COM SFWR 707: Formal Specification Techniques€¦ · COM SFWR 707: Formal Speciﬁcation...

COM SFWR 707:Formal

SpecificationTechniques

Dr. R. Khedri

Outline

Introduction

Basic Definitions

L-terms,Interpretation,L-formulas, andSatisfiability

Constructions

ElementaryEquivalence andIsomorphism

Theories

COM SFWR 707: Formal SpecificationTechniques

Dr. Ridha Khedri

Department of Computing and Software, McMaster UniversityCanada L8S 4L7, Hamilton, Ontario

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

1 Introduction

2 Basic DefinitionsMaps that preserve the interpretation of L

3 L-terms, Interpretation, L-formulas, and Satisfiability

4 ConstructionsL-Substructure (revisited)L–Quotient StructureDirect Product Structure

5 Elementary Equivalence and Isomorphism

6 TheoriesLogical Consequence

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Structures and Theories Introduction

In mathematical logic, we use first-order languages todescribe mathematical structures

Intuitively, a structure is a set that we wish to studyequipped with a collection of distinguished functions,relations, and elements

After that, we choose a language where we can talkabout them (Funct., rel., and elements) and nothingmore

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Structures and Theories Introduction

Example

When we study the ordered field of real numbers withthe exponential function

We study the structure 〈R,+, ·, exp, <, 0, 1〉What are the components of this structure?

We would use a language where we have symbols for+, ·, exp, <, 0, 1We can write statements such as:

∀(x , y | x , y ∈ R : exp(x) · exp(y) = exp(x + y) )

That we interpret as the assertion: exey = e(x+y) forall x and y in real numbers.

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions

Maps


Constructions


Theories

Structures and Theories Basic Definitions

Definition (Language)

A language L is given by specifying the following data:

1 a set of function symbols F and positive integers nf foreach f ∈ F

2 a set of relation symbols R and positive integers nR foreach R ∈ R

3 a set of constant symbols C.

τ = 〈F ,R, C, nF , nR〉

The numbers nf and nR tell us that f is a function ofnf variables and R is an nR -ary relation

Any or all of the sets F ,R and C may be empty

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions

Maps


Constructions


Theories


Example (Language)

〈{+,−, ·}, {}, {0, 1}, {(+, 2), (−, 2), (·, 2)}, {}〉 is thelanguage of rings Lr

〈{+,−, ·}, {<}, {0, 1}, {(+, 2), (−, 2), (·, 2)}, {(<, 2)}〉is the language of ordered rings Lor

The smallest language is that of pure indentity L=, inwhich no function, relation, or constant occur

It means τ = 〈{}, {}, {}, {}, {}〉

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions

Maps


Constructions


Theories


Definition (L-structure)

An L-structure M is given by the following data:

a nonempty set M called the universe, domain, orunderlying set of M;

a function f M : Mnf −→ M for each f ∈ F ;

a set RM ⊆ MnR for each R ∈ R;

an element cM ∈ M for each c ∈ C.

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions

Maps


Constructions


Theories


We refer to f M,RM, and cM as the interpretations ofthe symbols f ,R, and c

We often write the structure as

M = 〈M, f M,RM, cM : f ∈ F ,R ∈ R, andc ∈ C〉

We will use the notation A,B,M,N, · · · to refer to theunderlying sets of the structures A,B,M,N , · · ·

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions

Maps


Constructions


Theories


Example

Suppose that we are studying groups

We might use the language Lg = {·, e}F = {·}n(·) = 2C = {e}

An Lg -structure G = (G , ·G , eG)G = (R, ·, 1) is a Lg -structure

G = (N,+, 0) is a Lg -structureG = (N,+, 0) is NOT a group, but it is a Lg -structure

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions

Maps


Constructions


Theories

Structures and Theories Basic DefinitionsMaps that preserve the interpretation of L

Definition (L-embedding)

Suppose that M and N are L-structures with universes M,and N, respectively. An L-embedding η : M−→ N is aone-to-one map (i.e., injective) η : M −→ N that preservesthe interpretation of all the symbols of L.

Which means that:À ∀

(f , a1, · · · , anf

| f ∈ F ∧ a1, · · · , anf∈ M :

η(f M(a1, · · · , anf

))

= f N (η(a1), · · · , η(anf))

)Á ∀

(R, a1, · · · , amR

| R ∈ R ∧ a1, · · · , amR∈ M :

(a1, · · · , amR) ∈ RM ⇐⇒ (η(a1), · · · , η(amR

)) ∈ RN)

Â ∀(c | c ∈ C : η(cM) = cN

)

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions

Maps


Constructions


Theories


Definition (L-isomorphism)

A bijective L-embedding is called an L-isomorphism.

