Formale Systeme II: Theorie

Theories

SS 2018

Prof. Dr. Bernhard Beckert · Dr. Mattias Ulbrich

KIT – Die Forschungsuniversitat in der Helmholtz-Gemeinschaft www.kit.edu

Theories and Satisfiability –Introduction

Beckert, Ulbrich – Formale Systeme II: Theorie 2/47

Different Questions to Ask

Deciding logics

Question: Is formula φ valid, i.e., φ satisfied in all possiblestructures.

(∀x .p(x))→ p(f (x)) is valid.

x > y → y < x not valid (uninterpreted symbols!)

Deciding theories

Question: Is formula φ satisfied structures with fixedinterpretation for symbols.

∃x . 2 · x2 − x − 1 = 0 ∧ x < 0 holds in R, . . .

. . . but not in Z.

Beckert, Ulbrich – Formale Systeme II: Theorie 3/47

Theories

Given a FOL signature ΣFmlΣ . . . set of closed FOL-formulas over Σ.

Definition: Theory

A theory T ⊂ FmlΣ is a set of formulas such that

1 T is closed under consequence: If T |= φ then φ ∈ T

2 T is consistent: false 6∈ T

A FOL structure (D, I ) is called a T -model of ψ ∈ FmlΣ if

1 D, I |= ψ and

2 D, I |= φ for all φ ∈ T

Beckert, Ulbrich – Formale Systeme II: Theorie 4/47

Theories II

A FOL structure (D, I ) is called a T -structure if D, I |= φ forall φ ∈ T .

A T -structure (D, I ) is a T -model of ψ ∈ FmlΣ if D, I |= ψ.

ψ ∈ FmlΣ is called T -satisfiable if it has a T -model.

ψ ∈ FmlΣ is called T -valid if every T -structure is a T -modelof ψ. ⇐⇒ T |= ψ ⇐⇒ ψ ∈ T

T is called complete if: φ ∈ FmlΣ =⇒ φ ∈ T or ¬φ ∈ T

|=T is used instead of T |=: S |=T φ defined as S ∪ T |= φ

Beckert, Ulbrich – Formale Systeme II: Theorie 5/47

Generating Theories

Axiomatisation

Theory T may be represented by a set Ax ⊂ FmlΣ of axioms.T is the consequential closure of Ax, we write:

T = T (Ax) := {φ | Ax |= φ}

T is “axiomatisable”.

Fixing a structure

Theory T may be represented by one particular structure (D, I ).T is the set of true formulas in (D, I ), we write:

T = T (D, I ) := {φ | (D, I ) |= φ}

Beckert, Ulbrich – Formale Systeme II: Theorie 6/47

Discussion

Every theory T (D, I ) is complete.

If Ax is recursive enumerable, then T (Ax) is recursiveenumerable.

If Ax is decidable, then T (Ax) needs not be decidable.

T (D, I ) needs not be recursive enumerable.

(D, I ) is not the only T (D, I )-model.(In general, two T (D, I )-models are not even isomorphic)

Beckert, Ulbrich – Formale Systeme II: Theorie 7/47

Free variables

When dealing with theories, formulas often have free variables.

Open and closed (reminder)

φ1 = ∀x .∃y .p(x , y) is closed, has no free variables,φ2 = ∃y .p(x , y) is open, has free variables FV (φ2) = {x}

FmloΣ ⊃ FmlΣ . . . set of open formulas

Existential closure ∃[·]For φ ∈ FmloΣ with FV = {x1, ..., xn} define:

∃[φ] := ∃x1. . . .∃xn. φ

φ ∈ FmloΣ is called T-satisfiable if ∃[φ] is T-satisfiable.

Beckert, Ulbrich – Formale Systeme II: Theorie 8/47

Axioms for Equality

Theorem

Equality can be axiomatised in first order logic.

