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Circumscription

Witold Lukaszewicz: Nonmonotonic reasoning: Formaliza-

tion of Common-Sense Reasoning, Ellis Horwood, 1990.

1

Second-order logic

Alphabet:

1. A denumerable set of n-ary (for each n≥0) function va-

riables: {φn1, φn

2, . . .}; 0-ary function variables are called

individual variables, and we denote them by x1, x2, . . ..

2. A denumerable set of n-ary (for each n≥0) predicate va-

riables: {Φn1, Φn

2, . . .}.

3. Truth constants: True and False.

4. Standard sentential connectives and quantifiers.

5. A countable set P of predicate constants.

6. A distinguished 2-ary predicate constant: “=”.

7. A countable set F of function constants.

2

Terms

(1) Each individual variable is a term.

(2) Each individual constant is a term.

(3) If f is an n-ary (n>0) function constant and α1, . . . αn are

terms, then f(α1, . . . αn) is a term.

(4) If φ is an n-ary (n>0) function variable and α1, . . . αn are

terms, then φ(α1, . . . αn) is a term.

Examples of terms:

x, f(a, y), φ(x, φ(b, z)).

3

Atomic formulas

(1) True, False and all proposition constants are atomic for-

mulas.

(2) If P is an n-ary (n>0) predicate constant and α1, . . . αn

are terms, then P(α1, . . . αn) is an atomic formula.

(3) If α and β are terms, then (α = β) is an atomic formula.

(4) Each proposition variable is an atomic formula.

(5) If Φ is an n-ary (n>0) predicate variable and α1, . . . αn are

terms, then Φ(α1, . . . αn) is an atomic formula.

Examples of atomic formulas:

P(φ(a), a), Φ(b, x, φ(x)), (b = φ(y)).

4

Formulas

(1) Each atomic formula is a formula.

(2) If A is a formula, then so is (¬A).

(3) If A and B are formulas, then (A ⊃ B), (A ∧ B), (A ∨ B)

and (A ≡ B) are formulas.

(4) If A is a formula and X is any variable, then (∀X(A)) and

(∃X(A)) are formulas.

Examples of formulas:

(∀xΦ(x)), (∀x∃Ψ(¬Ψ(x) ∨Φ(x)))

5

Semantics

Definition A frame for a second-order language LAL is a

pair M=〈D,m〉, where D is a non-empty set, the domain of

M, and m is a function which assigns to each n-ary (n≥0)

predicate constant in AL, different from “=”, an n-ary rela-

tion on D, to the constant “=”, the identity relation on D,

and to each n-ary (n≥0) function constant in AL, an n-ary

function from Dn into D.

Given a frame M=〈D,m〉, we write |M| to denote the domain

of M; M|K| stands for m(K), where K is any function or

predicate constant in AL.

6

Definition An assignment over M is any function which as-

signs to every n-ary (n≥0) function variable an n-ary function

from |M|n into |M|, and to each n-ary (n≥0) predicate varia-

ble an n-ary relation on M.

The set of all assignments over M will be denoted by As(M).

If a∈As(M) and X is any variable, then we write [a]X to

denote the set of those assignments over M which differs

from “a” at most on X.

7

Definition The value VMa (α) of a term α in a frame M with

respect to a∈As(M) is an element of |M| defined as follows:

(1) VMa (x) = a(x);

(2) VMa (a) = M|a|;

(3) VMa (f(α1, . . . αn)) =

= M|f |[VMa (α1), . . . , VM

a (αn)];

(4) VMa (φ(α1, . . . αn)) =

= a(φ)[VMa (α1), . . . , VM

a (αn)];

8

Definition The value VMa (A) of an atomic formula A in M

with respect to a∈As(M) is an element of {0, 1} defined as

follows:

(1) VMa (True) = 1; VM

a (False) = 0;

(2) VMa (p) = M|p|;

(3) VMa (P(α1, . . . αn)) =

= M|P |[VMa (α1), . . . , VM

a (αn)];

(4) VMa (α = β) = 1 iff VM

a (α) = VMa (β);

(5) VMa (Φ0) = a(Φ0);

(6) VMa (Φ(α1, . . . αn)) =

= a(Φ)[VMa (α1), . . . , VM

a (αn)].

