Crash course in IDP
Part 1:
predicate logic
20211208
Slides by Jo Devriendt & Benjamin Callewaert
0. Sets, relations and functions
Let's quiz the homework :)
 What is a set?
 A collection of elements
 Does the order matter in a set?
 No
 What about duplicates?
 None allowed
 Given a set S with n elements, how many subsets does S have?
 2^n (each element can either be in or out the subset)
Let's quiz the homework :)
 What is a tuple?
 A finite list of elements
 Does the order matter in a tuple?
 Yes
 What about duplicates?
 Allowed
 What is a cartesian product?
 The set of all possible tuples over some given sets.
 Given a set S with n elements, how many tuples belong to the kfold cartesian product over S?
 n^k
 What is the 0fold cartesian product?
 The set with exactly one element  the empty tuple: { () }
0. Sets, relations and functions
Let's quiz the homework :)
 What is a relation?
 A subset of some given cartesian product.
 Given a set S with n elements, how many relations exist over the kfold cartesian product over S?
 2^(n^k), as n^k is the size of the cartesian product, and 2^(n^k) is the number of subsets of the cartesian product, which is exactly the set of all relations over this cartesian product
0. Sets, relations and functions
Let's quiz the homework :)
 What is a function?
 A mapping from some set to some other
 What is its signature, arity, domain, codomain?
 A function f with signature
has an arity k, codomain S, and domain
 A function f with signature
 Given a set S with n elements, how many functions exist with signature ?
 n^(n^k), as n^k is the number of tuples in the domain, and we can pick any value from codomain S for each tuple, so we can pick n values for each of the n^k tuples
0. Sets, relations and functions
 A vocabulary declares the types, predicates and functors of the specification
 Predicates and functors have an associated signature over the types
 in IDPZ3, predicates are treated as functors to Bool
1. Vocabularies and structures in FO(.)
vocabulary People {
type Person // type
parentOf: (Person * Person) > Bool // predicate
man: (Person) > Bool // predicate
woman: (Person) > Bool // predicate
brotherOf: (Person * Person) > Bool // predicate
sisterOf: (Person * Person) > Bool // predicate
age: (Person) > Int // functor
eldestDaughter: () > Person // functor
}
builtin type
constant: functor with arity 0
1. Vocabularies and structures in FO(.)
 A structure gives interpretations to symbols in the vocabulary
 type > set (contents are called domain elements)
 predicate > relation
 functor > function
structure Simpsons : People {
Person := {marge, homer, lisa, bart, maggie}
parentOf := { (marge, bart), (marge, lisa), (marge, maggie),
(homer, bart), (homer, lisa), (homer, maggie) }
man := { (homer), (bart) }
woman := ??
brotherOf := ??
sisterOf := ??
age := { (homer) > 36, (marge) > 34, (bart) > 10,
(lisa) > 8, (maggie) > 1 }
eldestDaughter := ??
}
domain element
structure Simpsons : People {
Person := {marge, homer, lisa, bart, maggie}
parentOf := { (marge, bart), (marge, lisa), (marge, maggie),
(homer, bart), (homer, lisa), (homer, maggie) }
man := { (homer), (bart) }
woman := ??
brotherOf := ??
sisterOf := ??
age := { (homer) > 36, (marge) > 34, (bart) > 10,
(lisa) > 8, (maggie) > 1 }
eldestDaughter := ??
}
vocabulary People {
type Person // type
parentOf: (Person * Person) > Bool // predicate
man: (Person) > Bool // predicate
woman: (Person) > Bool // predicate
brotherOf: (Person * Person) > Bool // predicate
sisterOf: (Person * Person) > Bool // predicate
age: (Person) > Int // functor
eldestDaughter: () > Person // functor
}
set
relation
relation
function
structure Simpsons : People {
Person := {marge, homer, lisa, bart, maggie}
parentOf := { (marge, bart), (marge, lisa), (marge, maggie),
(homer, bart), (homer, lisa), (homer, maggie) }
man := { (homer), (bart) }
woman := { (marge), (lisa), (maggie) }
brotherOf := { (bart, lisa), (bart, maggie) }
sisterOf := { (lisa, bart), (maggie, bart), (lisa, maggie),
(maggie, lisa) }
age := { (homer) > 36, (marge) > 34, (bart) > 10,
(lisa) > 8, (maggie) > 1 }
eldestDaughter := { () > lisa }
}
vocabulary People {
type Person // type
parentOf: (Person * Person) > Bool // predicate
man: (Person) > Bool // predicate
woman: (Person) > Bool // predicate
brotherOf: (Person * Person) > Bool // predicate
sisterOf: (Person * Person) > Bool // predicate
age: (Person) > Int // functor
eldestDaughter: () > Person // functor
}
vocabulary MapCol {
type Country
type Color
border: (Country * Country) > Bool
colorOf: (Country) > Color
}
structure Correct : MapCol {
Country := { BE, NL, DE, LU, FR }
Color := { R, G, B, Y}
border := { (NL, BE), (NL, DE), (BE, LU),
(BE, FR), (LU, FR), (LU, DE), (DE, FR) }
colorOf := { (BE)>B, (NL)>R,
(DE)>G, (LU)>Y, (FR)>R }
}
structure Nonsense : MapCol {
Country := { BE, NL, DE, LU, FR }
Color := { R, G, B, Y}
border := { (BE, BE), (FR, NL) }
colorOf := { (BE)>R, (NL)>R, (DE)>R, (LU)>R, (FR)>R }
}
??
