HOME

TheInfoList



OR:

Finite model theory is a subarea of
model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
. Model theory is the branch of
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
which deals with the relation between a formal language (syntax) and its interpretations (semantics). Finite model theory is a restriction of model theory to interpretations on finite structures, which have a finite universe. Since many central theorems of model theory do not hold when restricted to finite structures, finite model theory is quite different from model theory in its methods of proof. Central results of classical model theory that fail for finite structures under finite model theory include the compactness theorem, Gödel's completeness theorem, and the method of ultraproducts for first-order logic (FO). While model theory has many applications to mathematical algebra, finite model theory became an "unusually effective" instrument in computer science. In other words: "In the history of mathematical logic most interest has concentrated on infinite structures. ..Yet, the objects computers have and hold are always finite. To study computation we need a theory of finite structures." Thus the main application areas of finite model theory are: descriptive complexity theory, database theory and formal language theory.


Axiomatisability

A common motivating question in finite model theory is whether a given class of structures can be described in a given language. For instance, one might ask whether the class of cyclic graphs can be distinguished among graphs by a FO sentence, which can also be phrased as asking whether cyclicity is FO-expressible. A single finite structure can always be axiomatized in first-order logic, where axiomatized in a language ''L'' means described uniquely up to isomorphism by a single ''L''-sentence. Similarly, any finite collection of finite structures can always be axiomatized in first-order logic. Some, but not all, infinite collections of finite structures can also be axiomatized by a single first-order sentence.


Characterisation of a single structure

Is a language ''L'' expressive enough to axiomatize a single finite structure ''S''?


Problem

A structure like (1) in the figure can be described by FO sentences in the
logic of graphs In the mathematical fields of graph theory and finite model theory, the logic of graphs deals with formal specifications of graph properties using sentences of mathematical logic. There are several variations in the types of logical operation that ...
like # Every node has an edge to another node: \forall_x \exists_y G(x, y). # No node has an edge to itself: \forall_ (G(x, y) \Rightarrow x \neq y). # There is at least one node that is connected to all others: \exists_x \forall_y (x \neq y \Rightarrow G(x, y)). However, these properties do not axiomatize the structure, since for structure (1') the above properties hold as well, yet structures (1) and (1') are not isomorphic. Informally the question is whether by adding enough properties, these properties together describe exactly (1) and are valid (all together) for no other structure (up to isomorphism).


Approach

For a single finite structure it is always possible to precisely describe the structure by a single FO sentence. The principle is illustrated here for a structure with one binary relation R and without constants: # say that there are at least n elements: \varphi_1 = \bigwedge_ \neg (x_i = x_j) # say that there are at most n elements: \varphi_2 = \forall_y \bigvee_ (x_i = y) # state every element of the relation R: \varphi_3 = \bigwedge_ R(x_i, x_j) # state every non-element of the relation R: \varphi_4 = \bigwedge_ \neg R(x_i, x_j) all for the same tuple x_1 .. x_n, yielding the FO sentence \exists_ \dots \exists_ (\varphi_1 \land \varphi_2 \land \varphi_3 \land \varphi_4).


Extension to a fixed number of structures

The method of describing a single structure by means of a first-order sentence can easily be extended for any fixed number of structures. A unique description can be obtained by the disjunction of the descriptions for each structure. For instance, for two structures A and B with defining sentences \varphi_A and \varphi_B this would be :\varphi_A \lor \varphi_B.


Extension to an infinite structure

By definition, a set containing an infinite structure falls outside the area that FMT deals with. Note that infinite structures can never be discriminated in FO, because of the
Löwenheim–Skolem theorem In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-ord ...
, which implies that no first-order theory with an infinite model can have a unique model up to isomorphism. The most famous example is probably Skolem's theorem, that there is a countable non-standard model of arithmetic.


Characterisation of a class of structures

Is a language ''L'' expressive enough to describe exactly (up to isomorphism) those finite structures that have certain property ''P''?


Problem

The descriptions given so far all specify the number of elements of the universe. Unfortunately most interesting sets of structures are not restricted to a certain size, like all graphs that are trees, are connected or are acyclic. Thus to discriminate a finite number of structures is of special importance.