9

Definition The value VMa (A) of a non-atomic formula A in

M with respect to a∈As(M) is an element of {0, 1} defined

as follows:

(1) VMa (¬B) = 1−VM

a (B);

(2) VMa (B ⊃ C) = 1 iff VM

a (B) = 0 or VMa (C) = 1;

(3) VMa (B ∨ C) = 1 iff VM

a (B) = 1 or VMa (C) = 1;

(4) VMa (B ∧ C) = 1 iff VM

a (B) = 1 and VMa (C) = 1;

(5) VMa (B ≡ C) = 1 iff VM

a (A) = VMa (B);

(6) VMa (∀X B) = 1 iff VM

a’(B) = 1, for every a’ ∈ [a]X;

(7) VMa (∃X B) = 1 iff VM

a’(B) = 1, for some a’ ∈ [a]X.

10

A formula A is said to be satisfied in M by an assignment a∈

As(M) iff VMa (A) = 1.

A is satisfiable iff VMa (A) = 1, for some M and some a∈

As(M).

A is true in M iff VMa (A) = 1, for any a∈ As(M). In this case

we say that M is a model of A.

A is valid iff A is true in every frame.

Theorem The validity problem for second-order logic is un-

decidable. The problem is even not semi-decidable.

11

A second-order theory is a computable set of formulas. The

formulas from a theory T are called the axioms (premisses)

of T .

A frame in which all the axioms of a theory T are true is

called a model of T .

A theory T is called satisfiable iff it has a model.

A formula A is entailed by a theory T , written T |= A, iff A

is true in every model of T .

Theories T and T ’ are said to be equivalent, written T ⇔ T ’,

iff they have the same models.

Theorem If T = {A1, . . . , An}, then

T ⇔ {A1 ∧ · · · ∧An}.

12

THEOREM (Godel) There is no complete deduction sys-

tem for second-order logic.

Predicate and function expressions

Definition An n-ary predicate expression U is any expression

of the form

λx1 . . . xn A(x1, . . . , xn) (n ≥ 0)

where x1, . . . , xn are individual variables and A(x1, . . . , xn) is a

formula of first- or second-order logic.

13

U = λx1 . . . xn A(x1, . . . , xn)

An occurrence of X is said to be free in U iff X 6∈ {x1, . . . , xn}

and X is free in A(x1, . . . , xn).

Any individual variable which is free in U is called a parameter

in U .

If no parameters occur in U , then U is called a predicate

expression without parameters. Otherwise U is called a pre-

dicate expression with parameters.

λx P(x); λxy [Φ(x) ∨Q(y)]; λx Q(x, y)

14

Definition Let U be a predicate expression of the form

λx1 . . . xn A(x1, . . . , xn). The extension EMa (U) of U in a frame

M with respect to a∈As(M) is an n-ary relation R on |M| given

by

〈d1, . . . , dn〉 ∈ R iff

VMa(x1/d1,...xn/dn)

(A) = 1

where a(x1/d1, . . . xn/dn) is the assignment on M which is

like “a”, except perhaps on the variables x1, . . . , xn; here it

assigns d1, . . . , dn, respectively.

15

Note:

(1) For each n-ary predicate constant P :

EMa [λx1 . . . xn P(x1, . . . , xn)] = M|P |.

(2) For each n-ary predicate variable Φ:

EMa [λx1 . . . xn Φ(x1, . . . , xn)] = a(Φ).

λx A(x) stands for λx1 . . . xn A(x1, . . . , xn).

16

Let U = λx A(x), where x = (x1, . . . , xn), and let α =

(α1, . . . , αn) be an n-tuple of terms. The application of U

to α, written U(α), is the formula A(α).

The application is correct iff each αi is free for xi in A(x).

Examples

(1) If U = λxy P(x, y) and α = (z, f(b)), then

U(α) = P(z, f(b)). This is correct application.

(2) If U = λxy ∃z P(x, y, z) and α = (z, f(a)),

then U(α) = ∃z P(z, f(a), z). This is incorect application.

17

A substitution of predicate expressions for predicate variables

is a set

{Φ1 ← U1, . . . , Φn ← Un} (n ≥ 0)

where Φ1, . . . , Φn are distinct predicate variables, U1, . . . , Un

are predicate expressions and the corresponding Φi and Uihave the same arities.