This means: Given signature Σ, there is a set EqΣ ⊂ FmlΣ thataxiomatise equality:

φ≈ is formula φ with interpreted “=” replaced by uninterpred “≈”.

S |= φ ⇐⇒ S≈ |=T (EqΣ) φ≈

FOL with equality cannot be more expressive than FOL withoutbuilt-in equality.

Beckert, Ulbrich – Formale Systeme II: Theorie 9/47

Axioms for Equality

Axioms EqΣ:

∀x . x ≈ x (Reflexivity)

∀x1, x1, . . . , xn, x′n.

x1 ≈ x ′1 ∧ . . . ∧ xn ≈ x ′n → f (x1, ..., xn) ≈ f (x ′1, . . . , x′n)

for any function f in Σ with arity n. (Congruency)

∀x1, x1, . . . , xn, x′n.

x1 ≈ x ′1 ∧ . . . ∧ xn ≈ x ′n → p(x1, ..., xn)↔ p(x ′1, . . . , x′n)

for any predicate p in Σ with arity n. (Congruency)(This includes predicate ≈)

Symmetry and transitivity of ≈ are consequences of EqΣ

Exercise

Beckert, Ulbrich – Formale Systeme II: Theorie 10/47

Satisfiability Modulo Theories

SMT solvers

A lot of research in recent years:(Simplify), Z3, CVC4, Yices, MathSAT, SPT, . . .Some for many theories, others only for a single theory.

(Common input format SMT-Lib 2)

FmlQF ⊂ Fmlo . . . the set of quantifier-free formulas

Interesting questions for a theory T :

SAT: Is φ ∈ Fmlo a T -satisfiable formula?

QF-SAT: Is φ ∈ FmlQF a T -satisfiable formula?

Beckert, Ulbrich – Formale Systeme II: Theorie 11/47

Decision Procedure

Decision Procedure

A decision procedure DPT for a theory T is a deterministicalgorithm that always terminates.It takes a formula φ as input and returns SAT if φ is T -satisfiable,UNSAT otherwise.

N.B.:

φ is T -valid ⇐⇒ ¬φ is not T -satisfiable.

DPT can also be used to decide validity!

Beckert, Ulbrich – Formale Systeme II: Theorie 12/47

Decision Procedures

Theory QF-SAT SATEquality YES YESUninterpreted functions YES co-SEMIInteger arithmeticLinear arithmeticReal arithmeticBitvectors YES YESFloating points YES YES

Beckert, Ulbrich – Formale Systeme II: Theorie 13/47

Natural Arithmetic

Beckert, Ulbrich – Formale Systeme II: Theorie 14/47

Natural Numbers

Standard model of natural numbers

Let ΣN = ({+, ∗, 0, 1}, {<}).

N = (N, IN ) with “obvious” meaning:

IN ({

+∗<

})(a, b) = a

{+·<

}b, IN (0) = 0, IN (1) = 1

T (N ) is the set of all sentences over ΣN which are true in thenatural numbers.

Godel’s Incompleteness Theorem

“Any consistent formal system F within which a certain amount ofelementary arithmetic can be carried out is incomplete.”

Natural number arithmetic is not axiomatisable (w/ a r.e. set of axioms)

Beckert, Ulbrich – Formale Systeme II: Theorie 15/47

Peano Arithmetic

Natural number arithmetic is not axiomatisable . . .Let’s approximate.

The Peano Axioms PA1 ∀x(x + 1 6 .= 0)

2 ∀x∀y(x + 1.

= y + 1→ x.

= y)

3 ∀x(x + 0.

= x)

4 ∀x∀y(x + (y + 1).

= (x + y) + 1)

5 ∀x(x ∗ 0.

= 0)

6 ∀x∀y(x ∗ (y + 1).

= (x ∗ y) + x)

7 For any φ ∈ FmlΣN

(φ(0) ∧ ∀x(φ(x)→ φ(x + 1)))→ ∀x(φ)

That’s an infinite (yet recursive) set of Axioms.