2. Terms and atoms
maggie
eldestDaughter()
age(eldestDaughter())
 A term is either
 a domain element
 a constant
 an application of a functor to the right amount and type of terms
2. Terms and atoms
 An atom is either
 an equality between two terms
 a disequality between two terms
 an application of a predicate to the right amount and type of terms
eldestDaughter() = maggie
maggie ~= eldestDaughter()
age(eldestDaughter()) ~= 0
brotherOf(eldestDaughter(), bart)
NOTE: FO(.) is typed!
age(0) = maggie
2. Terms and atoms
vocabulary People {
type Person := {homer, marge, bart, lisa, maggie}
parentOf: (Person * Person) > Bool // predicate
man: (Person) > Bool // predicate
woman: (Person) > Bool // predicate
brotherOf: (Person * Person) > Bool // predicate
sisterOf: (Person * Person) > Bool // predicate
age: (Person) > Int // functor
eldestDaughter: () > Person // functor
}
homer
man
0 ~= 1
bart = brotherOf(lisa)
parentOf(marge, eldestDaughter())
age(lisa) = eldestDaughter()
age(eldestDaughter())
age(age(age(age(lisa))))
age(bart) ~= age(eldestDaughter())
sisterOf(eldestDaughter(),bart)
eldestDaughter()
parentOf(bart, eldestDaughter())
term, atom or bug?
term (domain element)
bug: predicate without input
atom, maps to true/false (0 and 1 ar domain elements of builtIn type Int)
bug, brotherOf takes 2 arguments
atom (false in above structure, but that does not matter)
bug, eldestDaughter() maps to Person, age maps to Int
term, maps to Int
bug, age maps to Int and takes Person as input, so can not be applied to itself
atom
atom
term
atom (correct predicate)
2. Terms and atoms
A structure is a "possible world". Terms and atoms are statements which can be evaluated in a structure. This evaluation under a structure is (again) called an interpretation.
Informally, a structure is a solution, interpretations are values, and predicates & functors are variables. Statements such as terms, atoms, formulas have a derived value.
2. Terms and atoms
 A term is either
 a domain element
 a constant
 an application of a functor to the right amount and type of terms
Given a structure U, the interpretation of
 a domain element is itself
 a constant C() is C's interpretation in U
 a functor application f(t1, ..., tk) is the value obtained by applying the interpretation of f in U to the interpretation of (t1, ..., tk) in U
maggie
eldestDaughter()
age(eldestDaughter())
age := {(homer) > 36,
(marge) > 34,
(bart) > 10,
(lisa) > 8,
(maggie) > 1}
eldestDaughter :=
{() > lisa}
Interpretations of terms are always domain elements of some type
2. Terms and atoms
Given a structure U, the interpretation of
 an equality is true iff both terms have the same interpretation
 a disequality is true iff both terms have a different interpretation
 a predicate application p(t1, ..., tk) is true iff the interpretation of (t1, ..., tk) in U is an element of the interpretation of p in U
 An atom is either
 an equality between two terms
 a disequality between two terms
 an application of a predicate to the right amount and type of terms
eldestDaughter() = maggie
maggie ~= eldestDaughter()
age(eldestDaughter()) ~= 0
brotherOf(eldestDaughter(), bart)
Interpretations of atoms are truth values:
true or false
2. Terms and atoms
structure Simpsons : People {
Person := {marge, homer, lisa, bart, maggie}
parentOf := { (marge, bart), (marge, lisa), (marge, maggie),
(homer, bart), (homer, lisa), (homer, maggie) }
man := { (homer), (bart) }
woman := { (marge), (lisa), (maggie) }
brotherOf := { (bart, lisa), (bart, maggie) }
sisterOf := { (lisa, bart), (maggie, bart), (lisa, maggie),
(maggie, lisa) }
age := { (homer) > 36, (marge) > 34, (bart) > 10,
(lisa) > 8, (maggie) > 1 }
eldestDaughter := { () > lisa }
}
eldestDaughter() = maggie
maggie ~= eldestDaughter()
age(eldestDaughter()) ~= 0
brotherOf(eldestDaughter(), bart)
false
true
true
false
2. Terms and atoms
homer
0 ~= 1
eldestDaughter()
brotherOf(bart,lisa)
age(eldestDaughter())
age(bart) ~= age(eldestDaughter())
parentOf(bart, eldestDaughter())
if not bug, what is interpretation in Simpsons?