Definition The application of a substitution θ = {Φ1 ←U1, . . . , Φn ← Un} to a formula A, written Aθ, is the formula

obtained from A by the following construction:

(1) Replace each free occurrence of Φi in A by ⌊Ui⌋.

(2) In the expression resulted from (1), which in general is

not a formula, successively replace each part of the form

⌊Ui⌋(α) by the application of Ui to α.

18

The application of θ to A is proper iff

(i) All the applications (of predicate expressions to terms)

performed in (2) are correct.

(ii) No free occurrence of any variable in any of U1, . . . , Un

becomes bound in Aθ.

Example Let

A = ∀xΦ(x) ⊃ ∀xΦ(b); θ = {Φ← λyP(x, y)}.

Applying θ to A, we first get

∀x ⌊λy P(x, y)⌋(x) ⊃ ∀x ⌊λy P(x, y)⌋(b)

and then

∀x P(x, x) ⊃ ∀x P(x, b).

19

Example Let

A = ∀xΦ(x)⊃Φ(y); θ = {Φ← λx∃yQ(x, y)}.

Applying θ to A, we first get

∀x ⌊λx ∃y Q(x, y)⌋(x) ⊃ ⌊λx ∃y Q(x, y)⌋(y)

and then

∀x ∃y Q(x, y) ⊃ ∃y Q(y, y).

This is an improper application because the application of

λx ∃y Q(x, y) to y is not correct (y is not free for x in

∃y Q(x, y)).

20

Example Let

A = ∀xy(Φ(x, y) ∨Ψ(x, y))

θ = {Φ← λxyP(x, y), Ψ← λuvQ(u, v)}.

Applying θ to A we first obtain

∀xy(⌊λxyP(x, y)⌋(x, y) ∨ ⌊λuvQ(u, v)⌋(x, y))

and then

∀xy(P(x, y) ∨Q(x, y)).

This is a proper application.

21

Theorem For any formula A and any substitution θ =

{Φ1 ← U1, . . . , Φn ← Un}, the formula

(∀Φ1 . . . ΦnA) ⊃ Aθ

is valid, provided that the application of θ to A is proper.

Corollary For any sentence of the form ∀Φ1 . . . ΦnA and any

substitution θ = {Φ1 ← U1, . . . , Φn ← Un},

∀Φ1 . . . ΦnA |= Aθ

provided that the application of θ to A is proper.

22

Definition An n-ary function expression F is any expression

of the form

λx1 . . . xn α(x1, . . . , xn) (n ≥ 0)

where x1, . . . , xn are individual variables and α(x1, . . . , xn) is a

term.

The following are function expressions:

λxf(x); λxy f(g(x), y); λxφ(x, y).

An n-ary function expression is meant to represent an n-ary

function.

23

Circumscription

We distinguish between:

• a predicate, thought of as a particular relation specified

over a particular domain, and

• predicate expressions, including predicate constants and

predicate variables, which can be used to denote this re-

lation.

Recall:

• An n-ary predicate constant P can be identified with the predicate

expression λx1 . . . xn P (x1 . . . xn).

• An n-ary predicate variable Φ can be identified with the predicate

expression λx1 . . . xn Φ(x1 . . . xn).

• The predicate denoted by a predicate expression U is called the exten-

sion of U .

24

General idea of circumscription

The objects (tuples of objects) that can be shown to satisfy

a certain predicate are the only objects (tuples of objects)

enjoying it.

More specifically: Given a theory T and an n-ary predicate

constant P , to circumscribe P in T is to assume that the

extension of P is minimal in the sense that it cannot be

made smaller without contradicting T .

Circumscription is thus a form of minimization: to circum-

scribe a predicate constant is to minimize its extension.

25

Let T consist of

Red(a) ∧On(a, b).

Circumscribing Red in T , we expect to infer

∀x Red(x) ⊃ x = a.

Circumscribing On in T , we expect to derive

∀xy. On(x, y) ⊃ x = a ∧ y = b.

Circumscription can be also used to minimize extensions of

several predicates.

