[PPT] - Weak monadic second-order theory of one successor (WS1S) PowerPoint Presentation

SLIDE 1

Weak monadic second-order theory of one successor (WS1S)

Presentation for Seminar on Decision Procedures

Susanne van den Elsen

Universit¨ at des Saarlandes

January 25th, 2013

Susanne van den Elsen (UdS) WS1S January 25th, 2013 1 / 52

SLIDE 2

Introduction

Expressiveness and decidability

expressiveness decidability/complexity

Susanne van den Elsen (UdS) WS1S January 25th, 2013 2 / 52

SLIDE 3

Introduction

Expressiveness and decidability

expressiveness decidability/complexity

Boolean Logic NP- complete

Susanne van den Elsen (UdS) WS1S January 25th, 2013 2 / 52

SLIDE 4

Introduction

Expressiveness and decidability

expressiveness decidability/complexity

Boolean Logic NP- complete Quantified Boolean Logic PSPACE- complete

Susanne van den Elsen (UdS) WS1S January 25th, 2013 2 / 52

SLIDE 5

Introduction

Expressiveness and decidability

expressiveness decidability/complexity

Boolean Logic NP- complete Quantified Boolean Logic PSPACE- complete First Order Logic ∀x.P(x) undecidable! Second Order Logic ∃R.∀x.R(x, x)

Susanne van den Elsen (UdS) WS1S January 25th, 2013 2 / 52

SLIDE 6

Introduction

Expressiveness and decidability

expressiveness decidability/complexity

Boolean Logic NP- complete Quantified Boolean Logic PSPACE- complete First Order Logic ∀x.P(x) undecidable! Second Order Logic ∃R.∀x.R(x, x) First-order theories

Susanne van den Elsen (UdS) WS1S January 25th, 2013 2 / 52

SLIDE 7

Introduction

Monadic second-order logic (MSO)

Monadic second-order logic (MSO): fragment of second-order logic
nly quantification over sets
Weak monadic second-order logic (WMSO): fragment of MSO
nly quantification over finite sets
WS1S: WMSO of one successor

Susanne van den Elsen (UdS) WS1S January 25th, 2013 3 / 52

SLIDE 8

Introduction

Monadic second-order logic (MSO)

Monadic second-order logic (MSO): fragment of second-order logic
nly quantification over sets
Weak monadic second-order logic (WMSO): fragment of MSO
nly quantification over finite sets
WS1S: WMSO of one successor

Susanne van den Elsen (UdS) WS1S January 25th, 2013 3 / 52

SLIDE 9

Introduction

Monadic second-order logic (MSO)

Monadic second-order logic (MSO): fragment of second-order logic
nly quantification over sets
Weak monadic second-order logic (WMSO): fragment of MSO
nly quantification over finite sets
WS1S: WMSO of one successor

Susanne van den Elsen (UdS) WS1S January 25th, 2013 3 / 52

SLIDE 10

Introduction

Expressiveness and decidability

expressiveness decidability/complexity

Boolean Logic NP- complete Quantified Boolean Logic PSPACE- complete WS1S non-elementary

22. . .2n

O(n) First Order Logic ∀x.P(x) undecidable! Second Order Logic ∃R.∀x.R(x, x) REG

Susanne van den Elsen (UdS) WS1S January 25th, 2013 4 / 52

SLIDE 11

Introduction

Application of WS1S

WS1S implemented (e.g. in MONA, MOSEL, . . . )
Express properties of programs manipulating linked datastructures (e.g. lists)

root

.next .next .next .next .next .next

node1 node2

Express that node2 is reachable from node1:

Reachable(node1, node2) := node1, node2 ∈ {x, y | y = x.next}∗ Closed(S, next) := ∀x∀y(x ∈ S ∧ y = x.next → y ∈ S) node1, node2 ∈ next∗ := ∀S.(node1 ∈ S ∧ Closed(S, next) → node2 ∈ S)

Susanne van den Elsen (UdS) WS1S January 25th, 2013 5 / 52

SLIDE 12

Introduction

Application of WS1S

WS1S implemented (e.g. in MONA, MOSEL, . . . )
Express properties of programs manipulating linked datastructures (e.g. lists)

root

.next .next .next .next .next .next

node1 node2

Express that node2 is reachable from node1:

Reachable(node1, node2) := node1, node2 ∈ {x, y | y = x.next}∗ Closed(S, next) := ∀x∀y(x ∈ S ∧ y = x.next → y ∈ S) node1, node2 ∈ next∗ := ∀S.(node1 ∈ S ∧ Closed(S, next) → node2 ∈ S)

Susanne van den Elsen (UdS) WS1S January 25th, 2013 5 / 52

SLIDE 13

Introduction

Application of WS1S

WS1S implemented (e.g. in MONA, MOSEL, . . . )
Express properties of programs manipulating linked datastructures (e.g. lists)

root

.next .next .next .next .next .next

node1 node2

Express that node2 is reachable from node1:

Reachable(node1, node2) := node1, node2 ∈ {x, y | y = x.next}∗ Closed(S, next) := ∀x∀y(x ∈ S ∧ y = x.next → y ∈ S) node1, node2 ∈ next∗ := ∀S.(node1 ∈ S ∧ Closed(S, next) → node2 ∈ S)

Susanne van den Elsen (UdS) WS1S January 25th, 2013 5 / 52

SLIDE 14

Introduction

Application of WS1S

x y

(a) x and y are separate

x

(b) x is acyclic

x y

(c) no garbage

Figure: Properties that can be expressed in terms of reachability.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 6 / 52