term: homer (interpretation odomain element is itself)
atom: true
term: lisa
atom: true
term: 8
atom: true
atom: false
3. Quantorfree formulas
brotherOf(eldestDaughter(), bart)
~brotherOf(eldestDaughter(), bart)
~brotherOf(eldestDaughter(), bart) & bart ~= eldestDaughter()
brotherOf(eldestDaughter(), bart)  bart ~= eldestDaughter()
eldestDaughter()=lisa => woman(eldestDaughter())
eldestDaughter()=lisa <=> woman(eldestDaughter())
A QF formula is either
 an atom
 a negation of a formula
 a conjunction of two formulae
 a disjunction of two formulae
 an implication of two formulae
 an equivalence of two formulae
Order of operations:
3. Quantorfree formulas
A QF formula is either
 an atom
 a negation of a formula
 a conjunction of two formulae
 a disjunction of two formulae
 an implication of two formulae
 an equivalence of two formulae
Given a structure U, the interpretation of
 negation = "not"
 conjunction = "and"
 disjunction = "or"
 implication = "if ... then ..."
 equivalence is true iff both formula have the same interpretation
Tricky!
P  Q  P => Q 

false  false  true 
false  true  true 
true  false  false 
true  true  true 
P  Q  P <=> Q 

false  false  true 
false  true  false 
true  false  false 
true  true  true 
Truth table
3. Quantorfree formulas
rain()  cloudy()  rain() => cloudy() 

false  false  true 
false  true  true 
true  false  false 
true  true  true 
rain() => cloudy()
"If it rains, then it is cloudy."
This is still true even if there are clouds but no rain!
3. Quantorfree formulas
rain()  cloudy()  ~rain()  cloudy() 

false  false  true 
false  true  true 
true  false  false 
true  true  true 
rain() => cloudy()
"If it rains, then it is cloudy."
Note that the implication is equivalent to the following disjunction:
~rain()  cloudy()
3. Quantorfree formulas
Logical Equivalence: Two formulas F and G are logically equivalent F ≡ G if the truth values of both formulas F and G are always the same.
 (P ∧ Q) ≡ (Q ∧ P) and (P ∨ Q) ≡ (Q ∨ P) (commutativity)
 ((P ∧ Q) ∧ R) ≡ (P ∧ (Q ∧ R)) and ((P ∨ Q) ∨ R) ≡ (P ∨ (Q ∨ R)) (associativity)
 ¬(¬P) ≡ P (double negation)

(P ⇒ Q) ≡ (¬Q ⇒ ¬P) (contraposition)
 (P ⇒ Q) ≡ (¬P ∨ Q)
 (P ⇔ Q) ≡ ((P ⇒ Q) ∧ (Q ⇒ P))
 ¬(P ∧ Q) ≡ (¬P ∨ ¬Q) and ¬(P ∨ Q) ≡ (¬P ∧ ¬Q)
 (P ∧ (Q ∨ R)) ≡ ((P ∧ Q) ∨ (P ∧ R)) (distributivity of ∧ over ∨)
 (P ∨ (Q ∧ R)) ≡ ((P ∨ Q) ∧ (P ∨ R)) (distributivity of ∨ over ∧)
3. Quantorfree formulas
 (P ∧ Q) ≡ (Q ∧ P)
 ((P ∨ Q) ∨ R) ≡ (P ∨ (Q ∨ R))
 ¬(¬P) ≡ P
 (P ⇒ Q) ≡ (¬Q ⇒ ¬P)
 (P ⇒ Q) ≡ (¬P ∨ Q)
 (P ⇔ Q) ≡ ((P ⇒ Q) ∧ (Q ⇒ P))
 ¬(P ∧ Q) ≡ (¬P ∨ ¬Q)
 (P ∧ (Q ∨ R)) ≡ ((P ∧ Q) ∨ (P ∧ R))
Same meaning  equivalent
Same meaning  equivalent
Same meaning  equivalent
Verify via truth tables
3. Quantorfree formulas
age(eldestDaughter())~=10
sisterOf(eldestDaughter(),lisa)
man(homer)  woman(homer)
sisterOf(bart,maggie)  (brotherOf(bart,lisa) & brotherOf(bart,marge))
woman(homer) <=> ~man(homer)
age(eldestDaughter())=10 <=> age(bart)=10
age(eldestDaughter())=10 => age(bart)=10
age(eldestDaughter())=8 => age(bart)=10 & age(maggie)=1
What is the interpretation of the following QF formulas?