26

All versions of circumscription share the following common

characteristics:

• Unlike other non-monotonic formalisms, circumscription allows us

to formalize non-monotonic reasoning directly in the language of

classical first-order logic.

• It is always the task of the user to specify predicates which are to

be minimized. Circumscription provides a general method which can

be applied to arbitrarily chosen predicates.

• Circumscription is based on syntactic manipulations. Given a the-

ory T and a list of predicate constants P1, . . . Pn, circumscription of

P1, . . . Pn in T amounts to implicitly adding a special second-order

sentence (a set of first-order formulas, in some versions of circum-

scription) capturing the desired minimization.

27

Abnormality formalism

(1) How the common-sense knowledge, including non-monotonic

rules, is to be represented in the framework of circum-

scription?

(2) Which predicates are to be minimized to assure intuitively

acceptable conclusions?

Ab − abnormal

∀x. Bird(x) ∧ ¬Ab(x) ⊃ Flies(x)

28

(1) Bird(Tweety)

(2) ∀x. Bird(x) ∧ ¬Ab(x) ⊃ Flies(x).

Conclusions:

∀x¬Ab(x)

∀x. Bird(x) ⊃ Flies(x)

Flies(Tweety).

(3) Bird(Clyde) ∧ ¬Flies(Clyde)

Conclusions:

∀x. x 6= Clyde ⊃ ¬Ab(x)

∀x. Bird(x) ∧ x 6= Clyde ⊃ Flies(x).

29

(1) Bird(Tweety)

(2) ∀x. Bird(x) ∧ ¬Ab(x) ⊃ Flies(x)

(3) Penguin(Joe)

(4) ∀x. Penguin(x) ⊃ Bird(x).

Conclusion:

Flies(Joe).

How to block this conclusion?

∀x. Penguin(x) ⊃ ¬Flies(x).

∀x. Penguin(x) ⊃ Ab(x).

30

∀x. Penguin(x) ∧ ¬Ab(x) ⊃ ¬Flies(x).

This is incorrect representation.

∀x. Penguin(x) ∧ ¬Ab1(x) ⊃ ¬Flies(x).

Ab and Ab1 should be jointly minimized.

Unfortunately, there is a conflict.

Minimization1: ¬Ab(Joe) and Flies(Joe)

Minimization2: ¬Ab1(Joe) and ¬Flies(Joe).

∀x. Penguin(x) ⊃ Ab(x).

31

There are non-monotonic rules requiring many-place abnor-

mality predicates:

“In the absence of evidence to the contrary, assume

that x gives y a gift, provided that y has a birthday

and x is a friend of y”.

∀xy. Birthday(y)∧Friend(x, y)∧¬Ab(x, y) ⊃ Gives−Gift(x, y).

Here Ab(x, y) should be read: “x behaves abnormally with

respect to y in the situation where y has a birthday and x is

a friend of y”.

32

Predicate circumscription(McCarthy 1980)

L - a fixed first-order language with equality.

The objects under consideration are finite sets of sentences

from L, referred to as theories.

Recall that every such a theory can be identified with a sin-

gle sentence, namely with the conjunction of all its members.

We write T(P1, . . . Pn) to indicate that some (but not neces-

sarily all) of predicate constants occurring in T are among

P1, . . . Pn.

33

L2 - a second-order language whose alphabet is that of L,

together with an infinite set of n-place (n≥0) predicate va-

riables.

Let U and V be n-ary predicate expressions. We write

U ≤ V for ∀x. U(x) ⊃ V (x)

where x = (x1, . . . xn) is a new n-tuple of individual variables

and ∀x stands for ∀x1 · · · ∀xn.

Note: U ≤ V states that the extension of U is a subset (not

necessarily proper) of the extension of V .

34

Let U and V , where U = (U1, . . . , Un) and V = (V1, . . . , Vn),

be similar tuples of predicate expressions, i.e. Ui and Vi have

the same arities, for each 1 ≤ i ≤ n. We write U ≤ V to

denote the sentence

n∧

i=1

[Ui ≤ Vi] i.e.n∧

i=1

[∀x. Ui(x) ⊃ Vi(x)].

U ≤ V states that the extension of each member of U is a

subset (not necessarily proper) of the extension of the corre-

sponding member of V .