SLIDE 15

Introduction

Overview of this presentation

WS1S

Susanne van den Elsen (UdS) WS1S January 25th, 2013 7 / 52

SLIDE 16

Introduction

Overview of this presentation

WS1S Syntax Semantics

Susanne van den Elsen (UdS) WS1S January 25th, 2013 7 / 52

SLIDE 17

Introduction

Overview of this presentation

WS1S Regular languages Syntax Semantics

Susanne van den Elsen (UdS) WS1S January 25th, 2013 7 / 52

SLIDE 18

Introduction

Overview of this presentation

WS1S Finite automata Regular languages Syntax Semantics

Susanne van den Elsen (UdS) WS1S January 25th, 2013 7 / 52

SLIDE 19

Introduction

Overview of this presentation

WS1S Finite automata Regular languages Syntax Semantics WS1S on words

Susanne van den Elsen (UdS) WS1S January 25th, 2013 7 / 52

SLIDE 20

Introduction

Overview of this presentation

WS1S Finite automata Regular languages Syntax Semantics WS1S on words Decision procedure

Susanne van den Elsen (UdS) WS1S January 25th, 2013 7 / 52

SLIDE 21

Introduction

Overview of this presentation

WS1S Finite automata Regular languages Syntax Semantics WS1S on words Decision procedure Complexity

22. . .2n

O(n)

Susanne van den Elsen (UdS) WS1S January 25th, 2013 7 / 52

SLIDE 22

Weak monadic second-order theory of one successor

Section 2 Weak monadic second-order theory of one successor

Susanne van den Elsen (UdS) WS1S January 25th, 2013 8 / 52

SLIDE 23

Weak monadic second-order theory of one successor Syntax

Syntax

Syntax (S1S) First order variable set V1 = {x1, x2, . . .} Second-order variable set V2 = {X1, X2, . . .} Terms t: t ::= 0 | x, for x ∈ V1 Formulas φ: φ ::= S(t, t) | t ∈ X | ¬φ | φ1 ∧ φ2 | ∃x.φ | ∃X.φ, for x ∈ V1 and X ∈ V2 Syntax (WS1S) Same as S1S, but only quantification over finite sets.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 9 / 52

SLIDE 24

Weak monadic second-order theory of one successor Semantics

Semantics

On structure ω, S, < Interpretation σ = σ1, σ2, where σ1 : V1 → N and σ2 : V2 → N ∈ 2N, with N is finite.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 10 / 52

SLIDE 25

Weak monadic second-order theory of one successor Semantics

Semantics

Semantics Semantics of terms: [0]σ1 = 0; [x]σ1 = σ1(x); Satisfiability: σ | = t ∈ X ⇐ ⇒ σ(t) ∈ σ(X); σ | = S(t, t′) ⇐ ⇒ σ(t) + 1 = σ(t′); σ | = ¬φ ⇐ ⇒ σ | = φ; σ | = φ1 ∧ φ2 ⇐ ⇒ σ | = φ1 and σ | = φ2; σ | = ∃x.φ ⇐ ⇒ σ[n/x] | = φ for some n ∈ N; σ | = ∃X.φ ⇐ ⇒ σ[N/X] | = φ for some finite N ∈ 2N. Validity: | = φ ⇐ ⇒ σ | = φ, for all interpretations σ

Susanne van den Elsen (UdS) WS1S January 25th, 2013 11 / 52

SLIDE 26

Weak monadic second-order theory of one successor Semantics

Syntactic sugar

Example Closed(Z, R) := ∀x∀y(x ∈ Z ∧ R(x, y) → y ∈ Z) z1, z2 ∈ R∗ := ∀Z.(z1 ∈ Z ∧ Closed(Z, R) → z2 ∈ Z) x ≤ y := x < y := first(x) := last(x) := X ⊆ Y := X = Y := X = ∅ := Sing(X) := Succ(X, Y ) := ;

Susanne van den Elsen (UdS) WS1S January 25th, 2013 12 / 52

SLIDE 27

Weak monadic second-order theory of one successor Semantics

Syntactic sugar

Example Closed(Z, R) := ∀x∀y(x ∈ Z ∧ R(x, y) → y ∈ Z) z1, z2 ∈ R∗ := ∀Z.(z1 ∈ Z ∧ Closed(Z, R) → z2 ∈ Z) x ≤ y := ∀X.(y ∈ X ∧ ∀z.∀z′(z ∈ X ∧ S(z′, z) → z′ ∈ X) → X(x)) i.e. x, y ∈ S∗ x < y := first(x) := last(x) := X ⊆ Y := X = Y := X = ∅ := Sing(X) := Succ(X, Y ) := ;

Susanne van den Elsen (UdS) WS1S January 25th, 2013 12 / 52

SLIDE 28

Weak monadic second-order theory of one successor Semantics

Syntactic sugar

Example Closed(Z, R) := ∀x∀y(x ∈ Z ∧ R(x, y) → y ∈ Z) z1, z2 ∈ R∗ := ∀Z.(z1 ∈ Z ∧ Closed(Z, R) → z2 ∈ Z) x ≤ y := ∀X.(y ∈ X ∧ ∀z.∀z′(z ∈ X ∧ S(z′, z) → z′ ∈ X) → X(x)) i.e. x, y ∈ S∗ x < y := x ≤ y ∧ ¬y ≤ x first(x) := last(x) := X ⊆ Y := X = Y := X = ∅ := Sing(X) := Succ(X, Y ) := ;