structure Simpsons : People {
Person := {marge, homer, lisa, bart, maggie}
parentOf := { (marge, bart), (marge, lisa), (marge, maggie),
(homer, bart), (homer, lisa), (homer, maggie) }
man := { (homer), (bart) }
woman := { (marge), (lisa), (maggie) }
brotherOf := { (bart, lisa), (bart, maggie) }
sisterOf := { (lisa, bart), (maggie, bart), (lisa, maggie),
(maggie, lisa) }
age := { (homer) > 36, (marge) > 34, (bart) > 10,
(lisa) > 8, (maggie) > 1 }
eldestDaughter := { () > lisa }
}
true
false
true
false
true
false
true
true
3. Quantorfree formulas
age(eldestDaughter())~=10
sisterOf(eldestDaughter(),lisa)
man(homer)  woman(homer)
sisterOf(bart,maggie)  (brotherOf(bart,lisa) & brotherOf(bart,marge))
woman(homer) <=> ~man(homer)
age(eldestDaughter())=10 <=> age(bart)=10
age(eldestDaughter())=10 => age(bart)=10
age(eldestDaughter())=8 => age(bart)=10 & age(maggie)=1
What is the interpretation of the following QF formulas?
Are these brackets necessary?
Order of operations:
~ > & >  > =>/<= > <=>
3. Quantorfree formulas
age(eldestDaughter())~=10
sisterOf(eldestDaughter(),lisa)
man(homer)  woman(homer)
sisterOf(bart,maggie)  (brotherOf(bart,lisa) & brotherOf(bart,marge))
woman(homer) <=> ~man(homer)
age(eldestDaughter())=10 <=> age(bart)=10
age(eldestDaughter())=10 => age(bart)=10
age(eldestDaughter())=8 => age(bart)=10 & age(maggie)=1
structure Simpsons : People {
Person := {homer, marge, bart, lisa, maggie}
parentOf := ??
man := ??
woman := ??
brotherOf := ??
sisterOf := ??
age := ??
eldestDaughter := ??
}
Given the above formulas, complete the following structure with interpretations such that all of the formulas are true:
4. Formulas with quantors
Each introduces a variable ranging over some type:
Quantors allow statements over a range of elements at the same time. There are two basic quantors:
These variables can be used as terms in a subsequent formula:
4. Formulas with quantors
forall is true iff the subsequent formula is true for all domain elements in the range.
exists is true iff the subsequent formula is true for at least one domain element in the range.
Interpretations of formulas are truth values:
true or false
In IDP, 'forall' is written as '!' and 'exists' is written as '?'.
forall can be seen as a "big and" (conjunction) over its range
exists can be seen as a "big or" (disjunction) over its range
! x in Person: ~man(x)  ~woman(x)
(~man(homer)  ~woman(homer))
& (~man(marge)  ~woman(marge))
& (~man(bart)  ~woman(bart))
& (~man(lisa)  ~woman(lisa))
& (~man(maggie)  ~woman(maggie))
is equivalent to
? x in Person: ~(man(x)  woman(x))
~(man(homer)  woman(homer))
 ~(man(marge)  woman(marge))
 ~(man(bart)  woman(bart))
 ~(man(lisa)  woman(lisa))
 ~(man(maggie)  woman(maggie))
is equivalent to
This is a good approximation of how IDP handles quantors internally
4. Formulas with quantors
! x in Person: ~man(x)  ~woman(x)
? x in Person: ~(man(x)  woman(x))
! x in Person: ? y in Person: parentOf(y,x)
! x in Person: (? y in Person: sisterOf(x,y)) => woman(x)
What is the interpretation of these formulas?
