“Many-sorted logic”的意思、由来-开放百科全书

Many-sorted logic can reflect formally our intention not to handle the universe as a homogeneous collection of objects, but to partition it in a way that is similar to types in typeful programming. Both functional and assertive "parts of speech" in the language of the logic reflect this typeful partitioning of the universe, even on the syntax level: substitution and argument passing can be done only accordingly, respecting the "sorts".

There are more ways to formalize the intention mentioned above; a many-sorted logic is any package of information which fulfills it. In most cases, the following are given:

The domain of discourse of any structure of that signature is then fragmented into disjoint subsets, one for every sort.

Example

When reasoning about biological organisms, it is useful to distinguish two sorts:

and

. While a function

makes sense, a similar function

usually does not. Many-sorted logic allows one to have terms like

, but to discard terms like

as syntactically ill-formed.

Algebraization

The algebraization of many-sorted logic is explained in an article by Caleiro and Gonçalves,^[1] which generalizes abstract algebraic logic to the many-sorted case, but can also be used as introductory material.

Order-sorted logic

While many-sorted logic requires two distinct sorts to have disjoint universe sets, order-sorted logic allows one sort

to be declared a subsort of another sort

, usually by writing

or similar syntax. In the above example, it is desirable to declare

Wherever a term of some sort

is required, a term of any subsort of

may be supplied instead. For example, assuming a function declaration

, and a constant declaration

, the term

is perfectly valid and has the sort

. In order to supply the information that the mother of a dog is a dog in turn, another declaration

may be issued; this is called function overloading, similar to overloading in programming languages.

Order-sorted logic can be translated into unsorted logic, using a unary predicate

for each sort

, and an axiom

for each subsort declaration

. The reverse approach was successful in automated theorem proving: in 1985, Christoph Walther could solve a then benchmark problem by translating it into order-sorted logic, thereby boiling it down an order of magnitude, as many unary predicates turned into sorts.^[2]

In order to incorporate order-sorted logic into a clause-based automated theorem prover, a corresponding order-sorted unification algorithm is necessary, which requires for any two declared sorts

their intersection

to be declared, too: if

and

are variables of sort

and

, respectively, the equation

has the solution

, where

In his framework, subsort declarations are propagated to complex type expressions.

As a programming example, a parametric sort

may be declared (with

being a type parameter as in a C++ template), and from a subsort declaration

the relation

is automatically inferred, meaning that each list of integers is also a list of floats.

Schmidt-Schauß generalized order-sorted logic to allow for term declarations.^[5]

As an example, assuming subsort declarations

and

, a term declaration like

allows to declare a property of integer addition that could not be expressed by ordinary overloading.

See also

References

1. ^{{cite book| author=Carlos Caleiro, Ricardo Gonçalves| chapter=On the algebraization of many-sorted logics| title=Proc. 18th int. conf. on Recent trends in algebraic development techniques (WADT)| year=2006| volume=| pages=21–36| publisher=Springer| editor=| series=| isbn=978-3-540-71997-7| url=http://sqig.math.ist.utl.pt/pub/CaleiroC/06-CG-manysorted.pdf}}
2. ^{{cite journal|first1=Christoph|last1=Walther|title=A Mechanical Solution of Schubert's Steamroller by Many-Sorted Resolution|journal=Artif. Intell.|volume=26|number=2|pages=217–224|url=http://www.inferenzsysteme.informatik.tu-darmstadt.de/media/is/publikationen/Schuberts_Steamroller_by_Many-Sorted_Resolution-AIJ-25-2-1985.pdf|year=1985|doi=10.1016/0004-3702(85)90029-3}}
3. ^{{cite conference|first1=Gert|last1=Smolka|title=Logic Programming with Polymorphically Order-Sorted Types|booktitle=Int. Workshop Algebraic and Logic Programming|publisher=Springer|series=LNCS|volume=343|pages=53–70|date=Nov 1988}}
4. ^{{citation|first1=Gert|last1=Smolka|title=Logic Programming over Polymorphically Order-Sorted Types|publisher=Univ. Kaiserslautern, Germany|date=May 1989}}
5. ^{{cite book|first1=Manfred|last1=Schmidt-Schauß|title=Computational Aspects of an Order-Sorted Logic with Term Declarations|publisher=Springer|series=LNAI|volume=395|date=Apr 1988}}

Example

Algebraization

Order-sorted logic

See also

References

External links