Susanne van den Elsen (UdS) WS1S January 25th, 2013 12 / 52

SLIDE 29

Weak monadic second-order theory of one successor Semantics

Syntactic sugar

Example Closed(Z, R) := ∀x∀y(x ∈ Z ∧ R(x, y) → y ∈ Z) z1, z2 ∈ R∗ := ∀Z.(z1 ∈ Z ∧ Closed(Z, R) → z2 ∈ Z) x ≤ y := ∀X.(y ∈ X ∧ ∀z.∀z′(z ∈ X ∧ S(z′, z) → z′ ∈ X) → X(x)) i.e. x, y ∈ S∗ x < y := x ≤ y ∧ ¬y ≤ x first(x) := ¬∃y.S(y, x) last(x) := X ⊆ Y := X = Y := X = ∅ := Sing(X) := Succ(X, Y ) := ;

Susanne van den Elsen (UdS) WS1S January 25th, 2013 12 / 52

SLIDE 30

Weak monadic second-order theory of one successor Semantics

Syntactic sugar

Example Closed(Z, R) := ∀x∀y(x ∈ Z ∧ R(x, y) → y ∈ Z) z1, z2 ∈ R∗ := ∀Z.(z1 ∈ Z ∧ Closed(Z, R) → z2 ∈ Z) x ≤ y := ∀X.(y ∈ X ∧ ∀z.∀z′(z ∈ X ∧ S(z′, z) → z′ ∈ X) → X(x)) i.e. x, y ∈ S∗ x < y := x ≤ y ∧ ¬y ≤ x first(x) := ¬∃y.S(y, x) last(x) := ¬∃y.S(x, y) X ⊆ Y := X = Y := X = ∅ := Sing(X) := Succ(X, Y ) := ;

Susanne van den Elsen (UdS) WS1S January 25th, 2013 12 / 52

SLIDE 31

Weak monadic second-order theory of one successor Semantics

Syntactic sugar

Example Closed(Z, R) := ∀x∀y(x ∈ Z ∧ R(x, y) → y ∈ Z) z1, z2 ∈ R∗ := ∀Z.(z1 ∈ Z ∧ Closed(Z, R) → z2 ∈ Z) x ≤ y := ∀X.(y ∈ X ∧ ∀z.∀z′(z ∈ X ∧ S(z′, z) → z′ ∈ X) → X(x)) i.e. x, y ∈ S∗ x < y := x ≤ y ∧ ¬y ≤ x first(x) := ¬∃y.S(y, x) last(x) := ¬∃y.S(x, y) X ⊆ Y := ∀x.(x ∈ X → x ∈ Y ) X = Y := X = ∅ := Sing(X) := Succ(X, Y ) := ;

Susanne van den Elsen (UdS) WS1S January 25th, 2013 12 / 52

SLIDE 32

Weak monadic second-order theory of one successor Semantics

Syntactic sugar

Example Closed(Z, R) := ∀x∀y(x ∈ Z ∧ R(x, y) → y ∈ Z) z1, z2 ∈ R∗ := ∀Z.(z1 ∈ Z ∧ Closed(Z, R) → z2 ∈ Z) x ≤ y := ∀X.(y ∈ X ∧ ∀z.∀z′(z ∈ X ∧ S(z′, z) → z′ ∈ X) → X(x)) i.e. x, y ∈ S∗ x < y := x ≤ y ∧ ¬y ≤ x first(x) := ¬∃y.S(y, x) last(x) := ¬∃y.S(x, y) X ⊆ Y := ∀x.(x ∈ X → x ∈ Y ) X = Y := X ⊆ Y ∧ Y ⊆ X X = ∅ := Sing(X) := Succ(X, Y ) := ;

Susanne van den Elsen (UdS) WS1S January 25th, 2013 12 / 52

SLIDE 33

Weak monadic second-order theory of one successor Semantics

Syntactic sugar

Example Closed(Z, R) := ∀x∀y(x ∈ Z ∧ R(x, y) → y ∈ Z) z1, z2 ∈ R∗ := ∀Z.(z1 ∈ Z ∧ Closed(Z, R) → z2 ∈ Z) x ≤ y := ∀X.(y ∈ X ∧ ∀z.∀z′(z ∈ X ∧ S(z′, z) → z′ ∈ X) → X(x)) i.e. x, y ∈ S∗ x < y := x ≤ y ∧ ¬y ≤ x first(x) := ¬∃y.S(y, x) last(x) := ¬∃y.S(x, y) X ⊆ Y := ∀x.(x ∈ X → x ∈ Y ) X = Y := X ⊆ Y ∧ Y ⊆ X X = ∅ := ∀Z.X ⊆ Z Sing(X) := Succ(X, Y ) := ;

Susanne van den Elsen (UdS) WS1S January 25th, 2013 12 / 52

SLIDE 34

Weak monadic second-order theory of one successor Semantics

Syntactic sugar

Example Closed(Z, R) := ∀x∀y(x ∈ Z ∧ R(x, y) → y ∈ Z) z1, z2 ∈ R∗ := ∀Z.(z1 ∈ Z ∧ Closed(Z, R) → z2 ∈ Z) x ≤ y := ∀X.(y ∈ X ∧ ∀z.∀z′(z ∈ X ∧ S(z′, z) → z′ ∈ X) → X(x)) i.e. x, y ∈ S∗ x < y := x ≤ y ∧ ¬y ≤ x first(x) := ¬∃y.S(y, x) last(x) := ¬∃y.S(x, y) X ⊆ Y := ∀x.(x ∈ X → x ∈ Y ) X = Y := X ⊆ Y ∧ Y ⊆ X X = ∅ := ∀Z.X ⊆ Z Sing(X) := ∃Y .(Y ⊆ X ∧ Y = X ∧ ¬∃Z.(Z ⊆ Y ∧ Z = Y )) Succ(X, Y ) := ;

