Parameter-Synthesis Problems for One-Counter Automata Guillermo A. - - PowerPoint PPT Presentation

Parameter-Synthesis Problems for One-Counter Automata

Guillermo A. P´ erez

(slides by Ritam Raha) INFINITY 2020

Outline

• 1. One-Counter Automata (Parametric) and Synthesis Problem
• 2. Previous Approach
• 3. Approach with Alternating Two-Way Automata for a subclass
• 4. (Failed) Approach with Partial Observation Games

Guillermo A. P´ erez 2/27

One-Counter Automata

1 n = 5 2 n = max(0, n - 5) 3 if n = 0: 4

while n < 10:

5

n = n + 1

6

n = n + 100

7

# make progress

8 else: 9

assert(False)

Guillermo A. P´ erez 3/27

One-Counter Automata

1 n = 5 2 n = max(0, n - 5) 3 if n = 0: 4

while n < 10:

5

n = n + 1

6

n = n + 100

7

# make progress

8 else: 9

assert(False) false

5 −5 = 0 ≥ 10 +100 +1 ≥ 1

◮ Configurations: (q, c), c ≥ 0;

Guillermo A. P´ erez 3/27

One-Counter Automata

1 def foobar(x): 2

n = 5

3

n = max(0, n - x)

4

if n = 0:

5

while n < 10:

6

n = n + 1

7

n = n + 100

8

# make progress

9

else:

10

assert(False)

Guillermo A. P´ erez 4/27

One-Counter Automata

1 def foobar(x): 2

n = 5

3

n = max(0, n - x)

4

if n = 0:

5

while n < 10:

6

n = n + 1

7

n = n + 100

8

# make progress

9

else:

10

assert(False) false

5 −x = 0 ≥ 10 +100 +1 ≥ 1

Guillermo A. P´ erez 4/27

Parametric One-Counter Automata

false X = {x1, . . . , xn}

5 −x1 = 0 ≥ x2 +x3 +1 ≥ 1

Guillermo A. P´ erez 5/27

Parametric One-Counter Automata

false X = {x1, . . . , xn}

5 −x1 = 0 ≥ x2 +x3 +1 ≥ 1

Definition (Succinct OCA with Parameters)

A = (Q, qin, T, δ, X) δ : T → Op with Op the union of

Guillermo A. P´ erez 5/27

Parametric One-Counter Automata

false X = {x1, . . . , xn}

5 −x1 = 0 ≥ x2 +x3 +1 ≥ 1

Definition (Succinct OCA with Parameters)

A = (Q, qin, T, δ, X) δ : T → Op with Op the union of

• CU := {+a : a ∈ Z}
• CT := {= 0, ≥ a, = a : a ∈ Z}

Guillermo A. P´ erez 5/27

Parametric One-Counter Automata

false X = {x1, . . . , xn}

5 −x1 = 0 ≥ x2 +x3 +1 ≥ 1

Definition (Succinct OCA with Parameters)

A = (Q, qin, T, δ, X) δ : T → Op with Op the union of

• CU := {+a : a ∈ Z}
• PU := {+x, −x : x ∈ X}
• CT := {= 0, ≥ a, = a : a ∈ Z}
• PT := {= x, ≥ x : x ∈ X}

Guillermo A. P´ erez 5/27

Parametric One-Counter Automata

false X = {x1, . . . , xn}

5 −x1 = 0 ≥ x2 +x3 +1 ≥ 1

Definition (Succinct OCA with Parameters)

A = (Q, qin, T, δ, X) δ : T → Op with Op the union of

• CU := {+a : a ∈ Z}
• PU := {+x, −x : x ∈ X}
• CT := {= 0, ≥ a, = a : a ∈ Z}
• PT := {= x, ≥ x : x ∈ X}

Non-parametric: X = ∅

Guillermo A. P´ erez 5/27

One-Counter Automata

Models CU PU ZT PT Non-Parametric SOCA ✓ ✘ ✓ ✘ Parametric OCAPT {−1, 0, 1} ✘ ✓ ✓ SOCAP ✓ ✓ ✓ ✓

Guillermo A. P´ erez 6/27

Decision Problems

Non-Parametric: Parametric:

Guillermo A. P´ erez 7/27

Decision Problems

Non-Parametric: ∃ρ such that (qin, 0)

