer science, denotational semantics is an approach of formalizing the meanings of programming languages by constructing mathematical objects that describe the meanings of expressions from the languages. Other approaches provide formal semantics of programming languages including axiomatic semantics and operational semantics.
In semiotics, denotation is the surface or the literal meaning. The definition most likely to appear in a dictionary.
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and has close relations to topology.
The distinctions between the three broad classes of approaches can sometimes be vague, but all known approaches to formal semantics use the above techniques, or some combination thereof.
Apart from the choice between denotational, operational, or axiomatic approaches, most variation in formal semantic systems arises from the choice of supporting mathematical formalism.
Some variations of formal semantics include the following:
Action semantics is a framework for the formal specification of semantics of programming languages invented by David Watt and Peter D. Mosses in the 1990s. It is a mixture of denotational, operational and algebraic semantics.
In computer science, algebraic semantics is a form of axiomatic semantics based on algebraic laws for describing and reasoning about program semantics in a formal manner.
Axiomatic semantics is an approach based on mathematical logic for proving the correctness of computer programs. It is closely related to Hoare logic.
For a variety of reasons, one might wish to describe the relationships between different formal semantics. For example:
It is also possible to relate multiple semantics through abstractions via the theory of abstract interpretation.
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Robert W. Floyd is credited with founding the field of programming language semantics in Floyd (1967).
In mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
In linguistics, syntax is the set of rules, principles, and processes that govern the structure of sentences in a given language, usually including word order. The term syntax is also used to refer to the study of such principles and processes. The goal of many syntacticians is to discover the syntactic rules common to all languages.
Semantics is the linguistic and philosophical study of meaning in language, programming languages, formal logics, and semiotics. It is concerned with the relationship between signifiers—like words, phrases, signs, and symbols—and what they stand for in reality, their denotation.
In computer science, specifically software engineering and hardware engineering, formal methods are a particular kind of mathematically based techniques for the specification, development and verification of software and hardware systems. The use of formal methods for software and hardware design is motivated by the expectation that, as in other engineering disciplines, performing appropriate mathematical analysis can contribute to the reliability and robustness of a design.
Lexical functional grammar (LFG) is a constraint-based grammar framework in theoretical linguistics. It posits two separate levels of syntactic structure, a phrase structure grammar representation of word order and constituency, and a representation of grammatical functions such as subject and object, similar to dependency grammar. The development of the theory was initiated by Joan Bresnan and Ronald Kaplan in the 1970s, in reaction to the theory of transformational grammar which was current in the late 1970s. It mainly focuses on syntax, including its relation with morphology and semantics. There has been little LFG work on phonology.
Head-driven phrase structure grammar (HPSG) is a highly lexicalized, constraint-based grammar developed by Carl Pollard and Ivan Sag. It is a type of phrase structure grammar, as opposed to a dependency grammar, and it is the immediate successor to generalized phrase structure grammar. HPSG draws from other fields such as computer science and uses Ferdinand de Saussure's notion of the sign. It uses a uniform formalism and is organized in a modular way which makes it attractive for natural language processing.
Operational semantics is a category of formal programming language semantics in which certain desired properties of a program, such as correctness, safety or security, are verified by constructing proofs from logical statements about its execution and procedures, rather than by attaching mathematical meanings to its terms. Operational semantics are classified in two categories: structural operational semantics formally describe how the individual steps of a computation take place in a computer-based system; by opposition natural semantics describe how the overall results of the executions are obtained. Other approaches to providing a formal semantics of programming languages include axiomatic semantics and denotational semantics.
A formal system is used to infer theorems from axioms according to a set of rules. These rules used to carry out the inference of theorems from axioms are known as the logical calculus of the formal system. A formal system is essentially an "axiomatic system". In 1921, David Hilbert proposed to use such system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined system of abstract thought.
Richard Merritt Montague was an American mathematician and philosopher.
In mathematics, semantics, and philosophy of language, the principle of compositionality is the principle that the meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them. This principle is also called Frege's principle, because Gottlob Frege is widely credited for the first modern formulation of it. The principle was never explicitly stated by Frege, and it was arguably already assumed by George Boole decades before Frege's work.
In linguistics, construction grammar is a family of theories which posit that human language consists of constructions, or learned pairings of linguistic forms with functions or meanings. Constructions can be individual words, morphemes, fixed expressions and idioms, and abstract grammatical rules such as the passive voice or ditransitive. Any linguistic pattern is considered to be a construction as long as some aspect of its form or its meaning cannot be predicted from its component parts, or from other constructions that are recognized to exist. In construction grammar, every utterance is understood to be a combination of multiple different constructions, which together specify its precise meaning and form.
In computer science, the syntax of a computer language is the set of rules that defines the combinations of symbols that are considered to be a correctly structured document or fragment in that language. This applies both to programming languages, where the document represents source code, and to markup languages, where the document represents data.
In linguistics and grammar, a quantifier is a type of determiner, such as all, some, many, few, a lot, and no, that indicates quantity.
Glue semantics, or simply Glue, is a linguistic theory of semantic composition and the syntax–semantics interface which assumes that meaning composition is constrained by a set of instructions stated within a formal logic. These instructions, called meaning constructors, state how the meanings of the parts of a sentence can be combined to provide the meaning of the sentence.
In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess." According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they aren't "about" anything at all. Rather, mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation. In contrast to logicism or intuitionism, formalism's contours are less defined due to broad approaches that can be categorized as formalist.
The categorical abstract machine (CAM) is a model of computation for programs that preserves the abilities of applicative, functional, or compositional style. It is based on the techniques of applicative computing.