Susanne van den Elsen (UdS) WS1S January 25th, 2013 12 / 52

SLIDE 35

Weak monadic second-order theory of one successor Semantics

Syntactic sugar

Example Closed(Z, R) := ∀x∀y(x ∈ Z ∧ R(x, y) → y ∈ Z) z1, z2 ∈ R∗ := ∀Z.(z1 ∈ Z ∧ Closed(Z, R) → z2 ∈ Z) x ≤ y := ∀X.(y ∈ X ∧ ∀z.∀z′(z ∈ X ∧ S(z′, z) → z′ ∈ X) → X(x)) i.e. x, y ∈ S∗ x < y := x ≤ y ∧ ¬y ≤ x first(x) := ¬∃y.S(y, x) last(x) := ¬∃y.S(x, y) X ⊆ Y := ∀x.(x ∈ X → x ∈ Y ) X = Y := X ⊆ Y ∧ Y ⊆ X X = ∅ := ∀Z.X ⊆ Z Sing(X) := ∃Y .(Y ⊆ X ∧ Y = X ∧ ¬∃Z.(Z ⊆ Y ∧ Z = Y )) Succ(X, Y ) := Sing(X) ∧ Sing(Y ) ∧ ((x ∈ X ∧ y ∈ Y ) → S(x, y)) ;

Susanne van den Elsen (UdS) WS1S January 25th, 2013 12 / 52

SLIDE 36

Weak monadic second-order theory of one successor WS1S0

WS1S0

Reduction of WS1S to simpler formalism WS1S0. Syntax Second-order variable set V2 = {X1, X2, . . .} Formulas φ ::= X ⊆ Y | Sing(X) | Succ(X, Y ) | ∃X.φ | ¬φ | φ1 ∧ φ2

Susanne van den Elsen (UdS) WS1S January 25th, 2013 13 / 52

SLIDE 37

Weak monadic second-order theory of one successor WS1S0

WS1S0

Example 0 ∈ X ∧ ∀p.p ∈ X ↔ (∃q.S(p, q) ∧ q ∈ Y ) ⇓ eliminate 0 (∃x0(x0 ∈ X ∧ ¬∃z.z < x0)) ∧ ∀p.p ∈ X ↔ (∃q.S(p, q) ∧ q ∈ Y ) ⇓ eliminate first order variables (∃X0.Sing(X0) ∧ (X0 ⊆ X ∧ ¬∃Z.Sing(Z) ∧ Z < X0))∧ ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y ))

Susanne van den Elsen (UdS) WS1S January 25th, 2013 14 / 52

SLIDE 38

Weak monadic second-order theory of one successor WS1S0

WS1S0

Example 0 ∈ X ∧ ∀p.p ∈ X ↔ (∃q.S(p, q) ∧ q ∈ Y ) ⇓ eliminate 0 (∃x0(x0 ∈ X ∧ ¬∃z.z < x0)) ∧ ∀p.p ∈ X ↔ (∃q.S(p, q) ∧ q ∈ Y ) ⇓ eliminate first order variables (∃X0.Sing(X0) ∧ (X0 ⊆ X ∧ ¬∃Z.Sing(Z) ∧ Z < X0))∧ ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y ))

Susanne van den Elsen (UdS) WS1S January 25th, 2013 14 / 52

SLIDE 39

Weak monadic second-order theory of one successor WS1S0

WS1S0

Example 0 ∈ X ∧ ∀p.p ∈ X ↔ (∃q.S(p, q) ∧ q ∈ Y ) ⇓ eliminate 0 (∃x0(x0 ∈ X ∧ ¬∃z.z < x0)) ∧ ∀p.p ∈ X ↔ (∃q.S(p, q) ∧ q ∈ Y ) ⇓ eliminate first order variables (∃X0.Sing(X0) ∧ (X0 ⊆ X ∧ ¬∃Z.Sing(Z) ∧ Z < X0))∧ ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y ))

Susanne van den Elsen (UdS) WS1S January 25th, 2013 14 / 52

SLIDE 40

Weak monadic second-order theory of one successor WS1S0

WS1S0

Example 0 ∈ X ∧ ∀p.p ∈ X ↔ (∃q.S(p, q) ∧ q ∈ Y ) ⇓ eliminate 0 (∃x0(x0 ∈ X ∧ ¬∃z.z < x0)) ∧ ∀p.p ∈ X ↔ (∃q.S(p, q) ∧ q ∈ Y ) ⇓ eliminate first order variables (∃X0.Sing(X0) ∧ (X0 ⊆ X ∧ ¬∃Z.Sing(Z) ∧ Z < X0))∧ ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y ))

Susanne van den Elsen (UdS) WS1S January 25th, 2013 14 / 52

SLIDE 41

Weak monadic second-order theory of one successor WS1S0

WS1S0

Example 0 ∈ X ∧ ∀p.p ∈ X ↔ (∃q.S(p, q) ∧ q ∈ Y ) ⇓ eliminate 0 (∃x0(x0 ∈ X ∧ ¬∃z.z < x0)) ∧ ∀p.p ∈ X ↔ (∃q.S(p, q) ∧ q ∈ Y ) ⇓ eliminate first order variables (∃X0.Sing(X0) ∧ (X0 ⊆ X ∧ ¬∃Z.Sing(Z) ∧ Z < X0))∧ ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y ))

Susanne van den Elsen (UdS) WS1S January 25th, 2013 14 / 52

SLIDE 42

Weak monadic second-order theory of one successor WS1S on words

MSO on words

WS1S0-formula φ(X1, . . . , Xn) with at most free variables X1, . . . , Xn
Interpretations as finite strings w in ({0, 1}n)∗
X1 is described along the first track, X2 along the second track, etc.
Letter c1, c2, . . . , cn at position p indicates that

p ∈ Xi ⇔ ci = 1 Example φ(X, Y ) : ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y )) Interpretation σ with σ(X) = {0, 3, 4, 5} and σ(Y ) = {1, 4, 5, 6} is denoted by w = X Y