structure Simpsons : People {
Person := {marge, homer, lisa, bart, maggie}
parentOf := { (marge, bart), (marge, lisa), (marge, maggie),
(homer, bart), (homer, lisa), (homer, maggie) }
man := { (homer), (bart) }
woman := { (marge), (lisa), (maggie) }
brotherOf := { (bart, lisa), (bart, maggie) }
sisterOf := { (lisa, bart), (maggie, bart), (lisa, maggie),
(maggie, lisa) }
age := { (homer) > 36, (marge) > 34, (bart) > 10,
(lisa) > 8, (maggie) > 1 }
eldestDaughter := { () > lisa }
}
true
false
true
false
4. Formulas with quantors
! x in Country: ! y in Country: border(x,y) => colorOf(x)~=colorOf(y)
Truth value of below formula in either structure?
structure Correct : MapCol {
Country := { BE, NL, DE, LU, FR }
Color := { R, G, B, Y}
border := { (NL, BE), (NL, DE), (BE, LU),
(BE, FR), (LU, FR), (LU, DE), (DE, FR) }
colorOf := { (BE)>B, (NL)>R,
(DE)>G, (LU)>Y, (FR)>R }
}
structure Nonsense : MapCol {
Country := { BE, NL, DE, LU, FR }
Color := { R, G, B, Y}
border := { (BE, BE), (FR, NL) }
colorOf := { (BE)>R, (NL)>R, (DE)>R, (LU)>R, (FR)>R }
}
4. Formulas with quantors
Homework I
age(eldestDaughter())~=10
sisterOf(eldestDaughter(),lisa)
man(homer)  woman(homer)
sisterOf(bart,maggie)  (brotherOf(bart,lisa) & brotherOf(bart,marge))
woman(homer) <=> ~man(homer)
age(eldestDaughter())=10 <=> age(bart)=10
age(eldestDaughter())=10 => age(bart)=10
age(eldestDaughter())=8 => age(bart)=10 & age(maggie)=1
 Adapt the Simpsons structure so that it makes all of the above true.
 Why does there not exist a single structure that makes all of the above false?
 Truth value of below formulas:
Crash course in IDP
Part 2:
extensions of predicate logic
20211215
5. Some IDP specifics
! x in Person: ~man(x)  ~woman(x).
? x in Person: ~(man(x)  woman(x)).
! x in Person: ? y in Person: parentOf(y,x).
! x in Person: (? y in Person: sisterOf(x,y)) => woman(x).
All constraints end with '.'
Shows where the constraint (and quantors inside it) end.
To use domain elements in formula, type interpretation must be given in vocabularium:
vocabulary People {
type Person := {homer, marge, bart, lisa, maggie}
parentOf: (Person * Person) > Bool // predicate
man: (Person) > Bool // predicate
woman: (Person) > Bool // predicate
brotherOf: (Person * Person) > Bool // predicate
sisterOf: (Person * Person) > Bool // predicate
age: (Person) > Int // functor
eldestDaughter: () > Person // functor
}
6. Cardinality aggregate

Counts the number of true formulas for given range
 introduces variable, similar to quantor
 Is a term: evaluates to Int
 can be used wherever one would use an Int term
#{x in Person : ~(man(x)  woman(x))} = 0.
"the number of persons that are not man or woman is zero"
E.g.:
#{x in Person : ~(man(x)  woman(x))}
"count"
range with variable x
subformula
! x in Person : man(x)  woman(x).
Equivalently:
7. Ifthenelse
 Switches the value of a term based on a condition
 Is a term: evaluates to some nonfixed type
 can be used wherever one would use a term of that type
 "=>" is an atom that evaluates to true/false
Score(CInsert) = (if Insert() then 3 else 0).
"the score of CInsert is 3 if we have an insert, else it is 0"
E.g.:
if Insert() then 3 else 0
case if true
condition
case if false
Insert() => Score(CInsert) = 3.
~Insert() => Score(CInsert) = 0.
Equivalently:
7. Ifthenelse
 Switches the value of a term based on a condition
 Is a term: evaluates to some nonfixed type
 can be used wherever one would use a term of that type
 "=>" is an atom that evaluates to true/false
if Insert() then 3 else 0
case if true
condition
case if false
Ifthenelse allow a simple way to encode (firsthit) decision tables. Can you see how?