ρ

− → qf Reach Parametric:

Guillermo A. P´ erez 7/27

Decision Problems

Non-Parametric: ∃ρ such that (qin, 0)

ρ

− → qf Reach [NP-complete (HKOW’09)] Parametric:

Guillermo A. P´ erez 7/27

Decision Problems

Non-Parametric: ∃ρ such that (qin, 0)

ρ

− → qf Reach [NP-complete (HKOW’09)] Parametric: ∃V : X → N s.t. ∃ρ, (qin, 0)

ρ

− →V qf Par-Reach

Guillermo A. P´ erez 7/27

Decision Problems

Non-Parametric: ∃ρ such that (qin, 0)

ρ

− → qf Reach [NP-complete (HKOW’09)] Parametric: ∃V : X → N s.t. ∃ρ, (qin, 0)

ρ

− →V qf Par-Reach [in NEXP (HKOW’09,LOW’15)]

Guillermo A. P´ erez 7/27

Decision Problems

Non-Parametric: ∃ρ such that (qin, 0)

ρ

− → qf Reach [NP-complete (HKOW’09)] For all infinite ρ, (qin, 0)

ρ

− → qf UnivReach Parametric: ∃V : X → N s.t. ∃ρ, (qin, 0)

ρ

− →V qf Par-Reach [in NEXP (HKOW’09,LOW’15)]

Guillermo A. P´ erez 7/27

Decision Problems

Non-Parametric: ∃ρ such that (qin, 0)

ρ

− → qf Reach [NP-complete (HKOW’09)] For all infinite ρ, (qin, 0)

ρ

− → qf UnivReach Parametric: ∃V : X → N s.t. ∃ρ, (qin, 0)

ρ

− →V qf Par-Reach [in NEXP (HKOW’09,LOW’15)] ∃V s.t. for all infinite ρ, (qin, 0)

ρ

− →V qf SynthReach

Guillermo A. P´ erez 7/27

Decision Problems

Non-Parametric: ∃ρ such that (qin, 0)

ρ

− → qf Reach [NP-complete (HKOW’09)] For all infinite ρ, (qin, 0)

ρ

− → qf UnivReach Parametric: ∃V : X → N s.t. ∃ρ, (qin, 0)

ρ

− →V qf Par-Reach [in NEXP (HKOW’09,LOW’15)] ∃V s.t. for all infinite ρ, (qin, 0)

ρ

− →V qf SynthReach

Guillermo A. P´ erez 7/27

UnivReach for SOCA

Model and problem

◮ Op = CU ∪ ZT ◮ UnivReach: all infinite paths reach qf

Guillermo A. P´ erez 8/27

UnivReach for SOCA

Model and problem

◮ Op = CU ∪ ZT ◮ UnivReach: all infinite paths reach qf ∃ infinite path avoiding qf = two Reach queries

Guillermo A. P´ erez 8/27

UnivReach for SOCA

Model and problem

◮ Op = CU ∪ ZT ◮ UnivReach: all infinite paths reach qf ∃ infinite path avoiding qf = two Reach queries (coNP)

Guillermo A. P´ erez 8/27

UnivReach for SOCA

Model and problem

◮ Op = CU ∪ ZT ◮ UnivReach: all infinite paths reach qf ∃ infinite path avoiding qf = two Reach queries (coNP) Hardness from reduction from co-subsetsum.

Proposition

The UnivReach problem for SOCA is coNP-complete.

Guillermo A. P´ erez 8/27

Decision Problems

Non-Parametric: ∃ρ such that (qin, 0)

ρ

− → qf Reach [NP-complete (HKOW’09)] For all infinite ρ, (qin, 0)

ρ

− → qf UnivReach Parametric: ∃V : XN s.t. ∃ρ, (qin, 0)

ρ

− →V qf Par-Reach [in NEXP (HKOW’09,LOW’15)] ∃V s.t. for all infinite ρ, (qin, 0)

ρ

− →V qf SynthReach

Guillermo A. P´ erez 9/27

Decision Problems

Non-Parametric: ∃ρ such that (qin, 0)

ρ

− → qf Reach [NP-complete (HKOW’09)] For all infinite ρ, (qin, 0)

ρ

− → qf UnivReach [coNP-complete] Parametric: ∃V : XN s.t. ∃ρ, (qin, 0)