Definition (Substructure/Extension)

If M ⊆ N and the inclusion map is an L-embedding, we sayeither that M is a substructure of N or that N is anextension of M.

Example

Z def= (Z,+Z , 0Z) is a substructure of

R def= (R,+R, 0R)

If η : Z −→ R is the function η(x)def= exp x , then η is

an L-embedding of (Z,+, 0) into (R, ·, 1).

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions

Maps


Constructions


Theories


Recall– Cardinal Numbers:

Definition

Two sets, A and B, are said to be equipollent, if there existsa bijective map A −→ B. In this case we write A ∼ B.

Theorem

Equipollence is an equivalence relation on the class of allsets.

Let I0 = ∅ and ∀(n | n ∈ N∗ : Indef= {1, 2, · · · , n} )

In ∼ Im ⇐⇒ n = m

To say that a set A has n elements means that A ∼ InWhen A ∼ In for some unique n ≥ 0, we say that A isfinite

A set that is not finite is infinite

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions

Maps


Constructions


Theories


Definition

The cardinal number (or cardinality) of a set A, denoted by|A|, is the equivalence class of A under the equivalencerelation of equipollence. |A| is an infinite or finite cardinalaccording as A is an infinite or finite set.

We shall identify the integer n ≥ 0 with the cardinalnumber |In| and write |In| = n

So, the cardinal number of a finite set is precisely thenumber of elements in the setWhat about the cardinality of a structure?

The cardinality of M is |M|, the cardinality of theuniverse of M

If η : M−→ N is an embedding then the cardinality ofN is at least the cardinality of M.

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions

Maps


Constructions


Theories


Definition

Let α and β be cardinal numbers. The sum α+ β is definedto be the cardinal number |A ∪ B|, where A and B aredisjoint sets such that |A| = α and |B| = β. The productαβ is defined to be the cardinal number |A× B|.

Theorem

If A is a set and P(A) its power, then |A| < |P(A)|.[Schroeder-Bernstein] If A and B are sets such that|A| ≤ |B| and |B| ≤ |A|, the |A| = |B|.The class of all cardinal numbers is linearly ordered by≤. If α and β are cardinal numbers, then exactly oneof the following is true:

α < β; α = β; β < α (Trichotomy Law)

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions

Maps


Constructions


Theories


What about the cardinality of a structure M?

The cardinality of M is |M|, i.e., the cardinality of theuniverse of M.

If η : M−→ N is an embedding, then the cardinalityof N is at least the cardinality of M.

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Structures and Theories L-terms,Interpretation, L-formulas, and Satisfiability

We use the language L to create formulas describingproperties of L-structures

Formulas will be strings of symbols built using

the symbols of L

variable symbols v1, v2, · · · , vn

the equality symbol =

the Boolean connectives ∧ ,Or ,¬

the quantifiers: ∃ and ∀

and parentheses ( , )

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories


Definition (L-terms)

The set of L-terms is the smallest set T such that

1 ∀(c | c ∈ C : c ∈ T )

2 ∀(i | i ∈ N : vi ∈ T )

3 ∀(f , t1, · · · , tn | f ∈ F ∧ t1, · · · , tn ∈ T :f (t1, · · · , tn) ∈ T )

Example

·(v1,−(v3, 1)), and · (+(v1, v2),+(v3, 1)) are Lr -terms

In the Lr -Structure (Z,+, ·, 0, 1), we think of the term

1 + (1 + (1 + 1)) as a name for the element 4(v1 + v2)(v3 + 1) is a name for the function(x , y , z) 7−→ (x + y)(z + 1)

Defining functions in this way can be done in any L-structure

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories


Interpretation of an L-term

Let M be an L-structure and that t is a term builtusing variables from v = (vi1 , · · · , vim)

We want to interpret t as a function tM : Mm −→ M

For s a subterm of t and a = (ai1 , · · · , aim) ∈ Mm, weinductively define sM(a) as follows.

1 If s is a constant symbol c , then sM(a) = cM.2 If s is the variable vij , then sM(a) = aij .3 If s is the term f (t1, · · · , tnf

), where f is a functionsymbol of L and t1, · · · , tn are terms, thensM(a) = fM

(tM1 (a), · · · , tMnf

(a)).

The function tM is defined by a 7−→ tM(a).