8. Arithmetic
With the use of mathematical symbols we can construct more complex numerical terms
Given two terms t1, t2, the following are valid numerical terms:
 t1 + t2 (addition)
 t1  t2 (subtraction)
  t1 (unary subtraction)
 t1 / t2 (division)
 t1 * t2 (multiplication)
 t1 % t2 (modulus)
 abs(t1) (absolute value)
Given two terms t1, t2, the following are valid numerical atoms:
 t1 = t2 (equality)
 t1 ~= t2 (disequality)
 t1 < t2 (less than)
 t1 =< t2 (less than or equal)
 t1 > t2 (greater than)
 t1 >= t2 (greater than or equal)
Watch out! => and <= are implications, not inequalities!
8. Arithmetic
age(eldestDaughter()) = age(Bart)2.
2*age(Bart) = age(eldestDaughter()).
age(eldestDaughter()) % 3 = 2.
! x, y in Person: parentOf(x,y) => age(x) > age(y)+18.
! x, y in Person: parentOf(x,y) => age(y) > age(x)+18.
What is the truth value of the following formulas?
structure Simpsons : People {
Person := {marge, homer, lisa, bart, maggie}
parentOf := { (marge, bart), (marge, lisa), (marge, maggie),
(homer, bart), (homer, lisa), (homer, maggie) }
man := { (homer), (bart) }
woman := { (marge), (lisa), (maggie) }
brotherOf := { (bart, lisa), (bart, maggie) }
sisterOf := { (lisa, bart), (maggie, bart), (lisa, maggie),
(maggie, lisa) }
age := { (homer) > 36, (marge) > 34, (bart) > 10,
(lisa) > 8, (maggie) > 1 }
eldestDaughter := { () > lisa }
}
true
false
true
false
true
8. Arithmetic
A "forall" is a "big and" over a range.
A "exists" is a "big or" over a range.
A "sum aggregate" / "sum lambda" is a "big sum" over a range.
condition ("filters" range)
value ("maps" to numeric value)
sum aggregate  old syntax
sum{x in Person : man(x) : age(x)} < 100.
sum(lambda x in Person: if man(x) then age(x) else 0) < 100.
sum lambda  new syntax
9. Definitions
 Ifthenelse allow to define functors
 for all inputs, determine the output of the functor
 (Inductive) definitions allow to define predicates
{
! x in Person: sick(x) < hasFever(x).
! x in Person: sick(x) < hasRunnyNose(x) & coughing(x).
}
"A person will be sick if they have a fever or if the have both a runny nose and are coughing. In all other cases, they will not be sick."
Rules covering different cases
If no rule applies, the defined symbol is false.
9. Definitions
 Ifthenelse allow to define functors
 for all inputs, determine the output of the functor
 (Inductive) definitions allow to define predicates
{
! x in Person: sick(x) < hasFever(x).
! x in Person: sick(x) < hasRunnyNose(x) & coughing(x).
}
"A person will be sick if they have a fever or if the have both a runny nose and are coughing. In all other cases, they will not be sick."
! x in Person: hasFever(x) => sick(x).
! x in Person: hasRunnyNose(x) & coughing(x) => sick(x).
! x in Person: sick(x) <=> hasFever(x).
! x in Person: sick(x) <=> hasRunnyNose(x) & coughing(x).
In what situations (structures!) are the below different?
 Ifthenelse allow to define functors
 for all inputs, determine the output of the functor
 (Inductive) definitions allow to define predicates
{
! x in Person: sick(x) < hasFever(x).
! x in Person: sick(x) < hasRunnyNose(x) & coughing(x).
}
"A person will be sick if they have a fever or if the have both a runny nose and are coughing. In all other cases, they will not be sick."
! x in Person: sick(x) <=> hasFever(x)  (hasRunnyNose(x) & coughing(x)).
Equivalently:
9. Definitions
 Ifthenelse allow to define functors
 for all inputs, determine the output of the functor
 (Inductive) definitions allow to define predicates
{
! x in Person: sick(x) < hasFever(x).
! x in Person: sick(x) < hasRunnyNose(x) & coughing(x).
}
head of rule
body of rule with description of a case
arrow '<' denoting a rule
defined predicate
forall quantification over the type of the predicate
brackets show which rules belong together
9. Definitions
 Has "wellfounded" semantics
 Allows expression of recursive / inductive concepts
{
! x in City: reachable(x) < x = kontich.
! x in City: reachable(x) < ? y in City: reachable(y) & rail(x,y).
