SOGrounder:
modelling and solving with Second-Order Logic
Matthias van der Hallen
28/10/2019
First-Order Logic
- A domain of non-logical objects: \(alice, bob, charly ...\)
- Predicates: \(Person(alice), Person(bob), Dog(charly)\)
- Functions: \(Owner(charly)=alice\)
Introduce variables representing non-logical objects:
\(\forall x : Person(x) \lor Dog(x).\)
"Everything is either a dog or a person."
Second-Order Logic
Also introduce variables representing predicates & functions:
\(\exists P : P(alice) \land P(bob).\)
"There is a set that contains both \(alice\) and \(bob\)."
Why?
Some knowledge cannot be expressed naturally in FO!
Why?
Some concepts are naturally expressed using predicates or functions
Graphs
\(Edge(a,b), Edge(c,b), Edge(c,d),\)
\(Edge(d,c), Edge(d,e), Edge(e,e).\)
Predicate representing its \(Edge\) relation:
Colorings
\(color(a)=red, color(b)=blue, color(c)=green\)
\(color(d)=blue, color(e)=red.\)
Coloring represented by the function:
Homomorphisms
\(h(f) = c, h(g) = d\)
Homomorphism represented by the function:
Schedules
\(shift(monday)=alice, shift(tuesday)=bob,\)
\(\ldots\)
Schedule represented by the function:
Why?
- Graphs
- Colorings
- Homomorphisms
- Schedules
- ...
Some concepts are naturally expressed using predicates or functions:
Many interesting problems use these concepts!
Graph Mining
Find a graph \(\mathcal{G}\) such that:
Core problem
- homomorphisms exist with the + examples
- no homomorphisms exist with the - examples
Graph Mining
Find a graph \(\mathcal{G}\) such that:
Core problem
- homomorphisms exist with the + examples
- no homomorphisms exist with the - examples
+ example
Candidate for \(\mathcal{G}\)
- example
Graph Mining
\(\lnot \exists f : \forall x, y : \mathcal{G}(x,y) \Rightarrow NegEx(x,y).\)
Find a graph \(\mathcal{G}\) such that:
- homomorphisms exist with the + examples
- no homomorphisms exist with the - examples
\(\exists f : \forall x, y : \mathcal{G}(x,y) \Rightarrow PosEx(x,y).\)
Theory
This requires support for Second-Order Logic!
Critical Friendship
Divide \(n\) people over \(k\) cars such that
nobody has to sit with someone they dislike.
Critical friendship pair:
Two distinct people s.t. if they would dislike each other, no proper division exists.
Critical Friendship
Observations:
- Model 'dislikes' as a graph
- Proper division is a \(k\)-coloring
- Two people \(p1\) and \(p2\) form a critical pair iff.
Any valid k-colouring maps \(p1\) and \(p2\) to the same car
Alice
Bob
David
Charles
Alice
Bob
David
Charles
Critical Friendship
Observations:
- Model 'dislikes' as a graph
- Proper division is a \(k\)-coloring
- Two people \(p1\) and \(p2\) form a critical pair iff.
Any valid k-colouring maps \(p1\) and \(p2\) to the same car
type Person. % The Person type, corresponding to nodes
type Car. % The Car type, corresponding to colours
dislike :: (Person, Person). % A predicate encoding dislikes
p1 :: Person. % A person constant
p2 :: Person. % A second person constant
p1 ≠ p2.
∃f :: (Person)→Car: ∀ a,b :: Person: dislike(a,b) ⇒ f(a)≠f(b).
∀f :: (Person)→Car: (∀ a,b :: Person: dislike(a,b) ⇒ f(a)≠f(b)) ⇒ f(p1)=f(p2).Model:
Critical Friendship
type Person. % The Person type, corresponding to nodes
type Car. % The Car type, corresponding to colours
dislike :: (Person, Person). % A predicate encoding dislikes
p1 :: Person. % A person constant
p2 :: Person. % A second person constant
p1 ≠ p2.
∃f :: (Person)→Car: ∀ a,b :: Person: dislike(a,b) ⇒ f(a)≠f(b).
∀f :: (Person)→Car: (∀ a,b :: Person: dislike(a,b) ⇒ f(a)≠f(b)) ⇒ f(p1)=f(p2).Model:
Person = {Alice; Bob; Charles; David}.
Car = {Astra; Berlingo}.
dislike = {Alice, Bob; Bob, David; Charles, David}.
p1 = Bob, p2 = Charles
Model expansion
How?