35

Let T(P1, . . . , Pn) be a theory. How to express the fact that

the extensions of P1, . . . , Pn are minimal with respect to T ,

i.e. none of them can be made smaller without contradicting

T?

∀Φ1 . . . Φn.[

T(Φ1, . . . , Φn) ∧∧n

i=1(Φi ≤ Pi)]

⊃∧n

i=1(Pi ≤ Φi).

Here Φ1, . . . , Φn are predicate variables with the same ari-

ties as P1, . . . , Pn, respectively. T(Φ1, . . . , Φn) is the sentence

obtained from T(P1, . . . , Pn) by replacing all occurrences of

P1, . . . , Pn by Φ1, . . . , Φn, respectively.

Notice: The presence of T(Φ1, . . . , Φn) in the above formula

assures that the extensions of P1, . . . , Pn are minimized rela-

tive to T .

36

Definition The circumscription schema of P1, . . . , Pn in T ,

denoted by CS(T ; P1, . . . , Pn), is the second-order formula

T(Φ1, . . . , Φn) ∧n∧

i=1

[Φi ≤ Pi]

⊃n∧

i=1

[Pi ≤ Φi]

or, equivalently

T(Φ1, . . . , Φn) ∧n∧

i=1

[∀x. Φi(x) ⊃ Pi(x)]

⊃

⊃n∧

i=1

[∀x. Pi(x) ⊃ Φi(x)]

where Φ1, . . . , Φn and T(Φ1, . . . , Φn) are defined as before.

37

Example Let T = {Red(a) ∧On(a, b)}.

CS(T ; Red) =

[Φ(a) ∧On(a, b) ∧ [∀x. Φ(x) ⊃ Red(x)]] ⊃

⊃ [∀x. Red(x) ⊃ Φ(x)].

Definition Let T(P1, . . . , Pn) be a theory and let CS(T ; P1, . . . , Pn)

be the set of all formulas obtainable from CS(T ; P1, . . . , Pn)

by proper substitutions of first-order predicate expressions for

Φ1, . . . , Φn. The predicate circumscription of P = (P1, . . . , Pn)

in T , written CIRCPR(T ; P ), is T ∪ CS(T ; P1, . . . , Pn).

The set of theorems derivable by circumscribing P = (P1, . . . , Pn)

in T is identified with the set of theorems classically derivable

from CIRCPR(T ; P).

38

Example Let T = {Red(a) ∧On(a, b)}.

CS(T ; Red) =

[Φ(a)∧On(a, b)∧[∀x. Φ(x) ⊃ Red(x)]] ⊃ [∀x. Red(x) ⊃ Φ(x)].

We show that

CIRCPR(T ; Red) |= ∀x. Red(x) ⊃ x = a (1)

Substituting λx x = a for Φ, we get

[a = a ∧On(a, b) ∧ [∀x. x = a ⊃ Red(x)]] ⊃

⊃ [∀x. Red(x) ⊃ x = a].

From this, we immediately conclude (1).

39

Example We circumscribe “=” in T = {P(a)}.

CS(T ; =) =

[P(a) ∧ [∀xy. Φ(x, y) ⊃ x = y]] ⊃ [∀xy. x = y ⊃ Φ(x, y)].

Putting {Φ← λxy False}, we get

[P(a) ∧ [∀xy. False ⊃ x = y]] ⊃ [∀xy. x = y ⊃ False]

which is equivalent to P(a) ⊃ ∀xy. x 6= y.

Since T ⊢ P(a), we finally conclude that CIRCPR(T ; =) is

inconsistent. But if we add equality axioms the above deri-

vation is blocked.

40

Semantics for predicate circumscription

Definition Let T(P1, . . . , Pn) be a theory, some of whose pre-

dicate constants are among P = (P1, . . . , Pn). A model M of

T is a P -submodel of a model N of T , written M≤P N, iff

(1) |M| = |N|;

(2) M|P | ⊆ N|P |, for every predicate constant P in P ;

(3) M|K| = N|K|, where K is any function constant or any

predicate constant not in P .

M is a P -minimal model of T iff every model of T which is a

P -submodel of M is identical with M.

41

Theorem If CIRCPR(T ; P) ⊢ A, then A is true in every

P -minimal model of T .