1

1 1 1 1 1 1 1

∈ ({0, 1}2)∗

Susanne van den Elsen (UdS) WS1S January 25th, 2013 15 / 52

SLIDE 43

Weak monadic second-order theory of one successor WS1S on words

MSO on words

WS1S0-formula φ(X1, . . . , Xn) with at most free variables X1, . . . , Xn
Interpretations as finite strings w in ({0, 1}n)∗
X1 is described along the first track, X2 along the second track, etc.
Letter c1, c2, . . . , cn at position p indicates that

p ∈ Xi ⇔ ci = 1 Example φ(X, Y ) : ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y )) Interpretation σ with σ(X) = {0, 3, 4, 5} and σ(Y ) = {1, 4, 5, 6} is denoted by w = X Y

1

1 1 1 1 1 1 1

∈ ({0, 1}2)∗

Susanne van den Elsen (UdS) WS1S January 25th, 2013 15 / 52

SLIDE 44

Weak monadic second-order theory of one successor WS1S on words

MSO on words

WS1S0-formula φ(X1, . . . , Xn) with at most free variables X1, . . . , Xn
Interpretations as finite strings w in ({0, 1}n)∗
X1 is described along the first track, X2 along the second track, etc.
Letter c1, c2, . . . , cn at position p indicates that

p ∈ Xi ⇔ ci = 1 Example φ(X, Y ) : ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y )) Interpretation σ with σ(X) = {0, 3, 4, 5} and σ(Y ) = {1, 4, 5, 6} is denoted by w = X Y

1

1 1 1 1 1 1 1

∈ ({0, 1}2)∗

Susanne van den Elsen (UdS) WS1S January 25th, 2013 15 / 52

SLIDE 45

Weak monadic second-order theory of one successor WS1S on words

MSO on words

Formally: represent w ∈ ({0, 1}n)∗ by structure

w = dom(w), 0, S, P1, . . . , Pn, where Pk = {i | (w(i))k = 1} Example φ(X, Y ) : ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y )) Interpretation σ with σ(X) = {0, 3, 4, 5} and σ(Y ) = {1, 4, 5, 6} is denoted by w = {0, . . . , 6}, 0, S, PX, PY and w | = φ. w = PX PY

1

2 3 4 5 6 1 1 1 1 1 1 1 1

Susanne van den Elsen (UdS)

WS1S January 25th, 2013 16 / 52

SLIDE 46

Weak monadic second-order theory of one successor WS1S on words

MSO on words

Formally: represent w ∈ ({0, 1}n)∗ by structure

w = dom(w), 0, S, P1, . . . , Pn, where Pk = {i | (w(i))k = 1} Example φ(X, Y ) : ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y )) Interpretation σ with σ(X) = {0, 3, 4, 5} and σ(Y ) = {1, 4, 5, 6} is denoted by w = {0, . . . , 6}, 0, S, PX, PY and w | = φ. w = PX PY

1

2 3 4 5 6 1 1 1 1 1 1 1 1

Susanne van den Elsen (UdS)

WS1S January 25th, 2013 16 / 52

SLIDE 47

Weak monadic second-order theory of one successor WS1S on words

Word models

w |

= φ(X1, . . . Xn) ⇔ φ holds in w with Pk as interpretation of Xk

L(φ) =def {w ∈ ({0, 1}n)∗ | w |

= φ(X1, . . . , Xn)} Example φ(X, Y ) : ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y )) w = PX PY

1

2 3 4 5 6 1 1 1 1 1 1 1 1

,

w ′ = PX PY

1

2 3 4 5 6 1 1 1 1 1 1 1 1 1 1 1 1

w |

= φ w ′ | = φ.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 17 / 52

SLIDE 48

Weak monadic second-order theory of one successor WS1S on words

Word models

w |

= φ(X1, . . . Xn) ⇔ φ holds in w with Pk as interpretation of Xk

L(φ) =def {w ∈ ({0, 1}n)∗ | w |

= φ(X1, . . . , Xn)} Example φ(X, Y ) : ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y )) w = PX PY

1

2 3 4 5 6 1 1 1 1 1 1 1 1

,

w ′ = PX PY

1

2 3 4 5 6 1 1 1 1 1 1 1 1 1 1 1 1

w |

= φ w ′ | = φ.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 17 / 52

SLIDE 49

Weak monadic second-order theory of one successor WS1S on words

Word models

Note: many words are associated with the same interpretation
An arbitrary number of 0, 0, . . . , 0 ∈ 0n letters may be added at the end of

the word Example φ(X, Y ) : ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y )) Interpretation σ with σ(X) = {0, 3, 4, 5} and σ(Y ) = {1, 4, 5, 6} is denoted by w, . . . , w ′, . . . w = PX PY

1

2 3 4 5 6 1 1 1 1 1 1 1 1

,

w ′ = PX PY

1

2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1

Susanne van den Elsen (UdS)

WS1S January 25th, 2013 18 / 52

SLIDE 50

Weak monadic second-order theory of one successor WS1S on words

Word models

Note: many words are associated with the same interpretation
An arbitrary number of 0, 0, . . . , 0 ∈ 0n letters may be added at the end of

the word Example φ(X, Y ) : ∀P.Sing(P) → (P ⊆ X ↔ (∃Q.Sing(Q) ∧ Succ(P, Q) ∧ Q ⊆ Y )) Interpretation σ with σ(X) = {0, 3, 4, 5} and σ(Y ) = {1, 4, 5, 6} is denoted by w, . . . , w ′, . . . w = PX PY