Beckert, Ulbrich – Formale Systeme II: Theorie 16/47

Peano Arithmetic

Peano arithmetic approximates natural arithmetic.

More T (PA)-models than T (N )-models

T (PA) is not complete.

=⇒ There are T (N )-valid formulas that are not T (PA)-validformulas.

There are artificial examples in T (N ) \ T (PA),but also actual mathematical theorems:

Beckert, Ulbrich – Formale Systeme II: Theorie 17/47

Beckert, Ulbrich – Formale Systeme II: Theorie 18/47

from: L. KIRBY and J. PARIS, ’Accessible Independence Results for Peano Arithmetic’ (1982)[2] R. L. GOODSTEIN, ’On the restricted ordinal theorem’, J. Symbolic Logic (1944)

Beckert, Ulbrich – Formale Systeme II: Theorie 19/47

Decision Procedures

Theory QF-SAT SATEquality YES YESUninterpreted functions YES co-SEMIInteger arithmetic NO1 NOLinear arithmeticReal arithmeticBitvectors YES YESFloating points YES YES

1 Yuri Matiyasevich. Enumerable sets are diophantine. Journal of SovieticMathematics, 1970.

Beckert, Ulbrich – Formale Systeme II: Theorie 20/47

Presburger Arithmetic

Let ΣP = ({0, 1,+}, {<}), the signature w/o multiplication.

The Presburger Axioms P

1 ∀x(x + 1 6 .= 0)

2 ∀x∀y(x + 1.

= y + 1→ x.

= y)

3 ∀x(x + 0.

= x)

4 ∀x∀y(x + (y + 1).

= (x + y) + 1)

5 For any φ ∈ FmlΣN

(φ(0) ∧ ∀x(φ(x)→ φ(x + 1)))→ ∀x(φ)

A subset of the Peano axioms (w/o those for multiplication).

Conventions:3

def= 1 + 1 + 1, 3x

def= x + x + x , etc.

Beckert, Ulbrich – Formale Systeme II: Theorie 21/47

Presburger Arithmetic

Mojzesz Presburger. Uber die Vollstandigkeit eines gewissenSystems der Arithmetik, Warsaw 1929

Theorem

He proved Presburger arithmetic to be

consistent,

complete, and

decidable.

We are interested in the 3rd property!

Beckert, Ulbrich – Formale Systeme II: Theorie 22/47

Quantifier Elimination

Definition

A theory T admits quantifier elimination (QE) if any formula

Q1x1 . . .Qnxn. φ(x1, . . . , xn, y1, . . . , ym) ∈ Fmlo

is T -equivalent to a quantifier-free formula

ψ(y1, . . . , ym) ∈ Fmlo .

Qi ∈ {∀,∃}

If T -ground instances in FmlQF ∩ Fml can be decided, QE gives usa decision procedure for T .

Beckert, Ulbrich – Formale Systeme II: Theorie 23/47

Quantifier Elimination

Lemma

If T admits QE for any formula

∃x . φ1(x , y1, . . . , ym) ∧ . . . ∧ φn(x , y1, . . . , ym) ∈ Fmlo

with φi literals, then T admits QE for any formula in Fmlo .

Literal: atomic formula or a negation of one.

Proof: (Easy) exercise.

Beckert, Ulbrich – Formale Systeme II: Theorie 24/47

Presburger and Quantifier Elimination

Does Presburger Arithmetic admits QE?

Almost ... However

∃x .y = x + x has no quantifier-free P-equivalent

Add predicates: {k |· : k ∈ N>0} “k divides ...”

∃x .y = x + x ↔ 2|y is P-valid

Presburger Arithmetic with divisibility admits QE.

