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Formal Methods in software development
a.a.2015/2016
Prof. Anna Labella
5/23/2017
1
The predicate calculus [H-R ch.2]
The need for a richer language
 Terms and formulas


quantifiers
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Examples

Every student preparing his homework must be left
alone
x (((x)  ψ(x))  χ(x))
 For every natural number there is number greater
than it
x (N(x)  y (N(y) G(y,x))
x y G(y,x)

There are people who do not like their dog
x (x, d(x))
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Syntax
Inductive term definition
 BNF


t::= x | c | f(t1,t2,…,tn)
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Syntax

Inductive definition of wff

BNF
::= p(t1,t2,…,tn) | t1=t2 | () | () |
() | () | x | x
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Peano’s arithmetics
Predicates: - = -, - ≤  Functions : - + -, - . -, s(-),…
 Constants: 0

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Exercises
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Syntax
Free and bounded variables
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Syntax
Open and closed formulas
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Syntax
Be careful with substitution!
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Syntax
Free and bounded variables
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Natural deduction 3
Proof rules for equality
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Natural deduction 3
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Natural deduction 3
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Exercises
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Natural deduction 1
Proof rules for universal quantification
where t is free for x in φ
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where x0 is a fresh variable
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Natural deduction 2
Proof rules for existential quantification
where t is free for x in φ
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where x0 is a fresh variable
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Examples of proofs
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where x0 is a fresh variable
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Examples of proofs
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Semantics
syntactical data
 F functional symbols- constants
 P predicate symbols
An interpretation M
 a non empty set (domain) A
 F  a set of functions fM on A (An)
 P  a set of relations PM on A (An)
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Semantics
A subset is the interpretation of a 1-ary
predicate
A n-ary relation is the interpretation of a
n-ary predicate
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An example: arithmetics
syntactical data
F: f1(-,-), f2(-,-), f0(-), c
P: - = -, P1(-,-)
An interpretation M
 Natural numbers (domain)
 Functions : - + -, - . -, s(-), 0
 Predicates: - = -, - ≤ 5/23/2017
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Semantics
environments
Assigning values
 l: var  elements of A
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An example: arithmetics 2
 f1(c,x)=x always true in the interpretation
 f2(c,x)=x sometimes true sometimes false in the
interpretation depending on the assignment
x(f1(c,x)=x) true in the interpretation
 x(f2(c,x)=x) true in the interpretation
 x(f2(c,x)=x) false in the interpretation
 x(f1(c,x)=f1(c,x)) valid

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Semantics
An interpretation M
is a model of 
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M
|=l  :
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Semantics: satisfiability
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Exercises
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Semantics: validity
 is valid if for every intepretation and
for every environment
M |= 
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Properties
Soundness  |-   M |= 
 Completeness M |=    |- 
 Indecidability
 Compactness


Expressivity
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Second order logic
Existential
second order logic

Universal
second order logic
Peano’s arithmetics
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