Семантикалық кодтау: Тілдерді салыстыру және өрнектілігін анықтау

A semantics encoding is a translation between formal languages. For programmers, the most familiar form of encoding is the compilation of a programming language into machine code or byte code. Conversion between document formats are also forms of encoding. Compilation of TeX or LaTeX documents to PostScript are also commonly encountered encoding processes. Some high level preprocessors, such as OCaml's Camlp4, also involve encoding of a programming language into another. Formally, an encoding of a language A into language B is a mapping of all terms of A into B. If there is a satisfactory encoding of A into B, B is considered at least as powerful (or at least as expressive) as A.

An informal notion of translation is not sufficient to help determine expressivity of languages, as it permits trivial encodings such as mapping all elements of A to the same element of B. Therefore, it is necessary to determine the definition of a "good enough" encoding. This notion varies with the application. Commonly, an encoding is expected to preserve a number of properties.

This assumes the existence of a notion of reduction on both language A and language B. Typically, in the case of a programming language, reduction is the relation which models the execution of a program. We write for one step of reduction and for any number of steps of reduction. soundness For every terms of language A, if then completeness For every term of language A and every terms of language B, if then there exists some such that
This preservation guarantees that both languages behave the same way. Soundness guarantees that all possible behaviours are preserved while completeness guarantees that no behaviour is added by the encoding. In particular, in the case of compilation of a programming language, soundness and completeness together mean that the compiled program behaves accordingly to the high level semantics of the programming language.

This also assumes the existence of a notion of reduction on both language A and language B.

soundness for any term , if all reductions of converge, then all reductions of converge. completeness for any term , if all reductions of converge, then all reductions of converge. In the case of compilation of a programming language, soundness guarantees that the compilation does not introduce non termination such as endless loops or endless recursions. The completeness property is useful when language B is used to study or test a program written in language A, possibly by extracting key parts of the code: if this study or test proves that the program terminates in B, then it also terminates in A.

This assumes the existence of a notion of observation on both language A and language B. In programming languages, typical observables are results of inputs and outputs, by opposition to pure computation. In a description language such as HTML, a typical observable is the result of page rendering. soundness for every observable on terms of A, there exists an observable of terms of B such that for any term with observable , has observable completeness for every observable on terms of A, there exists an observable on terms of B such that for any term with observable , has observable .

This assumes the existence of notion of simulation on both language A and language B. In a programming languages, a program simulates another if it can perform all the same (observable) tasks and possibly some others. Simulations are used typically to describe compile time optimizations. soundness for every terms , if simulates then simulates completeness for every terms , if simulates then simulates
Preservation of simulations is a much stronger property than preservation of observations, which it entails. In turn, it is weaker than a property of preservation of bisimulations. As in previous cases, soundness is important for compilation, while completeness is useful for testing or proving properties.

This assumes the existence of a notion of equivalence on both language A and language B. Typically, this can be a notion of equality of structured data or a notion of syntactically different yet semantically identical programs, such as structural congruence or structural equivalence. soundness if two terms and are equivalent in A, then and are equivalent in B.
completeness if two terms and are equivalent in B, then and are equivalent in A.

This assumes the existence of a notion of distribution on both language A and language B. Typically, for compilation of distributed programs written in Acute, JoCaml or E, this means distribution of processes and data among several computers or CPUs. soundness if a term is the composition of two agents then must be the composition of two agents completeness if a term is the composition of two agents then must be the composition of two agents such that and .