1

2 3 4 5 6 1 1 1 1 1 1 1 1

,

w ′ = PX PY

1

2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1

Susanne van den Elsen (UdS)

WS1S January 25th, 2013 18 / 52

SLIDE 51

Expressiveness

Section 3 Expressiveness

Susanne van den Elsen (UdS) WS1S January 25th, 2013 19 / 52

SLIDE 52

Expressiveness B¨ uchi theorem

B¨ uchi theorem

Theorem (B¨ uchi theorem, [B¨ uchi, 1960] and [Elgot, 1961]) A language L ⊆ Σ∗ is regular ⇐ ⇒ it is expressible in weak monadic second-order logic on words.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 20 / 52

SLIDE 53

Expressiveness From finite automata to WS1S

From finite automata to WS1S

Lemma Every regular language L ⊆ Σ∗ is expressible in WS1S on words. Proof. From [Thomas, 1997].

Assume L ⊆ Σ∗ is regular.
Finite automaton A = S, Σ, s0, T, F such that L = L(A).
Assume S = {0, . . . , k}, where s0 = 0.
Over a word w = a0 . . . an−1 ∈ L, the WS1S sentence φ will state:

there is an accepting run r = r0r1 . . . rn, with r0 = 0, and rn ∈ F r0

a0

− → r1

a1

− → . . .

an−1

− − − → rn

Code r0r1 . . . rn−1 by a tuple X0, . . . , Xk of pairwise disjoint subsets of

{0, . . . , n − 1} such that Xi contains those positions of r where state i is assumed.

From rn−1, A should be able to reach a final state via last letter an−1.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 21 / 52

SLIDE 54

Expressiveness From finite automata to WS1S

From finite automata to WS1S

Proof continued. Thus A accepts the nonempty word w iff w | = ∃X0 . . . Xk

i=j

∀x¬

x ∈ Xi ∧ x ∈ Xj
∧

∀x

first(x) → x ∈ X0
∧

∀x∀y

S(x, y) →
i,a,j∈T
x ∈ Xi ∧ x ∈ Pa ∧ y ∈ Xj
∧

∀x

last(x) →
f ∈F;i,a,f ∈T
x ∈ Xi ∧ x ∈ Pa
.

If empty word is not accepted, clause such as ∃x.x = x should be added.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 22 / 52

SLIDE 55

Expressiveness From WS1S to finite automata

From WS1S to finite automata

Lemma Every WS1S definable language is recognisable by a finite automaton. Proof.

Assume L = L(φ) = {σ | σ |

= φ}, for WS1S formula φ.

Rewrite φ as an MSO0 formula φ′(X1, . . . , Xn), where

L(φ′) = {w ∈ ({0, 1}n)∗ | w | = φ}.

Construct automaton A such that L(A) = L(φ′) by induction on the

structure of φ′.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 23 / 52

SLIDE 56

Expressiveness From WS1S to finite automata

Example φWS1S : ∃X.∀p.p ∈ X φWS1S0 : ∃X.∀P.(Sing(P) → P ⊆ X) ∃X.¬∃P.¬(Sing(P) → P ⊆ X) ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X) Since φ has no free variables: L(φ) ⊆ ()∗

Susanne van den Elsen (UdS) WS1S January 25th, 2013 24 / 52

SLIDE 57

Expressiveness From WS1S to finite automata

Base case: Sing(X)

Proof continued. start 1

Figure: ASing(X)

Susanne van den Elsen (UdS) WS1S January 25th, 2013 25 / 52

SLIDE 58

Expressiveness From WS1S to finite automata

Base case: Sing(X)

Example φWS1S0 : ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X) 1 start 2 ⊥ P : 0 P :

1
P :
P :
1
Σ

Figure: φ1(P) : Sing(P)

Susanne van den Elsen (UdS) WS1S January 25th, 2013 26 / 52

SLIDE 59

Expressiveness From WS1S to finite automata

Base case: Succ(X1, X2)

Proof continued. start

1
1
Figure: ASucc(X1,X2)

Susanne van den Elsen (UdS) WS1S January 25th, 2013 27 / 52

SLIDE 60

Expressiveness From WS1S to finite automata

Base case: X1 ⊆ X2

Proof continued. start

,

1 1

,

1

Figure: AX1⊆X2

Susanne van den Elsen (UdS) WS1S January 25th, 2013 28 / 52

SLIDE 61

Expressiveness From WS1S to finite automata

Base case: X1 ⊆ X2

Example φWS1S0 : ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X) 3 start ⊥ P : X :

,

1

,

1 1

P :

X : 1

Σ

Figure: φ2(P, X) : P ⊆ X

Susanne van den Elsen (UdS) WS1S January 25th, 2013 29 / 52

SLIDE 62

Expressiveness From WS1S to finite automata

Step case: ¬ψ

Proof continued.

L(¬ψ) = L(ψ) = L(Aψ) = L(A¬ψ).
Complement automaton Aψ by flipping accepting and non-accepting states

Susanne van den Elsen (UdS) WS1S January 25th, 2013 30 / 52

SLIDE 63

Expressiveness From WS1S to finite automata

Step case: ¬ψ

Example φWS1S0 : ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X) 3 start ⊥ P : X :

,

1

,

1 1

P :

X : 1

Σ

Figure: φ3(P, X) : ¬P ⊆ X

Susanne van den Elsen (UdS) WS1S January 25th, 2013 31 / 52

SLIDE 64

Expressiveness From WS1S to finite automata

Step case: ψ1 ∧ ψ2

Proof continued.