Cooper’s algorithm ... Blackboard

Beckert, Ulbrich – Formale Systeme II: Theorie 25/47

Decision Procedures

Theory QF-SAT SATEquality YES YESUninterpreted functions YES co-SEMIInteger arithmetic NO NOLinear arithmetic YES YESReal arithmeticBitvectors YES YESFloating points YES YES

Beckert, Ulbrich – Formale Systeme II: Theorie 26/47

Real Arithmetic

Beckert, Ulbrich – Formale Systeme II: Theorie 27/47

Real arithmetic is decidable

Σ = ({+,−, ·, 0, 1}, {≤}), ϕ ∈ FmlΣ

Reminder:

N |= ϕ is not decidable, not even recursive enumerable (Godel).

Tarski-Seidenberg theorem (c. 1948)

R |= ϕ is decidable.Complexity is double exponential (c. 1988).

Idea: Quantifier elimination

Find formula ψ such that (∃x .ϕ(x , y))↔ ψ(y).Computer algebra systems do this: Redlog, Mathematica, (Z3)

Beckert, Ulbrich – Formale Systeme II: Theorie 28/47

Real arithmetic – Axioms

Real arithmetic has a finite axiomatisation R

+ is an Abelian group, · is an Abelian semigroup:

∀x , y , z . (x + y) + z = x + (y + z) ∀x , y , z . (x · y) · z = x · (y · z)∀x , y . x + y = y + x ∀x , y . x · y = y · x∀x . x + 0 = x ∧ 0 + x = x ∀x . x · 1 = x ∧ 1 · x = x∀x . x + (−x) = 0 ∧ (−x) + x = 0

Distributive Laws∀x , y , z . (x + y) · z = x · z + y · z ∧ z · (x + y) = z · x + z · y

Ordering∀x , y , z . x ≤ y → x + z ≤ y + z∀x , y . 0 ≤ x ∧ 0 ≤ y → 0 ≤ xy

Beckert, Ulbrich – Formale Systeme II: Theorie 29/47

Real closed fields

T (R) = T (R) is the set of FOL sentences that are true in R.

But there are also other interesting models of T (R):

Real numbers R,

Real algebraic numbers R ∩ Q(real numbers that are roots of polynomials with integer coeffs.)

Computable numbers(real numbers that can be approximated arbitrarily precisely.)

. . .

Beckert, Ulbrich – Formale Systeme II: Theorie 30/47

Semialgebraic sets

Semialgebraic set

S ⊆ Rn is called semialgebraic if it defined by a booleancombination of polynomial equations and inequalitites.

Boolean combination means: ∪,∩, {

Observation:

S is semialgebaric iff there is a quantifier-free FOL-formula ϕ(S)with n free variables x1, . . . , xn such that

(s1, . . . , sn) ∈ S ⇐⇒ R, [x1 7→ s1, . . . , xn 7→ sn] |= ϕ(S)

Beckert, Ulbrich – Formale Systeme II: Theorie 31/47

Tarski-Seidenberg Theorem

Definition: Projection πn : Rn → Rn−1

πn((s1, . . . , sn)) := (s1, . . . , sn−1)

πn(S) := {πn(s) | s ∈ S} (extended to 2R)

(s1, . . . , sn−1) ∈ πn(S) ⇐⇒ R, [x1 7→ s1, . . . , xn−1 7→ sn−1] |= ∃xn. ϕ(S)

Tarski-Seidenberg Theorem (Projektionssatz)

Let S ⊆ Rn be semialgebraic.Then πn(S) ∈ Rn−1 is also semialgebraic.

Beckert, Ulbrich – Formale Systeme II: Theorie 32/47

Example

Single variable, single quadratic equation

Let Squad be the solutions of ax2 + bx + c = 0.(is semialgebraic: ax2 + bx + c ∈ R[a, b, c , x ])

Due to Tarski-Seidenberg, there must be an equiv. quantifier-freeformula ϕ(π4(Squad)) with free variables a, b, c .

∃x .ax2 + bx + c = 0

⇐⇒

(a 6= 0 ∧ b2 − 4ac ≥ 0)

∨ (a = 0 ∧ (b = 0→ c = 0))

(∃x .x3 + a2x2 + a1x + a0 = 0 is trivally equivalent to true.