ρ

− →V qf Par-Reach [in NEXP (HKOW’09,LOW’15)] ? ∃V s.t. for all infinite ρ, (qin, 0)

ρ

− →V qf SynthReach

Guillermo A. P´ erez 9/27

Outline

• 1. One-Counter Automata (Parametric) and Synthesis Problem
• 2. Previous Approach
• 3. Approach with Alternating Two-Way Automata for a subclass
• 4. Approach with Partial Observation Games

Guillermo A. P´ erez 10/27

PAD & Extensions

◮ Presburger arithmetic (PA) := N, 0, 1, +, <

Guillermo A. P´ erez 11/27

PAD & Extensions

◮ Presburger arithmetic (PA) := N, 0, 1, +, < ◮ Presburger arithmetic with Divisibility (PAD) := PA + | (a | b ⇐ ⇒ ∃c ∈ Z : b = ac)

Guillermo A. P´ erez 11/27

PAD & Extensions

◮ Presburger arithmetic (PA) := N, 0, 1, +, < ◮ Presburger arithmetic with Divisibility (PAD) := PA + | (a | b ⇐ ⇒ ∃c ∈ Z : b = ac)

Theorem (Robinson’49, Lipshitz’81)

Full PAD is undecidable; one alternation suffices for undecidability.

Guillermo A. P´ erez 11/27

PAD & Extensions

◮ Presburger arithmetic (PA) := N, 0, 1, +, < ◮ Presburger arithmetic with Divisibility (PAD) := PA + | (a | b ⇐ ⇒ ∃c ∈ Z : b = ac)

Theorem (Robinson’49, Lipshitz’81)

Full PAD is undecidable; one alternation suffices for undecidability.

Theorem (Lipshitz’78, Lechner-Ouaknine-Worrell’15)

The existential fragment of PAD (EPAD) is decidable in NEXP.

Guillermo A. P´ erez 11/27

∀∃RPAD & Undecidability

◮ ∀∃RPAD := ∀z1 . . . ∀zn∃x1 . . . ∃xm. ϕ(x, z)

◮ divisibilities of the form f (z) | g(x, z)

Guillermo A. P´ erez 12/27

∀∃RPAD & Undecidability

◮ ∀∃RPAD := ∀z1 . . . ∀zn∃x1 . . . ∃xm. ϕ(x, z)

◮ divisibilities of the form f (z) | g(x, z)

◮ ∀∃RPAD+ := ∀∃RPAD with ¬ not allowed before divisibility.

Guillermo A. P´ erez 12/27

∀∃RPAD & Undecidability

◮ ∀∃RPAD := ∀z1 . . . ∀zn∃x1 . . . ∃xm. ϕ(x, z)

◮ divisibilities of the form f (z) | g(x, z)

◮ ∀∃RPAD+ := ∀∃RPAD with ¬ not allowed before divisibility.

Claim (Bozga-Iosif’05, Lechner’15)

SynthReach for SOCAP is decidable by a reduction to ∀∃RPAD+.

Guillermo A. P´ erez 12/27

∀∃RPAD & Undecidability

◮ ∀∃RPAD := ∀z1 . . . ∀zn∃x1 . . . ∃xm. ϕ(x, z)

◮ divisibilities of the form f (z) | g(x, z)

◮ ∀∃RPAD+ := ∀∃RPAD with ¬ not allowed before divisibility.

Claim (Bozga-Iosif’05, Lechner’15)

SynthReach for SOCAP is decidable by a reduction to ∀∃RPAD+. ∀∃RPAD ≡ ∀∃RPAD+

Guillermo A. P´ erez 12/27

∀∃RPAD & Undecidability

◮ ∀∃RPAD := ∀z1 . . . ∀zn∃x1 . . . ∃xm. ϕ(x, z)

◮ divisibilities of the form f (z) | g(x, z)

◮ ∀∃RPAD+ := ∀∃RPAD with ¬ not allowed before divisibility.

Claim (Bozga-Iosif’05, Lechner’15)

SynthReach for SOCAP is decidable by a reduction to ∀∃RPAD+. ∀∃RPAD ≡ ∀∃RPAD+ Idea: We have to rewrite ¬(a | b)

Guillermo A. P´ erez 12/27

∀∃RPAD & Undecidability

◮ ∀∃RPAD := ∀z1 . . . ∀zn∃x1 . . . ∃xm. ϕ(x, z)

◮ divisibilities of the form f (z) | g(x, z)

◮ ∀∃RPAD+ := ∀∃RPAD with ¬ not allowed before divisibility.