L(ψ1 ∧ ψ2) = L(ψ1) ∩ L(ψ2) = L(Aψ1) ∩ L(Aψ2) = L(Aψ1∧ψ2).
Product of automata Aψ1 and Aψ2.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 32 / 52

SLIDE 65

Expressiveness From WS1S to finite automata

Step case: ψ1 ∧ ψ2

Example φWS1S0 : ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X)

1, 3 start 2, 3 ⊥, 3 1, ⊥ 2, ⊥ ⊥, ⊥ P : X :

,
1
P :

X :

1

1

P :

X :

1
P :

X :

,
1
P :

X :

1

1

P :

X :

1
P :

X :

,
1
,
1

1

P :

X :

1
P :

X :

,
1
P :

X :

1
,
1

1

Σ

Figure: φ4(P, X) : Sing(P) ∧ ¬P ⊆ X

Susanne van den Elsen (UdS) WS1S January 25th, 2013 33 / 52

SLIDE 66

Expressiveness From WS1S to finite automata

Example φWS1S0 : ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X)

1, 3 start 2, ⊥ ⊥, ⊥ P : X :

,

1

P :

X : 1 1

P :

X : 1

P :

X :

,

1

P :

X : 1

,

1 1

Σ

Figure: φ4(P, X) : Sing(P) ∧ ¬P ⊆ X after minimization

Susanne van den Elsen (UdS) WS1S January 25th, 2013 34 / 52

SLIDE 67

Expressiveness From WS1S to finite automata

Step case: ∃Xi.ψ

Proof continued.

A∃Xi.ψ acts like Aψ except it guesses the values of the set Xi.
Projection of Aψ on Xi.
Let ψ(X1, . . . , Xn).
Aψ with Σ = {0, 1}n−1 × {0, 1}.
Word w ∈ Σ∗ can be identified by a pair u, v, with u ∈ ({0, 1}n−1)∗ and

v ∈ ({0, 1})∗, and |u| = |v| = |w|.

pri(L(ψ)) = {u ∈ ({0, 1}n−1)∗ | ∃v ∈ {0, 1}∗ such that u, v ∈ L(ψ)}
A∃Xi.ψ by removing track of Xi from the automaton Aψ.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 35 / 52

SLIDE 68

Expressiveness From WS1S to finite automata

Step case: ∃Xi.ψ

Proof continued.

A∃Xi.ψ acts like Aψ except it guesses the values of the set Xi.
Projection of Aψ on Xi.
Let ψ(X1, . . . , Xn).
Aψ with Σ = {0, 1}n−1 × {0, 1}.
Word w ∈ Σ∗ can be identified by a pair u, v, with u ∈ ({0, 1}n−1)∗ and

v ∈ ({0, 1})∗, and |u| = |v| = |w|.

pri(L(ψ)) = {u ∈ ({0, 1}n−1)∗ | ∃v ∈ {0, 1}∗ such that u, v ∈ L(ψ)}
A∃Xi.ψ by removing track of Xi from the automaton Aψ.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 35 / 52

SLIDE 69

Expressiveness From WS1S to finite automata

Step case: ∃Xi.ψ

Proof continued.

A∃Xi.ψ acts like Aψ except it guesses the values of the set Xi.
Projection of Aψ on Xi.
Let ψ(X1, . . . , Xn).
Aψ with Σ = {0, 1}n−1 × {0, 1}.
Word w ∈ Σ∗ can be identified by a pair u, v, with u ∈ ({0, 1}n−1)∗ and

v ∈ ({0, 1})∗, and |u| = |v| = |w|.

pri(L(ψ)) = {u ∈ ({0, 1}n−1)∗ | ∃v ∈ {0, 1}∗ such that u, v ∈ L(ψ)}
A∃Xi.ψ by removing track of Xi from the automaton Aψ.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 35 / 52

SLIDE 70

Expressiveness From WS1S to finite automata

Step case: ∃Xi.ψ

Example (A problem.) start X1 : X2 :

1

1 1

X1 :

X2 :

1

1 1 Σ

Figure: Aψ with ψ(X1, X2) : 0 ∈ X1 ∧ 1 ∈ X2

start X1 :

1
Σ

Σ

Figure: Aφ with φ(X1) : ∃X2.0 ∈ X1 ∧ 1 ∈ X2

10 ∈ L(Aφ) and 10 |

= φ.

1 ∈ L(Aφ), although 1 |

= φ.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 36 / 52

SLIDE 71

Expressiveness From WS1S to finite automata

Step case: ∃Xi.ψ

Example (A solution.) start X1 : X2 : 1 1 1

X1 :

X2 : 1 1 1 Σ

Figure: Aψ with ψ(X1, X2) : 0 ∈ X1 ∧ 1 ∈ X2

∃s ∈ S reachable from s0 on 1, such that an accepting state s′ ∈ F can be

reached from s on a string of 0-letters in the track of X1.

We characterise such s as accepting before projection.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 37 / 52

SLIDE 72

Expressiveness From WS1S to finite automata

Step case: ∃Xi.ψ

Proof continued.

Li = {u ∈ Σk | j-th track is all 0 for j = i}
L/Li is right quotient of L by Li:

L/Li = {w | ∃x ∈ Li such that wx ∈ L}.

L(∃Xi.ψ) = pri(L(ψ)/Li).