)Beckert, Ulbrich – Formale Systeme II: Theorie 33/47

Quantifier Elimination – Algorithm

1 Sufficient to look at ∃x .∧

i φi (y , x) for atomic φi .→ Excercise

2 Sufficient to consider φi of shape p(y , x){<>=

}0

for p ∈ R[y ][x ] → Why?

3 Every polynomial p ∈ R[x ] has finitely many connectedregions with same sign. → BoardChoose a set Rep of representatives.

4 ∃x .∧i

φi (x , y)↔∨

r∈Rep

∧i

φi (r , y)

Decision Technique

Cylindrical Algebraic Decomposition (CAD)

Beckert, Ulbrich – Formale Systeme II: Theorie 34/47

Quantifier Elimination – Linear Example

In R[z , x ]:ψ := ∃x .x > 2 ∧ x < 3 ∧ x > z

Interesting points for x : I = {2, 3, z}Interesting intervals: (−∞, 2), (2, 3), (3,∞), (2, z), . . .

Representatives:Rep =

{2, 3, z , “−∞”, “+∞”, 2+3

2 , 2+z2 , 3+z

2

}={

i1+i22 | i1, i2 ∈ I

}∪ {“−∞”, “+∞”}

For the example:ψ ↔

∨r∈Rep r > 2 ∧ r < 3 ∧ x > z

↔ 2.5 > z ∨ (z > 2 ∧ z < 4 ∧ 2 > z) ∨ (z > 1 ∧ z < 3 ∧ 3 > z)↔ z < 3

Beckert, Ulbrich – Formale Systeme II: Theorie 35/47

Decision Procedures

Theory QF-SAT SATEquality YES YESUninterpreted functions YES co-SEMIInteger arithmetic NO NOLinear arithmetic YES YESReal arithmetic YES YESBitvectors YES YESFloating points YES YES

Beckert, Ulbrich – Formale Systeme II: Theorie 36/47

Divison

Adding division (the inverse ·−1) does not increse expressive power.

Consider Σdiv = Σ ∪ {·−1}.Let quantifier-free ϕ ∈ FmlqfΣdiv

contain a division by t:

ϕ[t−1] ↔((∃y .y = t−1 ∧ ϕ[y ]) ∨ (t = 0 ∧ ϕ[n])

)(1)

n is a fresh free variable for the value of “0−1”

Let ψ ∈ FmlΣdivcontain divisions.

Obtain ψ′ ∈ FmlΣ by applying (1) to literals in ψ.

R |= ψ ⇐⇒ R |= ∀n.ψ′

Underspecification: ψ is true in R if it is true for all possiblevaluations of “0−1”: R |= 1

0 = 10 , R 6|= 1

0 = 20

Beckert, Ulbrich – Formale Systeme II: Theorie 37/47

Combining Theories

Beckert, Ulbrich – Formale Systeme II: Theorie 38/47

Combining Theories

What if we have two (or more) theories within one formula?

f (a) = g(a + 1) ∧ g(a + b) > f (a) satisfiable?

Decision procedures exist for linear integers, and for uninterpretedfunctions.

Goal

Find decision procedures for combinations of theories.

Combinations of theoriesLet T1 ⊆ FmlΣ1 and T2 ⊆ FmlΣ2 be theories.The combined theory T1,2 ∈ FmlΣ1∪Σ2 is defined as:

T1,2def= T (T1 ∪ T2)

Beckert, Ulbrich – Formale Systeme II: Theorie 39/47

Purification

f (a) = g(a + 1) ∧ g(a + b) > f (a) (1)

Purification

Extract expressions using fresh constants and equalities.Make each atomic formula belong to one theory only.

f (a) = g(y) ∧ y = a + 1 ∧z = g(u) ∧ u = a + b ∧ w = f (a) ∧ z > w

is equisatisfiable to (1).