}
This is what's being defined!
induction step
base case
No equivalent nondefinition formula exists in predicate logic
9. Definitions
10. Other extensions
// range notation for type interpretation
type ExRes := {0..3}
// functor interpretation with "else"
TmaxT := { (Fluoroloy_A02) > 150} else 200
// sum via lambda
TotalCost() = sum(lambda c in component : Score(c) + Cost(Material(c))).
// quantification over predicate
? (x,y) in rangemap_CostIndex_CostIndex: x < Quantity() =< y.
// builtin truth values
true.
false.
 And more, see docs.idpz3.be/en/stable/IDPLanguage.html
 When in doubt, ask expert at KU Leuven
 they have a responsibility to teach the language
11. The hard part: writing your own constraints
Let's try three examples in the interactive consultant:
interactiveconsultant.idpz3.be
 graph coloring
 nqueens
 sudoku
vocabulary V {
type Country
type Color
border: (Country * Country) > Bool
colorOf: (Country) > Color
}
theory T : V {
// two countries with a border should have a different color
// ...
}
structure S : V {
Country := { BE, NL, DE, LU, FR }
Color := { R, G, B, Y}
border := { (NL, BE), (NL, DE), (BE, LU),
(BE, FR), (LU, FR), (LU, DE), (DE, FR) }
}
Known interpretation
No known interpretation: this is what we want to solve
"Two countries with a border should have a different color."
This is a statement about all ('!') pairs (two variables: x,y) of countries:
11. The hard part: writing your own constraints
! x in Country: ! y in Country: ... something about x, y ...
! x in Country: ! y in Country: border(x,y) => ... something ...
What do we know about these two countries? That if they have a border, then something must be true:
What must be true? The color of the countries should be different:
! x in Country: ! y in Country: border(x,y) => colorOf(x) ~= colorOf(y).
vocabulary V {
type Index := {1..4}
type Diag := {1..7}
n : () > Index
toDiag1: (Index * Index) > Diag
toDiag2: (Index * Index) > Diag
queen: (Index * Index) > Bool
}
structure S : V {
n := 4
}
theory T : V {
! x, y in Index: toDiag1(x, y) = x  y + n().
! x, y in Index: toDiag2(x, y) = x + y  1.
// every row has exactly one queen
// every column has exactly one queen
// every diagonal has at most one queen
}
"Every row has exactly one queen."
This is a statement about every ('!') row (one variable: x):
! x in Index: ... something about x ...
"has ... queen" > queens reside on squares, which are combinations of rows and columns. We already introduced a row (it's 'x'), now we need to express something about a column 'y'.
! x in Index: ... y in Index ... something about x and y...
"exactly one" suggests a cardinality aggregate to introduce y.
! x in Index: #{y in Index: ... something about x and y ... } = 1.
Now we have: for each row x, there exists exactly one column y.
... where there is a queen!
! x in Index: #{y in Index: queen(x,y) } = 1.
vocabulary V {
type Row := {1..4} // The rows of the grid (1 to 4)
type Column := {1..4} // The columns of the grid (1 to 4)
type Block := {1..4} // 4 blocks of 2x2 where the numbers need to be entered
type Number := {1..4} // The numbers of the grid (1 to 4)
blockOf: (Row * Column) > Block
// This declares the block of each cell.
// This means that blockOf(r,c)=b if and only if
// b is the block of the cell on row r and column c.
solution: (Row * Column) > Number
// The solution: a number assigned to every cell.
// A cell is represented by its row and column.
// For example: solution(1,2) = 3 means that the
// cell on row 1 and column 2 contains a 3.
}
theory T : V {
// On every row every number is present.
// In every column every number is present.
// In every block every number is present.
}
structure S : V {
// Fix which cells lie in which block
blockOf := {(1,1)>1, (1,2)>1, (2,1)>1, (2,2)>1,
(3,1)>2, (3,2)>2, (4,1)>2, (4,2)>2,
(1,3)>3, (1,4)>3, (2,3)>3, (2,4)>3,
(3,3)>4, (3,4)>4, (4,3)>4, (4,4)>4}
}
Homework II: truth value of below formulas
! x in Person: !y in Person: brotherOf(x,y) => sisterOf(y,x).
! x in Person: !y in Person: brotherOf(x,y) <= sisterOf(y,x).
! x in Person: #{y in Person: parentOf(y,x)} < 2.
? x in Person: ! y in Person: ! z in Person: parentOf(y,x) & parentOf(z,x) => y=z.
true
false
false
true
Crash course in IDP
Part 3:
modeling full problem domains
20211222
Contents
 Have a look at the homeworks
 A handful of slides on modeling advise
 Improve omniseal and lipseal specifications
 Assignment on modeling a simple problem domain
 Feedback: what one thing would you prefer to change the most for this crash course?
