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
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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.
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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
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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
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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}
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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)
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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.
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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.
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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
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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?
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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!
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Decision Procedures
Theory QF-SAT SATEquality YES YESUninterpreted functions YES co-SEMIInteger arithmeticLinear arithmeticReal arithmeticBitvectors YES YESFloating points YES YES
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Natural Arithmetic
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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)
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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.
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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:
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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)
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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.
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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.
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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!
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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 .
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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.
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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
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Decision Procedures
Theory QF-SAT SATEquality YES YESUninterpreted functions YES co-SEMIInteger arithmetic NO NOLinear arithmetic YES YESReal arithmeticBitvectors YES YESFloating points YES YES
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Real Arithmetic
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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)
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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
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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.)
. . .
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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)
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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.
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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.
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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)
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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
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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
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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
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Combining Theories
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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)
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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”.
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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)
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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)
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Example
from: D. Kroning, O.Strichman: Decision Procedures, Springer Verlag
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Example
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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
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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.
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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”.
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