Note: This resembles the construction of the “short CNF”.

Beckert, Ulbrich – Formale Systeme II: Theorie 40/47

Convex Theories

Definition

A Σ theory T is convex if for every conjunctive ϕ ∈ FmlΣ

(ϕ→⋃

i=1 xi = yi ) is T -valid for some finite n > 1implies that

(ϕ→ xi = yi ) is T -valid for some i ∈ {1, . . . , n}

where xi , yi , for i ∈ {1, . . . , n}, are variables.

Examples:

Linear arithmetic over R is convex.

Linear arithmetic over N is not convex:

x1 = 1 ∧ x2 = 2 ∧ 1 ≤ x3 ∧ x3 ≤ 2→ (x3 = x1 ∨ x3 = x2)

Beckert, Ulbrich – Formale Systeme II: Theorie 41/47

Nelson-Oppen Combination Procedure

In order for the Nelson–Oppen procedure to be applicable, thetheories T1,T2 must comply with the following restrictions:

1 T1,T2 are quantifier-free first-order theories with equality.

2 There is a decision procedure for each of the theories

3 The signatures are disjoint, i.e., for all Σ1 ∩ Σ2 = ∅4 T1,T2 are theories are stably infinite: Every T -satisfiable

formula has an infinite model (e.g., linear arithmetic over R,but not the theory of finite-width bit vectors).

(Generalisation to more than two theories: simple, see literature)

Beckert, Ulbrich – Formale Systeme II: Theorie 42/47

Example

from: D. Kroning, O.Strichman: Decision Procedures, Springer Verlag

Beckert, Ulbrich – Formale Systeme II: Theorie 43/47

Example

Beckert, Ulbrich – Formale Systeme II: Theorie 44/47

Nelsson-Oppen Algorithm – convex case

T1,T2 convex theories with with the Nelsson-Oppen properties.Assume convex (conjunctive) problem.

τ bridges between T1 and T2 and is a conjunction of equalitiesover variables

After purification: ϕ1 ∈ Fml1, ϕ2 ∈ Fml2, τ ⊆ Fml=

1 If ϕ1 ∧∧τ is T1-unsatisfiable, return UNSAT

2 If ϕ2 ∧∧τ is T2-unsatisfiable, return UNSAT

3 “learn” new equalities:τ := τ ∪

⋃{x = y | ϕ1 ∧ τ → x = y is T1-valid}

∪⋃{x = y | ϕ2 ∧ τ → x = y is T2-valid}

4 If nothing was “learnt”, return SAT

5 Go to 1

Beckert, Ulbrich – Formale Systeme II: Theorie 45/47

Soundness

This algorithm is a decision procedure for T1/2.

To show: ϕ1 ∧ ϕ2 is satisfiable ⇐⇒ algorithm returns SAT .

Proof sketch on blackboard

see also: D. Kroning, O. Strichman: Decision Procedures, SpringerVerlag. Section 10.3.3.

Beckert, Ulbrich – Formale Systeme II: Theorie 46/47

Non-convex theories

1 If ϕ1 ∧ τ is T1-unsatisfiable, return UNSAT

2 If ϕ2 ∧ τ is T2-unsatisfiable, return UNSAT

3 “learn” new equalities:τ := τ ∧

∧{x = y | ϕ1 ∧ τ → x = y is T1-valid}

∧∧{x = y | ϕ2 ∧ τ → x = y is T2-valid}

4 If nothing was “learnt”, split: If there exists i such that

ϕi → (x1 = y1 ∨ . . . ∨ xk = yk) andϕi 6→ (xj = yj)

then apply Nelson–Oppen recursively to adding xi = yi to thedifferent τ .If any of these subproblems is satisfiable, return “Satisfiable”.Otherwise, return “Unsatisfiable”.

Beckert, Ulbrich – Formale Systeme II: Theorie 47/47