Susanne van den Elsen (UdS) WS1S January 25th, 2013 38 / 52

SLIDE 73

Expressiveness From WS1S to finite automata

Example φWS1S0 : ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X)

4 start 5 6 P : X :

,

1

P :

X : 1 1

P :

X : 1

P :

X :

,

1

P :

X : 1

,

1 1

Σ

Figure: φ4(P, X) : Sing(P) ∧ ¬P ⊆ X prepared for projection

Susanne van den Elsen (UdS) WS1S January 25th, 2013 39 / 52

SLIDE 74

Expressiveness From WS1S to finite automata

Example φWS1S0 : ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X) 4 start 6 5 X : 0 , 1 X : 1 X : 0 X : 0 , 1 X : 0 , 1 Σ

Figure: φ5(X) : ∃P.(Sing(P) ∧ ¬P ⊆ X)

Susanne van den Elsen (UdS) WS1S January 25th, 2013 40 / 52

SLIDE 75

Expressiveness From WS1S to finite automata

Example φWS1S0 : ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X) 4 start 4, 5 4, 6 4, 5, 6 X :

X : 1

Σ X : 0 X :

1
Σ

Figure: φ5(X) : ∃P.(Sing(P) ∧ ¬P ⊆ X) after determinization

Susanne van den Elsen (UdS) WS1S January 25th, 2013 41 / 52

SLIDE 76

Expressiveness From WS1S to finite automata

Example φWS1S0 : ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X) 4 start 4, 5 4, 6 4, 5, 6 X : 0 X : 1 Σ X : 0 X :

1
Σ

Figure: φ6(X) : ¬∃P.(Sing(P) ∧ ¬P ⊆ X)

Susanne van den Elsen (UdS) WS1S January 25th, 2013 42 / 52

SLIDE 77

Expressiveness From WS1S to finite automata

Example φWS1S0 : ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X) 4 start 4, 5 4, 6 4, 5, 6 () () () () () ()

Figure: φ : ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X)

Susanne van den Elsen (UdS) WS1S January 25th, 2013 43 / 52

SLIDE 78

Decidability

Section 4 Decidability

Susanne van den Elsen (UdS) WS1S January 25th, 2013 44 / 52

SLIDE 79

Decidability Decision procedure

Decision procedure

Satisfiability: Given WS1S sentence φ determine whether φ is satisfiable or not.

1. Convert φ to MSO0 formula φ′.
2. Construct automaton Aφ′ such that L(Aφ′) = L(φ′) = L(φ).
3. Check whether L(Aφ′) = ∅.
4. If not, φ is satisfiable, otherwise, φ is unsatisfiable.

Validity: Given WS1S formula φ, decide whether φ is valid or not.

1. Check whether ¬φ is unsatisfiable (L(¬φ) = ∅, and hence L(φ) = ()∗).
2. If so, φ is valid, otherwise, φ is invalid.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 45 / 52

SLIDE 80

Decidability Decision procedure

Decision procedure

Example 4 start 4, 5 4, 6 4, 5, 6 () () () () () ()

Figure: φ : ∃X.¬∃P.(Sing(P) ∧ ¬P ⊆ X)

L(Aφ) = ∅: no w such that w | = φ. Hence, φ is unsatisfiable

Susanne van den Elsen (UdS) WS1S January 25th, 2013 46 / 52

SLIDE 81

Decidability Decision procedure

Checking language emptiness

Given automaton A, check whether L(A) = ∅.

Regard A as a directed graph
If there is a word w ∈ L(A) then there is an accepting run of w on A
Then there is a path from the initial state to an accepting state
Do breadth-first search on the graph from the initial state
If an accepting state is visited, return false (L(A) = ∅)
Otherwise, return true (L(A) = ∅).

Susanne van den Elsen (UdS) WS1S January 25th, 2013 47 / 52

SLIDE 82

Complexity

Section 5 Complexity

Susanne van den Elsen (UdS) WS1S January 25th, 2013 48 / 52

SLIDE 83

Complexity

Start with deterministic base case automata.
∧: Intersection of Aφ1 and Aφ2: O(|Aφ1| · |Aφ2|)
¬: Complementation of Aφ′: O(|Aφ′|) for deterministic Aφ′
∃: Projection of Aφ′: O(|Aφ′|) but introduces nondeterminism!
Determinization of nondeterministic Aφ′: O(2|Aφ′|): exponential blow-up!
. . . ∀∃∀ . . . : quantifier alternation results in exponential blow-ups.

∀X.∃Y .φ ≡ ¬∃X.¬∃Y .φ

If |Aφ| = n then

|A¬∃Y .φ| = O(2|n|) |A¬∃X.¬∃Y .φ| = O(22|n|)

Checking language emptiness Aφ: O(|Aφ|)

Susanne van den Elsen (UdS) WS1S January 25th, 2013 49 / 52

SLIDE 84

Complexity

Theorem by Meyer [Meyer, 2002]. Theorem The complexity of deciding WS1S-formula φ with length n is

22. . .2n

O(n) That is, it is non-elementary.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 50 / 52

SLIDE 85

References

Section 6 References

Susanne van den Elsen (UdS) WS1S January 25th, 2013 51 / 52

SLIDE 86

References

References I

B¨ uchi, J. R. (1960). Weak second-order arithmetic and finite automata. Zeitschrift f¨ ur mathematischen Logik und Grundlagen der Mathematik, 6:66–92. Elgot, C. C. (1961). Decision problems of finite automata design and related arithmetics. Transactions of the American Mathematical Society, 98:21–53. Meyer, M. (2002). Automata, Logics, and Infinite Games, volume 2500 of LNCS, chapter Decidability of S1S and S2S, pages 207–230. Springer, Berlin. Thomas, W. (1997). Handbook of Formal Languages Vol. 3: Beyond Words, chapter Languages, Automata, and Logic, pages 389–456. Springer, Berlin.

Susanne van den Elsen (UdS) WS1S January 25th, 2013 52 / 52