SOGrounder: A system built on the traditional ground-and-solve approach
Ground-and-solve
Step 1: Translate (grounding) the high-level language to a simpeler low-level language.
Ground-and-solve
Step 2: Use a general-purpose solver to find interpretations for this translation.
Ground-and-solve
Step 3: Back-translate to an interpretation for the high-level model.
Grounding
First-Order Logic
Translation to SAT:
SAT
Formulas over propositional variables
\(x \lor \lnot (y \land z)\)
First-Order Logic
Translation to SAT:
- Push negations: \(\lnot \exists x : \phi \rightsquigarrow \forall x : \lnot \phi\),
\(\lnot (x \land y) \rightsquigarrow \lnot x \lor \lnot y\), ... - Unnest: \(f(g(x)) = z \rightsquigarrow \exists y : g(x) = y \land f(y)=z\)
- Replace functions \(f/n\) by predicates \(f'/(n+1)\)
- Ground first-order quantifications:
- Replace ground predicate applications \(P(a)\) by propositional variables \(p_a\)
\(\forall x : \phi \rightsquigarrow \bigwedge_{c \in type(x)} \phi[x/c]\).
\(\exist x : \phi \rightsquigarrow \bigvee_{c \in type(x)} \phi[x/c]\).
First-Order Logic
Example
type Person = {Alice; Bob}.
type Car = {Astra; Berlingo}.
dislike :: (Person, Person). % A predicate encoding dislikes
sitsIn :: (Person)→Car
∀ a, b :: Person : dislike(a,b) ⇒ sitsIn(a) ≠ sitsIn(b).∀ a, b :: Person : ¬dislike(a,b) ∨ sitsIn(a) ≠ sitsIn(b).∀ a, b :: Person : ¬dislike(a,b) ∨
(∃ sa, sb :: Car : sitsIn(a) = sa ∧ sitsIn(b) = sb ∧ ≠(sa,sb)).∀ a, b :: Person : ¬dislike(a,b) ∨
(∃ sa, sb :: Car : sitsIn(a,sa) ∧ sitsIn(b,sb) ∧ ≠(sa,sb)).(¬dislike(Alice,Bob) ∨
((sitsIn(Alice,Astra) ∧ sitsIn(Bob,Astra) ∧ ≠(Astra, Astra )) ∨
(sitsIn(Alice,Astra) ∧ sitsIn(Bob,Berlingo) ∧ ≠(Astra, Berlingo)) ∨
(sitsIn(Alice,Berlingo) ∧ sitsIn(Bob,Astra) ∧ ≠(Berlingo, Astra )) ∨
(sitsIn(Alice,Berlingo) ∧ sitsIn(Bob,Berlingo) ∧ ≠(Berlingo, Berlingo)))) ∧
...\((\lnot d_{Alice,Bob} \lor (s_{Alice,Astra} ∧ s_{Bob,Berlingo}) ∨ (s_{Alice,Berlingo} ∧ s_{Bob,Astra})) \land ...\)
Second-Order Logic
Translation to Quantified Boolean Formulas:
Introduces quantifiers (\(\forall, \exists\)) for propositional variables
\(\forall x\exists y\forall z . x \lor \lnot (y \land z)\)
Second-Order Logic
Translation to Quantified Boolean Formulas:
Ground second-order quantifications \(\forall P : \phi\), \(\exist P : \phi \):
- Ensure all predicate and function names are unique (renaming)
- Ground \(\phi\)
- Based on the type \(T_1 \times \ldots \times T_n\) of \(P\), introduce propositions \(p_{t_1,\ldots,t_n}\) for \(t_1 \in T_1\), etc. and quantify them correctly.
- For functions, introduce constraints ensuring existence and uniqueness
Second-Order Logic
Example
type Person = {Alice; Bob}.
type Car = {Astra; Berlingo}.