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories


Example

Let L = {f , g , c}, where n(f ) = 1, n(g) = 2, and c is aconstant symbol

take the L-terms t1 = g(v1, c), t2 =f (g(c , f (v1))), and t3 = g(f (g(v1, v2)), g(v1, f (v2)))

Let M be the structure (R, exp,+, 1); that is

fM = expgM = +cM = 1

Give the following

tM1 (a1)tM2 (a1)tM3 (a1, a2)

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories


Example

In breeding a strain of cattle, which can be black or brown,monochromatic or spotted , it is known that black isdominant and brown recessive and that monochromatic isdominant over spotted.

1 Give the possible types of cattle in this herd.

2 Due to dominance, in crossing a black spotted one witha brown monochromatic one, we expect a blackmonochromatic one. Give the “operation” ∗ thatsymbolizes this phenomenon.

3 Define the structure C def= (C , ∗).

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories


Definition (L-formula)

We say that φ is an atomic L-formla if φ is either

1 t1 = t2, where t1 and t2 are terms, or

2 R(t1, · · · , tnR), where R ∈ R and t1, · · · , tnR

are terms.

Definition

The set of L-formulas is the smallest set W containing theatomic formulas such that

1 if φ is in W, then¬φ is in W,

2 if φ and ψ are in W, then φ ∧ ψ and φ ∨ ψ are in W,and

3 if φ is in W, then ∃(vi |: φ ) and ∀(vi |: φ ) are inW.

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories


Definition (sentence)

We call a formula a sentence if it has no free variables.

Definition (Satisfiability)

Let φ be a formula with free variables fromv = (vi1 , · · · , vim) and let a = (ai1 , · · · , qim) ∈ Mm. Weinductively define M |= φ(a) as follows.

If φ is t1 = t2, then M |= φ(a) if tM1 (a) = tM2 (a)

If φ is R(t1, · · · , tnR), then M |= φ(a) if

(tM1 , · · · , tMnR) ∈ RR

If φ is ¬ψ, then M |= φ(a) if M 6|= ψ(a)

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories


Definition (Satisfiability–Continued–)

If φ is ψ ∧ θ, then M |= φ(a) if M |= ψ(a) andM |= θ(a)

If φ is ψ ∨ θ, then M |= φ(a) if M |= ψ(a) orM |= θ(a)

If φ is ∃(vj |: ψ(v , vj) ), then M |= φ(a) if there isb ∈ M such that M |= φ(a, b)

If φ is ∀(vj |: ψ(v , vj) ), if∀(b | b ∈ M : M |= φ(a, b) )

If M |= φ(a) we say that M satisfies φ(a) or φ(a) is true inM.

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories


Proposition

Suppose that M is a substructure of N , a ∈ M, and φ(v) isa quantifier-free formula. Then, M |= φ(a) if and only ifN |= φ(a).

Proof.

Claim:If t(v) is a term and b ∈ M, then tM(b) = tN (b). This isproved by induction on terms.Then, we prove the proposition by induction on formulas(atomic ones and then for composite ones)

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions

L-Substructure

L-Quotient Structure

Direct ProductStructure


Theories

Structures and Theories ConstructionsL-Substructure (revisited)

Definition

Let M and N be L-structures with M ⊆ N. Then M is a

(L-)substructure of N , noted by M⊆ N , if

∀(f ,R | f ∈ F ∧ R ∈ R :

f M = (f N ∩Mnf +1)

∧ RM = (RN ∩MnR ))

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions

L-Substructure




Theories

Structures and Theories ConstructionsL–Quotient Structure

Consider an L-structure M and an equivalence relation θ.

Definition (Compatibility with a structure)

We say that θ is compatible with M if, given f ∈ F , andboth a and b in Mnf such that∀(i | 1 ≤ i ≤ nf : (ai , bi ) ∈ θ ), THEN(f M(a), f M(b)

)∈ θ.