12. The very hard part: deciding vocabulary of a problem domain
 Vocabulary introduces symbols for the concepts, objects, relationships, functions... in the problem domain
 A good vocabulary leads to simple and readable formulas
 picking the right vocabulary is hard!
 "refactor" the vocabulary + formulas when needed
13. Modeling advise
 Always start small!
 as in programming, extending what works is easier than figuring out why a huge program does not work
 Use a complete but small solution to quickly check the formulas as you write them.
 (partly) avoids writing formulas that are too strict.
 Use IDPZ3 to generate solutions after adding formulas. Check that these solutions make sense given the formulas you wrote so far.
 (partly) avoids writing formulas that are insufficiently strict.
13. Modeling advise
 Size of specification can be measured in two ways
 number of ground symbols
 the number of symbols in the interactive consultant
 number of ground constraints
 the number of constraints where all ranges are "replaced by their possible values"
 number of ground symbols
 Larger size => slower solving time
13. Modeling advise
 Some formulas represent constraints, others definitions
 definitions give meaning to intermediary symbols, constraints restrict set of solutions
 definitions can typically be substituted by their defining formula in the constraints
 useful to know the difference
sick() => takeMedicine().
tax() < 10000.
sick() <=> hasFever()  (hasRunnyNose() & coughing()).
tax() = (if income()<limit() then 0 else income()*0.4).
(if income()<limit() then 0 else income()*0.4) < 10000.
(hasFever()  (hasRunnyNose() & coughing())) => takeMedicine().
13. Modeling advise
 forall ! works best with implication => or disjunction 
 exists ? works best with conjunction &
 equivalence <=> mainly used in combination with forall ! to express definitions
"only people above 18 should drink alcohol"
! x in People: drinksalcohol(x) => age(x)>=18.
"there is someone over 18 who does not drink alcohol"
? x in People: age(x)>=18 & ~drinksalcohol(x).
"adults are those people older than 18"
! x in People: adult(x) <=> age(x)>=18.
"nobody is both a man and a woman"
! x in People: ~(man(x) & woman(x)).
"there exists someone who is neither man nor woman"
? x in People: ~(man(x)  woman(x)).
"nonbinary people are those who identify neither as man nor as woman"
! x in People: nonbinary(x) <=> ~(man(x)  woman(x)).
13. Modeling advise
Always start small!
Homework III: modeling from scratch
In this exercise we’ll try to model a simple scheduling problem. A company has five tasks that need to be executed: task1, task2, task3, task4 and task5. To do this the company has 6 employees: bart, ben, jo, deise, andrea and joost. But every employee is able to complete certain tasks. Bart can complete task1. Ben can complete task1 and task2. Jo can complete task2 and task4. Andrea can complete task1, task2, and taks3. Deise can complete task 3 and task5. And finally, Joost can complete all tasks. The aim of the company is to minimize the total salary cost. Only employees that execute a task get paid. The salary of the employees is very different. Bart earns 20 euro. Ben earns 30 euro, Jo gets 50 euro, Andrea gets 100 euro, Deise 120 euro and Joost 150 euro. Finally, there are a couple of constraints:

Every task needs to be executed.

An employee can only execute a task if he can complete it.

An employee can only work on 1 task.
The ultimate goal of the company is to minimize the total salary that needs to be paid.
Task

Create a vocabulary that can represent this problem. Think about when to use a predicate or a function, which types will be needed, etc …

Make a structure using this vocabulary that incorporates the info from the story above (bart earns 20 euros, Andrea can complete task1, task2, and task3, …)

Complete the structure with an arbitrary solution to test your constraints (written in the next step) .

Make sure that all constraints are satisfied by writing a good theory.

Find a solution that minimizes the total wage cost
TODO: incorporate feedback
 The jump from "evaluate these formulas" to "solve nqueens / sudoku" is too big.
 add intermediary examples, perhaps some without quantification
 explain that for cardinality, {x in T1: x in T2: ... } is written as {x in T1 , x in T2: ... } (for the nqueens diagonal constraint)
 Give some technique to translate English sentences to FO(.). E.g., look for quantors, what is being said about these quantors, use appropriate connectives ("=>" and "" for "!", "&" for "?"). Incorporate modeling advise in that section. Probably flesh this out as a separate workshop.
Crash course IDP
By krr
Crash course IDP
 453