dislike :: (Person, Person). % A predicate encoding dislikes
∀s :: (Person)→Car: (∀ a,b :: Person: dislike(a,b) ⇒ s(a)≠s(b)) ⇒ s(p1)=s(p2).\(\forall s_{Alice,Astra}, s_{Alice,Berlingo}, s_{Bob,Astra}, s_{Bob,Berlingo} :\)
\((\Psi) \Rightarrow (\lnot d_{Alice,Bob} \lor (s_{Alice,Astra} ∧ s_{Bob,Berlingo}) ∨ (s_{Alice,Berlingo} ∧ s_{Bob,Astra})) \land ...\)
\((s_{Alice, Astra} \lor s_{Alice,Berlingo} \lor s_{Bob,Astra} \lor s_{Bob,Berlingo}) \land \)
\((\lnot s_{Alice,Astra} \lor \lnot s_{Alice,Berlingo}) \land\)
\((\lnot s_{Bob,Astra} \lor \lnot s_{Bob,Berlingo})\)
\(\Psi\):
Caveats
- Some propositions introduced by quantifications represent irrelevant predicate applications
- Some solvers require input in conjunctive normal form:
Standard technique: Tseitin transformation
But
\(\bigwedge_i C_i\) where \(C_i = \bigvee_j p_{i,j}\)
Naive introduction of auxiliary propositions can introduce unwanted dependencies between propositions, hurting search
\((p_1 \land p_2) \lor p_3 \rightsquigarrow \exists t . (t \Leftrightarrow p_1 \land p_2) \land (t \lor p_3)\)
Advanced Grounding Techniques
- Reduced Grounding: RED
- Lifted Unit Propagation
- Ground with Bounds
Advanced Grounding Techniques
- Reduced Grounding: RED
Use knowledge about the truth value of sub-formulas to simplify formulas.
\[\forall a,b . dislike(a,b) \Rightarrow f(a) \neq f(b).\]
While grounding for \(a,b\), if \(dislike(a,b)\) can be evaluated, replace it and simplify the formula.
Advanced Grounding Techniques
- Reduced Grounding: RED
- Lifted Unit Propagation
Derive Certainly-True and Certainly-False values of unknown predicates using lifted (symbolic) reasoning.
\(\forall x . P(x) \Rightarrow Q(x).\)
If we know e.g. \(P(1)\), derive \(Q(1)\)
When deriving CT / CF value for second-order variable
\(\exist\): Process normally
\(\forall\): derive UNSAT
Advanced Grounding Techniques
- Reduced Grounding: RED
- Lifted Unit Propagation
- Ground-with-Bounds
Extend CT/CF concept to (sub-)formulas and manipulate them to reduce grounding.
\(\phi \lor (\forall x . P(x) \Rightarrow Q(x)).\)
\(Type(x) = \{1;2;3\}, P_{ct} = \{1;2\}, Q_{ct} = \{1\}\) \(\Rightarrow \forall_{ct} = \{1\}\)
Ground \(\forall\) for \(x \not\in \forall_{ct}\)
Solving
Consider a Quantified Boolean Formula in Prenex-CNF form:
\(Q_1 X_1, \ldots, Q_n X_n . \phi\) with \(Q_i \in \{\exist, \forall\}\)
Solving QBF
Consecutive variables \(x\) with the same quantifier \(Q\) are gathered in quantifier blocks \(X\)
\(Q_1 X_, \ldots, Q_n X_n\) often abbreviated to \(\Pi\)
\(\Pi\) introduces ordering: \(x \prec_{\Pi} x'\)
Solving QBF: QCDCL
Conflict / Solution driven clause/cube learning
Generalization of well-known SAT solving technique CDCL: QCDCL
Clause:
Cube:
- Disjunction of literals \(\bigvee l_i\)
- \(\Pi . \phi \Rightarrow C\)
- Conjunction of literals \(\bigvee l_i\)
- \(\Pi . C \Rightarrow \phi\)
Solving QBF: QCDCL
PCNF \(\phi\)
Extend Assignment \(A\)
\(\phi[A]\) = T/F
Propagation
T: Learn Cube
F: Learn Clause
No
UNSAT
\(\empty\) Clause
\(\empty\) Cube
SAT
Backtrack
Cube/Clause \(\neq\empty\)
Learn Cube/Clause: QRES
Resolution-Clause/Cube: From \(C_1 \cup \{p\}\) and \(C_2 \cup \{\overline{p}\}\), learn \(C_1 \cup C_2\) if \(p\) is \(\exist\) / \(\forall\)
Reduce Clause: Drop \(\forall\)-quantified \(l\) from \(C \cup \{l\}\) to learn \(C\) iff. \(C\) contains no \(\exist\)-quantified variable \(l'\) s.t. \(l \prec_{\Pi} l'\)
Reduce Cube: Drop \(\exist\)-quantified \(l\) from \(C \cup \{l\}\) to learn \(C\) iff. \(C\) contains no \(\forall\)-quantified variable \(l'\) s.t. \(l \prec_{\Pi} l'\)
QRES: Example
Consider prefix: \(\exist x, y \forall a\exist z\)
Start with cubes
\(\overline{x} \land \overline{y} \land \overline{a} \land \overline{z}\)
\(\overline{x} \land \overline{y} \land a \land z\)
red
\(\overline{x} \land \overline{y} \land \overline{a}\)
\(\overline{x} \land \overline{y} \land a\)
res
\(\overline{x} \land \overline{y}\)
red
red
\(\empty\)
Conclusion
- If we want to derive 'critical pairs', we can specify it in second-order logic
- To reason over these specifications, we can use the SOGrounder system
- It will ground the specification, resulting in a quantified boolean formula
- Generic, off-the-shelve solvers can determine satisfiability, with assignments or proofs
- We translate the solution back to the problem domain
Future work
Extend SOGrounder with:
- Support for aggregates
- Higher-Order predicates (Templating)
Thank you
Thank you
Replaced slides
Critical Friendship
Observations:
- Model 'dislikes' as a graph
- Proper division is a \(k\)-coloring
- Two people \(p1\) and \(p2\) form a critical pair iff.