A compatible equivalence relation on M is also calledcongruence on M and sometimes said to be congruentto M

NOTE: The compatibitity condition for a congruencepertains only to the algebraic part of the structure M

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions

L-Substructure




Theories


Theorem

Let θ be an equivalence relation on a monoid G def= (G , ·, e)

such that (a1, a2) ∈ θ and (b1, b2) ∈ θ implies(a1 · b1, a2 · b2) ∈ θ for all ai , bi ∈ G. Then the structureG/θ = (G/θ, ·/θ, e) is a monoid, where

G/θ is is the set of all equivalence classes of G over θ

·/θ is the binary operation defined by a ·/θ bdef= a · b,

where x denotes the equivalence class of x ∈ G.

e is the equivalence class of the identity element e

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions

L-Substructure




Theories


Definition

Given an L-structure M and an equivalence relation θ onM compatible with M, then we define the quotientstructure M/θ of M by θ as follows:

The support set of M/θ is M/θ. The equivalenceequivalence class x = {y | (x , y) ∈ θ}.For every f ∈ F , the corresponding L-operation f M/θ

on M/θ, which we denote f θ, is defined by

f θ(a1, · · · , anf

) def= f M

(a1, · · · , anf

)For every relation R ∈ R,

(a1, · · · , anR

)∈ Rθ if the

following holds:

∀(i | 1 ≤ i ≤ nR : ∃(bi | bi ∈ ai : (b1, · · · , bmR) ∈ RM ) )

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions

L-Substructure




Theories


The compatibility of θ with M insures that theoperation F θ is well-defined on equivalence classes

The properties of equivalence relations (transitivity inparticular) assure that Rθ is well-defined relation onM/θ

Thus, the quotient structure is an L-structure

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions

L-Substructure




Theories


Associated with the construction, from M and θ, ofM/θ is the quotient morphism (or, quotient map)

qθ : M −→ M/θ

which puts every element x ∈ M in its equivalence

class modulo θ (i.e., qθ(x)def= xθ)

Exercise

Show that qθ is surjective morphism.

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions

L-Substructure




Theories


Definition (Kernel)

Let η : M−→ N be a homorphism of the L-structures Mand N .By the kernel of η, noted ker(η), we mean the equivalencerelation θη on M defined by

(x , y) ∈ θη ⇐⇒ η(x) = η(y)

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions

L-Substructure




Theories

Structures and Theories Constructions DirectProduct Structure

Definition (Direct Product)

Let {Mi}I ={(

Mi , {f i}f ∈F , {R i}R∈R)

| i ∈ I}, be an

I -indexed family of L-structures. The direct product ΠIMi

of the family is defined as follows:

The support set if ΠIMi (i.e., the Cartesian Product ofMi )

Operations on the product are defined componentwise

Given R ∈ R, the relation RΠ on ΠIMi is defined as

follows:

(x1, · · · , xm) ∈ RΠ

⇐⇒ ∀(i | i ∈ I : (x1(i), · · · , xm(i)) ∈ R i ),

where m is the arity m(R) of R and(x1, · · · , xm) ∈ (ΠIAi )

m.

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions

L-Substructure




Theories


Clearly, ΠIMi =(ΠIMi , {f Π}f ∈F , {RΠ}R∈R

)as it is

defined has the same language as L as each of thestructures in the family {Mi}I .

The set I can be empty: the empty product Π∅ has asupport with one element e.

R∅ = {(e, · · · , e)}

If ∀(i , j | i , j ∈ I : Mi = N = Mj ), thenΠIMi = N |I | denoted N I .

N I def= ΠIMi is called I -direct power of the

L-structure N .

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions

L-Substructure




Theories


Exercise

Show that the projections

Πi : ΠIMi −→ Mi

are surjective homomorphisms.

Note: ∀(x , i | x ∈ ΠIMi ∧ i ∈ I : Πi (x) = xi )

Πi is said to the ith-projection of ΠIMi onto Mi

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Structures and Theories ElementaryEquivalence and Isomorphism

We next consider structures that satisfy the same sentences

Definition

We say that two L-structures M and N are elementarilyequivalent and write M≡ N if

∀(φ | φ is an L-sentence : M |= φ ⇐⇒ N |= φ ).

We let Th(M), the full theory of M, be the set ofL-sentences φ such that M |= φ

It is easy to see that M≡ N if and only ifTh(M) = Th(N)

Our next result shows that Th(M) is an isomorphisminvariant of M

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Structures and Theories ElementaryEquivalence and Isomorphism

Theorem

Suppose that j : M−→ N is an isomorphism. Then,M≡ N .

Proof highlights.

We show by induction on formulas thatM |= φ(a1, · · · , an) if and only ifN |= φ

(j(a1), · · · , j(an)

)for all formulas φ.

We first must show that terms behave well

Suppose that t is a term and the free variables in t arefrom v = (v1, · · · , vn). For a = (a1, · · · , an) ∈ M, welet j(a) denote (j(a1), · · · , j(an)). Thenj(tM(a)) = tN (j(a))We prove this by induction on terms.