Example Let T consist of:

Bird(Tweety)

∀x. Bird(x) ∧ ¬Ab(x) ⊃ Flies(x).

Consider the frame M, where:

|M|={TWEETY}; M|Flies|={};

M|Bird|=M|Ab|={TWEETY}.

M is an Ab-minimal model of T (we cannot make M|Ab| smal-

ler without contradicting T). Thus, since Flies(Tweety) is

false in M, this formula cannot be derived by circumscribing

Ab in T .

42

This example illustrates a more general phenomenon:

THEOREM: Neither positive nor negative new ground in-

stances of uncircumscribed predicates can be derived by pre-

dicate circumscription.

In view of the above theorem, predicate circumscription is

generally incapable to deal with non-monotonic rules!

Reason: While minimizing the extensions of circumscribed

predicates, predicate circumscription keeps the extensions of

all other predicates fixed.

43

Second-order circumscription(McCarthy, Lifschitz)

Second-order circumscription extends predicate circumscrip-

tion in three ways:

1. The process of minimization is expressed by a special cir-

cumscription axiom, being a single second-order sentence

rather than a set of first-order formulas.

2. The extensions of arbitrarily chosen predicate constants,

also of those not minimized, may be varied during the

minimization.

3. The extensions of arbitrarily chosen function constants

may be also allowed to vary in the process of minimiza-

tion.

44

Recall:

(1) If U and V are predicate expressions of the same arity,

then U ≤ V stands for ∀x. U(x) ⊃ V (x).

(2) If U = (U1, . . . , Un) and V = (V1, . . . , Vn) are similar tuples

of predicate expressions, then U ≤ V stands for

n∧

i=1

[Ui ≤ Vi] i.e.n∧

i=1

[∀x. Ui(x) ⊃ Vi(x)].

(3) We write U = V for (U ≤ V )∧ (V ≤ U), and U < V for

(U ≤ V ) ∧ ¬(V ≤ U).

45

Definition Let P be a tuple of distinct predicate constants,

let S be a tuple of distinct function and/or predicate con-

stants disjoint with P , and suppose that T(P , S) is a first- or

second-order theory. The second-order circumscription of P

in T with variable S, written CIRCSO(T ; P ; S), is the sentence

T(P , S) ∧ ∀ΦΨ ¬[T(Φ, Ψ) ∧Φ < P ]

where Φ and Ψ are tuples of variables similar to P and S,

respectively.

A formula A is said to be a consequence of second-order

circumscription of P in T with variable S iff

CIRCSO(T ; P ; S) |= A.

46

CIRCSO(T ; P ; S) can be rewritten as

T(P , S) ∧ ∀ΦΨ

(

T(Φ, Ψ) ∧ (Φ ≤ P) ⊃ (P ≤ Φ)

)

which, in turn, is an abbreviation for

T(P , S) ∧ ∀ΦΨ[(

T(Φ, Ψ) ∧∧n

i=1[∀x. Φi(x) ⊃ Pi(x)])

⊃

⊃∧n

i=1[∀x. Pi(x) ⊃ Φi(x)]]

.

47

Example Let T = {Bird(Tweety)∧

∧ [∀x. Bird(x) ∧ ¬Ab(x) ⊃ Flies(x)]}.

We take P = (Ab) and S = (Flies).

CIRCSO(T ; P ; S) = T(P , S) ∧ ∀ΦΨ(

Bird(Tweety) ∧ [∀x. Bird(x) ∧ ¬Φ(x) ⊃ Ψ(x)]∧

∧ [∀x. Φ(x) ⊃ Ab(x)]

)

⊃ ∀x. Ab(x) ⊃ Φ(x).

Taking {Φ←λxFalse; Ψ←λxBird(x)}, we get:

CIRCSO(T ; P ; S) |= ∀x. Ab(x) ⊃ False

CIRCSO(T ; P ; S) |= ∀x. ¬Ab(x)

CIRCSO(T ; P ; S) |= ∀x. Bird(x) ⊃ Flies(x)

CIRCSO(T ; P ; S) |= Flies(Tweety).