Approach

Instead of a general statement, the following is a sketch of a methodology to differentiate between structures that can and cannot be discriminated. 1. The core idea is that whenever one wants to see if a property ''P'' can be expressed in FO, one chooses structures ''A'' and ''B'', where ''A'' does have ''P'' and ''B'' doesn't. If for ''A'' and ''B'' the same FO sentences hold, then ''P'' cannot be expressed in FO. In short: :A \in P, B \not\in P and A \equiv B, where A \equiv B is shorthand for A \models \alpha \Leftrightarrow B \models \alpha for all FO-sentences α, and ''P'' represents the class of structures with property ''P''. 2. The methodology considers countably many subsets of the language, the union of which forms the language itself. For instance, for FO consider classes FO 'm''for each ''m''. For each ''m'' the above core idea then has to be shown. That is: :A \in P, B \not\in P and A \equiv_m B with a pair A, B for each m and α (in ≡) from FO 'm'' It may be appropriate to choose the classes FO 'm''to form a partition of the language. 3. One common way to define FO 'm''is by means of the quantifier rank qr(α) of a FO formula α, which expresses the depth of quantifier nesting. For example, for a formula in
prenex normal form A formula of the predicate calculus is in prenex normal form (PNF) if it is written as a string of quantifiers and bound variables, called the prefix, followed by a quantifier-free part, called the matrix. Together with the normal forms in pro ...
, qr is simply the total number of its quantifiers. Then FO 'm''can be defined as all FO formulas α with qr(α) ≤ ''m'' (or, if a partition is desired, as those FO formulas with quantifier rank equal to ''m''). 4. Thus it all comes down to showing A \models \alpha \Leftrightarrow B \models \alpha on the subsets FO 'm'' The main approach here is to use the algebraic characterization provided by Ehrenfeucht–Fraïssé games. Informally, these take a single partial isomorphism on ''A'' and ''B'' and extend it ''m'' times, in order to either prove or disprove A \equiv_m B, dependent on who wins the game.


Example

We want to show that the property that the size of an ordered structure A = (A, ≤) is even, can not be expressed in FO. 1. The idea is to pick A ∈ EVEN and B ∉ EVEN, where EVEN is the class of all structures of even size. 2. We start with two ordered structures A2 and B2 with universes A2 = and B2 = . Obviously A2 ∈ EVEN and B2 ∉ EVEN. 3. For ''m'' = 2, we can now show* that in a 2-move Ehrenfeucht–Fraïssé game on A2 and B2 the duplicator always wins, and thus A2 and B2 cannot be discriminated in FO i.e. A2 \models α ⇔ B2 \models α for every α ∈ FO 4. Next we have to scale the structures up by increasing ''m''. For example, for ''m'' = 3 we must find an A3 and B3 such that the duplicator always wins the 3-move game. This can be achieved by A3 = and B3 = . More generally, we can choose A''m'' = and B''m'' = ; for any ''m'' the duplicator always wins the ''m''-move game for this pair of structures*. 5. Thus EVEN on finite ordered structures cannot be expressed in FO. (*) Note that the proof of the result of the Ehrenfeucht–Fraïssé game has been omitted, since it is not the main focus here.


Zero-one laws

and, independently, proved a zero–one law for first-order sentences in finite models; Fagin's proof used the compactness theorem. According to this result, every first-order sentence in a relational signature \sigma is either
almost always In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
true or almost always false in finite \sigma-structures. That is, let be a fixed first-order sentence, and choose a random \sigma-structure G_n with domain \, uniformly among all \sigma-structures with domain \. Then in the limit as tends to infinity, the probability that models will tend either to zero or to one: :\lim_\operatorname _n\models Sin\. The problem of determining whether a given sentence has probability tending to zero or to one is PSPACE-complete. A similar analysis has been performed for more expressive logics than first-order logic. The 0-1 law has been shown to hold for sentences in
FO(LFP) In mathematical logic, fixed-point logics are extensions of classical predicate logic that have been introduced to express recursion. Their development has been motivated by descriptive complexity theory and their relationship to database query lan ...
, first-order logic augmented with a least fixed point operator, and more generally for sentences in the infinitary logic L^_, which allows for potentially arbitrarily long conjunctions and disjunctions. Another important variant is the unlabelled 0-1 law, where instead of considering the fraction of structures with domain \, one considers the fraction of isomorphism classes of structures with elements. This fraction is well-defined, since any two isomorphic structures satisfy the same sentences. The unlabelled 0-1 law also holds for L^_ and therefore in particular for FO(LFP) and first-order logic.


Descriptive complexity theory

An important goal of finite model theory is the characterisation of
complexity class In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms o ...
es by the type of
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
needed to express the languages in them. For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic. This connection between complexity and the logic of finite structures allows results to be transferred easily from one area to the other, facilitating new proof methods and providing additional evidence that the main complexity classes are somehow "natural" and not tied to the specific abstract machines used to define them. Specifically, each
logical system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A for ...
produces a set of queries expressible in it. The queries – when restricted to finite structures – correspond to the computational problems of traditional complexity theory. Some well-known complexity classes are captured by logical languages as follows: * In the presence of a linear order, first-order logic with a commutative, transitive closure operator added yields L, problems solvable in logarithmic space. * In the presence of a linear order, first-order logic with a transitive closure operator yields NL, the problems solvable in nondeterministic logarithmic space. * In the presence of a linear order, first-order logic with a least fixed point operator gives P, the problems solvable in deterministic polynomial time. * On all finite structures (regardless of whether they are ordered), Existential second-order logic gives NP ( Fagin's theorem).