Then, we proceed by induction on formulas.

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Logical Consequence

Structures and Theories Theories

Definition (Theory)

Let L be a language. An L-theory T is simply a set ofL-sentences. We say that M is a model of T and writeM |= T if

∀(φ | φ ∈ T : M |= φ )

Example

The set T = { ∀(x |: x = 0 ), ∃(x |: x 6= 0 )} is a theory.

Definition (Satisfiable Theory)

We say that a theory is satisfiable if it has a model.

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Logical Consequence


Definition (Elementary class)

We say that a class of L-structures K is an elementary classif there is an L-theory T such that K = {M | M |= T}.

One way to get a theory is to take Th(M), the fulltheory of an L-structure MIn this case, the elementary class of models of Th(M)is exactly the class of L-structures elementarilyequivalent to MMore typically, we have a class of structures in mindand try to write a set of properties T describing thesestructures

We call these sentences axioms for the elementary class

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Logical Consequence


Example (Infinite Sets)

Let L = ∅Consider the L-theory where we have, for each n, thesentence φn given by∃(x1 |: ∃(x2 |: · · · ∃(xn |:∧(i , j | i < j ≤ n : xi 6= xj ) ) ) · · · )

The sentence φn asserts that there are at least ndistinct elements

An L-structure M with universe M is a model of T ifand only if M is infinite

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Logical Consequence


Example (Linear Orders)

Let L = {<} where < is a binary relation symbol

The class of linear orders is axiomatized by theL-sentences

1 ∀(x |: ¬(x < x) )2 ∀(x , y , z |: (x < y ∧ y < z) =⇒ x < z )3 ∀(x , y |: x < y ∨ x = y ∨ y < x )

To get the theory of dense linear orders, we could add:

∀(x , y | x < y : ∃(z |: x < z ∧ z < y ) )

To get the theory of linear orders where every elementhas a unique successor, we could add:

∀(x |: ∃(y |: x < y ∧ ∀(z | x < z : z = y ∨ y < z ) ) )

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Logical Consequence


Example (Equivalence Relations)

Let L = {θ} where θ is a binary relation symbol

The theory of equivalence relations is given by thesentences

1 ∀(x |: (x , x) ∈ θ )2 ∀(x , y | (x , y) ∈ θ : (y , x) ∈ θ )3 ∀(x , y , z | (x , y) ∈ θ ∧ (y , z) ∈ θ : (x , z) ∈ θ )

To get the theory of equivalence relations where every

equivalence class has exactly two elements, we add

∀(x |: ∃

(y | x 6= y ∧ (x , y) ∈ θ

: ∀(z | (x , y) ∈ θ : z = x ∨ z = y )) )

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Logical Consequence


Example (Graphs)

Let L = {R} where R is a binary relation symbol

The theory of irreflexive graphs is axiomatized by1 ∀(x |: (x , x) 6∈ R )2 ∀(x , y | (x , y) ∈ R : (y , x) ∈ R )

Example (Groups)

Let L = {·, e}The class of groups is axiomatized by

1 ∀(x |: e · x = x · e = x )2 ∀(x , y , z |: x · (y · z) = (x · y) · z )3 ∀(x |: ∃(y |: x · y = y · x = e ) )

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Logical Consequence


Example (Ordered Abelian Groups)

Let L = {+, <, 0}The axioms for ordered groups are:

1 The axioms for additive groups

2 The axioms for linear orders

3 ∀(x , y , z | x < y : x + z < y + z )

The literature is full of theories given by their axioms: Rings,Semi-modules, Modules, Fields, Peano Arithmetic, etc.

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Logical Consequence

Structures and Theories Theories LogicalConsequence

Definition (Logical Consequence)

Let T be an L-theory and φ an L-sentence. We say that φis a logical consequence of T and write T |= φ if M |= φwhenever M |= T .

Example

Let L = {+, <, 0} and let T be the theory of OrderedAbelian Groups

∀(x | x 6= 0 : x + x 6= 0 ) is a logical consequence ofT

To show that T |= φ1 we find a model of T (i.e., M |= T )2 we show that M |= φ

To show that T 6|= φwe usually construct a couterexample

COM SFWR 707:Formal


Dr. R. Khedri

Outline

Introduction

Basic Definitions


Constructions


Theories

Logical Consequence

Date post:	20-Aug-2020
Category:	Documents
Upload:	others
View:	2 times
Download:	0 times

COM SFWR 707: Formal Specification Techniques€¦ · COM SFWR 707: Formal Speciﬁcation...

Documents