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Semantics for second-order circumscription

Theorem Let P , S and T(P , S) be as before. Let M and

N be models of T . M is a (P ; S)-submodel of N, written

M≤(P ;S)N, iff

(1) |M| = |N|;

(2) M|P | ⊆ N|P |, for every predicate constant P in P ;

(3) M|K| = N|K|, for any function or predicate constant K

not in P, S.

We write M<(P ;S) N iff M≤(P ;S) N but not N≤(P ;S) M. A model

M of T is (P ; S)-minimal iff T has no model N such that

N<(P ;S) M.

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MOD(P ;S)(T) - the class of all (P ; S)- minimal models of T .

MOD(T) - the class of all models of T .

Theorem For any T , P and S,

MOD(CIRCSO(T ; P ; S))=MOD(P ;S)(T).

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Satisfiability of second-order circumscription

Theorem Second-order circumscription need not preserve

satisfiability.

Definition A theory T is well-founded with respect to (P ; S)

iff, for every M∈MOD(T), there is N∈MOD(P ;S)(T) such that

N≤(P ;S) M.

Theorem If T is satisfiable and well-founded with respect

to (P ; S), then CIRCSO(T ; P ; S) is satisfiable.

Well-foundedness is not necessary for satisfiability of second-

order circumscription.

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A sentence is said to be universal iff its prenex normal form

contains no existential quantifiers. A theory is said to be

universal iff all of its axioms are universal.

Theorem If T is universal, then T is well-founded with re-

spect to any (P ; ()).

Corollary For any tuple of predicate constants P and any

satisfiable universal theory T , CIRCSO(T ; P ; ()) is satisfiable.

Theorem Second-order circumscription need not preserve

satisfiability for universal theories if function constants are

allowed to vary.

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The situation is different if varying constants are restricted

to predicate constants:

Theorem For any disjoint tuples P and S of predicate con-

stants, every universal theory is well-founded with respect to

(P ; S).

Corollary For any disjoint tuples P and S of predicate con-

stants and any satisfiable universal theory T , CIRCSO(T ; P ; S)

is satisfiable.

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Expressive power of second-order circumscription

Theorem If T(P , S) is well-founded with respect to (P ; S),

P ∈ P is an n-ary predicate constant, S is a tuple of predi-

cate constants disjoint with P , and α is an n-tuple of ground

terms, then

CIRCSO(T ; P ; S) |= P(α) iff T |= P(α).

Theorem If T(P , S) is well-founded with respect to (P ; S),

P and S are disjoint tuples of predicate constants, R is an

n-ary predicate constant not in P, S, and α is an n-tuple of

ground terms, then

(i) CIRCSO(T ; P ; S) |= R(α) iff T |= R(α)

(ii) CIRCSO(T ; P ; S) |= ¬R(α) iff T |= ¬R(α).

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Note: The only ground literals that may be obtained from

well-founded theories by using second-order circumscription

(without varying function constants) are negative instances

of circumscribed predicates and instances, both positive and

negative, of variable predicates.

This is not true if function constants are allowed to vary. In

such a case, both positive literals of circumscribed predi-

cates and new instances of non-variable predicates may

be obtainable.

No new formulas can be obtained by circumscribing “=”:

Theorem For any T , S and A:

CIRCSO(T ; =; S) |= A iff T |= A.

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The domain closure assumption, with respect to a theory T , is

the assumption that the only individuals to be considered are

those forced by the Herbrand universe of T . In the case where

T gives rise to a finite Herbrand universe {c1, . . . , cn}, the

domain closure assumption can be formalized by employing

the domain closure axiom (Reiter):

∀x. x = c1 ∨ · · · ∨ x = cn.

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Second-order circumscription cannot be used, at least for

well-founded theories and without varying functions, to ob-

tain the domain closure axioms:

Theorem If T(P , S) is well-founded with respect to (P ; S),

S contains no function constants, and α1, . . . , αn are ground

terms, then

CIRCSO(T ; P ; S) |= ∀x. x = α1 ∨ · · · ∨ x = αn

iff T |= ∀x. x = α1 ∨ · · · ∨ x = αn.

Frequently, second-order circumscription axiom can be repla-

ced by an equivalent first-order sentence. This allows to avoid

second-order logic and makes the formalism very attractive

for the purpose of practical applications (see Lukaszewicz:

“Non-Monotonic Reasoning” pp. 249-252).

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