Applications


Database theory

A substantial fragment of SQL (namely that which is effectively relational algebra) is based on first-order logic (more precisely can be translated in domain relational calculus by means of Codd's theorem), as the following example illustrates: Think of a database table "GIRLS" with the columns "FIRST_NAME" and "LAST_NAME". This corresponds to a binary relation, say G(f, l) on FIRST_NAME X LAST_NAME. The FO query , which returns all the last names where the first name is 'Judy', would look in SQL like this: select LAST_NAME from GIRLS where FIRST_NAME = 'Judy' Notice, we assume here, that all last names appear only once (or we should use SELECT DISTINCT since we assume that relations and answers are sets, not bags). Next we want to make a more complex statement. Therefore, in addition to the "GIRLS" table we have a table "BOYS" also with the columns "FIRST_NAME" and "LAST_NAME". Now we want to query the last names of all the girls that have the same last name as at least one of the boys. The FO query is , and the corresponding SQL statement is: select FIRST_NAME, LAST_NAME from GIRLS where LAST_NAME IN ( select LAST_NAME from BOYS ); Notice that in order to express the "∧" we introduced the new language element "IN" with a subsequent select statement. This makes the language more expressive for the price of higher difficulty to learn and implement. This is a common trade-off in formal language design. The way shown above ("IN") is by far not the only one to extend the language. An alternative way is e.g. to introduce a "JOIN" operator, that is: select distinct g.FIRST_NAME, g.LAST_NAME from GIRLS g, BOYS b where g.LAST_NAME=b.LAST_NAME; First-order logic is too restrictive for some database applications, for instance because of its inability to express transitive closure. This has led to more powerful constructs being added to database query languages, such as recursive WITH in SQL:1999. More expressive logics, like fixpoint logics, have therefore been studied in finite model theory because of their relevance to database theory and applications.


Querying and search

Narrative data contains no defined relations. Thus the logical structure of text search queries can be expressed in propositional logic, like in: ("Java" AND NOT "island") OR ("C#" AND NOT "music") Note that the challenges in full text search are different from database querying, like ranking of results.


History

* Trakhtenbrot 1950: failure of completeness theorem in first-order logic * Scholz 1952: characterisation of spectra in first-order logic * Fagin 1974: the set of all properties expressible in existential second-order logic is precisely the complexity class NP * Chandra, Harel 1979/80: fixed-point first-order logic extension for database query languages capable of expressing transitive closure -> queries as central objects of FMT * Immerman, Vardi 1982: fixed-point logic over ordered structures captures PTIME -> descriptive complexity ( Immerman–Szelepcsényi theorem) *
Ebbinghaus Ebbinghaus is a German surname. It may refer to: * Bernhard Ebbinghaus (Born 1961), a German sociologist * Heinz-Dieter Ebbinghaus (born 1939), a German mathematician * Hermann Ebbinghaus (1850–1909), a German psychologist ** the Ebbinghaus illu ...
, Flum 1995: first comprehensive book "Finite Model Theory" * Abiteboul, Hull, Vianu 1995: book "Foundations of Databases" * Immerman 1999: book "
Descriptive Complexity ''Descriptive Complexity'' is a book in mathematical logic and computational complexity theory by Neil Immerman. It concerns descriptive complexity theory, an area in which the expressibility of mathematical properties using different types of l ...
" * Kuper, Libkin, Paredaens 2000: book "Constraint Databases" * Darmstadt 2005/ Aachen 2006: first international workshops on "Algorithmic Model Theory"


Citations


References

* * * Glebskiĭ, Yu V., D. I. Kogan, M. I. Liogon'kiĭ, and V. A. Talanov. "Volume and fraction of satisfiability of formulae of the first-order predicate calculus." Kibernetika 2 (1969): 17-27. * * *


Further reading

*


External links

* Also suitable as a general introduction and overview. * Leonid Libkin
Introductory chapter of "Elements of Finite Model Theory"
. Motivates three main application areas: databases, complexity and formal languages. * Jouko Väänänen
A Short Course on Finite Model Theory
Department of Mathematics, University of Helsinki. Based on lectures from 1993-1994. * Anuj Dawar
Infinite and Finite Model Theory
slides, University of Cambridge, 2002. * Includes a list of open FMT problems. {{Mathematical logic