Claim (Bozga-Iosif’05, Lechner’15)

SynthReach for SOCAP is decidable by a reduction to ∀∃RPAD+. ∀∃RPAD ≡ ∀∃RPAD+ Idea: We have to rewrite ¬(a | b) ¬(a | b) ≡ b = aq + r where, 0 < r < b.

Guillermo A. P´ erez 12/27

∀∃RPAD & Undecidability

Theorem (Bozga-Iosif’05)

∀∃RPAD is undecidable. Idea: Using the single restricted alternation we define

• 1. LCM

Guillermo A. P´ erez 13/27

∀∃RPAD & Undecidability

Theorem (Bozga-Iosif’05)

∀∃RPAD is undecidable. Idea: Using the single restricted alternation we define

• 1. LCM
• 2. Square (x2)

Guillermo A. P´ erez 13/27

∀∃RPAD & Undecidability

Theorem (Bozga-Iosif’05)

∀∃RPAD is undecidable. Idea: Using the single restricted alternation we define

• 1. LCM
• 2. Square (x2)
• 3. Multiplication

Undecidable!!

Guillermo A. P´ erez 13/27

Decision Problems

Non-Parametric: ∃ρ such that (qin, 0)

ρ

− → qf Reach [NP-complete (HKOW’09)] For all infinite ρ, (qin, 0)

ρ

− → qf UnivReach [coNP-complete] Parametric: ∃V : X → N s.t. ∃ρ, (qin, 0)

ρ

− →V qf Par-Reach [in NEXP (HKOW’09,LOW’15)] ∃V s.t. for all infinite ρ, (qin, 0)

ρ

− →V qf SynthReach

Guillermo A. P´ erez 14/27

Outline

• 1. One-Counter Automata (Parametric) and Synthesis Problem
• 2. Previous Approach
• 3. Approach with Alternating Two-Way Automata for a subclass
• 4. Approach with Partial Observation Games

Guillermo A. P´ erez 15/27

Alternating Two-Way Automaton

q1 q2 q3 q4 q5 w = abω a a b b b b a

Guillermo A. P´ erez 16/27

Alternating Two-Way Automaton

q1 q2 q3 q4 q5 w = abω ∨ a b b b b a

Guillermo A. P´ erez 16/27

Alternating Two-Way Automaton

q1 q2 q3 q4 q5 w = abω ∧ a b b b b a

Guillermo A. P´ erez 16/27

Alternating Two-Way Automaton

q1 q2 q3 q4 q5 w = abω ∧ a

+ 1 + 1

b, +1 a, +1 b, +1 b, +1 b, +1

Guillermo A. P´ erez 16/27

Alternating Two-Way Automaton

q1 q2 q3 q4 q5 w = abω ∧ a

+ 1 + 1

b, +1 a, +1 b, +1 b, +1 b, +1 b, +1 a, +1

Guillermo A. P´ erez 16/27

Alternating Two-Way Automaton

q1 q2 q3 q4 q5 w = abω ∧ a

+ 1 + 1

b, +1 a, +1 b, +1 b, +1 b, −1

Guillermo A. P´ erez 16/27

Alternating Two-Way Automaton

q1 q2 q3 q4 q5 w = abω ∧ a

+ 1 + 1

b, +1 a, +1 b, +1 b, +1 b, −1

Theorem (Serre’06)

The non-emptiness problem for A2As is in PSPACE.

Guillermo A. P´ erez 16/27

From OCAPT to A2A

Op = {−1, 0, +1} ∪ ZT ∪ PT

Guillermo A. P´ erez 17/27

From OCAPT to A2A

Op = {−1, 0, +1} ∪ ZT ∪ PT

Proposition (Based on Bollig-Quaas-Sangnier’19)

For every OCAPT A we construct an A2A T of poly-size which accepts words corresponding to valuations that witness SynthReach is true.