Any valid k-colouring maps \(p1\) and \(p2\) to the same car
type Person. % The Person type, corresponding to nodes
type Car. % The Car type, corresponding to colours
dislike :: (Person, Person). % A predicate encoding dislikes
p1 :: Person. % A person constant
p2 :: Person. % A second person constant
p1 ≠ p2.
∃f :: (Person)→Car: ∀ a,b :: Person: dislike(a,b) ⇒ f(a)≠f(b).
∀f :: (Person)→Car: (∀ a,b :: Person: dislike(a,b) ⇒ f(a)≠f(b)) ⇒ f(p1)=f(p2).Model:
type Person. % The Person type, corresponding to nodes
type Car. % The Car type, corresponding to colours
dislike :: (Person, Person). % A predicate encoding dislikes
p1 :: Person. % A person constant
p2 :: Person. % A second person constant
p1 ≠ p2.
∃f :: (Person)→Car: ∀ a,b :: Person: dislike(a,b) ⇒ f(a)≠f(b).
∀f :: (Person)→Car: (∀ a,b :: Person: dislike(a,b) ⇒ f(a)≠f(b)) ⇒ f(p1)=f(p2).type Person. % The Person type, corresponding to nodes
type Car. % The Car type, corresponding to colours
dislike :: (Person, Person). % A predicate encoding dislikes
p1 :: Person. % A person constant
p2 :: Person. % A second person constant
p1 ≠ p2.
∃f :: (Person)→Car: ∀ a,b :: Person: dislike(a,b) ⇒ f(a)≠f(b).
∀f :: (Person)→Car: (∀ a,b :: Person: dislike(a,b) ⇒ f(a)≠f(b)) ⇒ f(p1)=f(p2).Ground-and-solve
Step 1: Translate the high-level language to a simpeler low-level language.
Step 2: Use a general-purpose solver to find interpretations for this translation.
Step 3: Back-translate to an interpretation for the high-level model.
First-Order Logic
Translation to SAT:
Formulas over propositional variables
\( x \lor \lnot y \land z \)
- Push negations: \(\lnot \exists x : \phi \rightsquigarrow \forall x : \lnot \phi\),
\(\lnot x \land y \rightsquigarrow \lnot x \lor \lnot y\), ... - Unnest: \(f(g(x)) = z \rightsquigarrow \exists y : g(x) = y \land f(y)=z\)
- Replace functions \(f/n\) by predicates \(f'/(n+1)\)
- Ground first-order quantifications:
- Replace ground predicate applications \(P(a)\) by propositional variables \(p_a\)
\(\forall x : \phi \rightsquigarrow \bigwedge_{c \in type(x)} \phi[x/c]\).
\(\exist x : \phi \rightsquigarrow \bigvee_{c \in type(x)} \phi[x/c]\).
Solving QBF: QCDCL
PCNF \(\phi\)
Extend Assignment
\(\phi[A]\) = T/F
Propagation
T: Learn Cube
F: Learn Clause
No
UNSAT
\(\empty\) Clause
\(\empty\) Cube
SAT
Backtrack
Cube/Clause \(\neq\empty\)
SOGrounder
By krr