Guillermo A. P´ erez 17/27

From OCAPT to A2A

Op = {−1, 0, +1} ∪ ZT ∪ PT

Proposition (Based on Bollig-Quaas-Sangnier’19)

For every OCAPT A we construct an A2A T of poly-size which accepts words corresponding to valuations that witness SynthReach is true. Idea:

Guillermo A. P´ erez 17/27

From OCAPT to A2A

Op = {−1, 0, +1} ∪ ZT ∪ PT

Proposition (Based on Bollig-Quaas-Sangnier’19)

For every OCAPT A we construct an A2A T of poly-size which accepts words corresponding to valuations that witness SynthReach is true. Idea: ◮ Encode valuations as parameter words

Guillermo A. P´ erez 17/27

From OCAPT to A2A

Op = {−1, 0, +1} ∪ ZT ∪ PT

Proposition (Based on Bollig-Quaas-Sangnier’19)

For every OCAPT A we construct an A2A T of poly-size which accepts words corresponding to valuations that witness SynthReach is true. Idea: ◮ Encode valuations as parameter words ◮ For every transition we build an A2A

Guillermo A. P´ erez 17/27

From OCAPT to A2A

Op = {−1, 0, +1} ∪ ZT ∪ PT

Proposition (Based on Bollig-Quaas-Sangnier’19)

For every OCAPT A we construct an A2A T of poly-size which accepts words corresponding to valuations that witness SynthReach is true. Idea: ◮ Encode valuations as parameter words ◮ For every transition we build an A2A ◮ Accept the reaching runs ◮ (. . . and runs that “die off”)

Guillermo A. P´ erez 17/27

OCAPT to A2A

Valuation to words:

Guillermo A. P´ erez 18/27

OCAPT to A2A

Valuation to words: V : X → N; Σ = X ∪ {};

Guillermo A. P´ erez 18/27

OCAPT to A2A

Valuation to words: V : X → N; Σ = X ∪ {};

• Guillermo A. P´

erez 18/27

OCAPT to A2A

Valuation to words: V : X → N; Σ = X ∪ {}; x1x2ω

Guillermo A. P´ erez 18/27

OCAPT to A2A

Valuation to words: V : X → N; Σ = X ∪ {}; x1x2ω ⇒ x1 → 2, x2 → 3;

Guillermo A. P´ erez 18/27

OCAPT to A2A

Valuation to words: V : X → N; Σ = X ∪ {}; x1x2ω ⇒ x1 → 2, x2 → 3; Op to A2A:

Guillermo A. P´ erez 18/27

OCAPT to A2A

Valuation to words: V : X → N; Σ = X ∪ {}; x1x2ω ⇒ x1 → 2, x2 → 3; Op to A2A:

q ∨ ∧ q′ violation validation s i m u l a t i

• n

Guillermo A. P´ erez 18/27

OCAPT to A2A

Valuation to words: V : X → N; Σ = X ∪ {}; x1x2ω ⇒ x1 → 2, x2 → 3; Op to A2A:

q ∨ ∧ q′ violation validation s i m u l a t i

• n

Accept reaching runs:

qf true Guillermo A. P´ erez 18/27

OCAPT to A2A

• Encoding Equality Test:

Guillermo A. P´ erez 19/27

OCAPT to A2A

• Encoding Equality Test:

q ∨ ∧ q′ true

• +1

+1 x

• X\{x}, +1

X\{x}, +1

Guillermo A. P´ erez 19/27

OCAPT to A2A

• Encoding Decrement:

Guillermo A. P´ erez 20/27

OCAPT to A2A

• Encoding Decrement:

∨ q q′ true

• −1

, 0 x, −1 first?

Guillermo A. P´ erez 20/27

OCAPT to A2A

We want to check all runs:

Guillermo A. P´ erez 21/27

OCAPT to A2A

We want to check all runs: q

• p1

− − → q1 ⇒ (q, , T1)

Guillermo A. P´ erez 21/27

OCAPT to A2A

We want to check all runs: q

• p1

− − → q1 ⇒ (q, , T1) q

• p2

− − → q2 ⇒ (q, , T2)

Guillermo A. P´ erez 21/27

OCAPT to A2A

We want to check all runs: q

• p1

− − → q1 ⇒ (q, , T1) q

• p2

− − → q2 ⇒ (q, , T2) (q, , T1 ∧ T2) ∈ T

Guillermo A. P´ erez 21/27

OCAPT to A2A

Proposition

SynthReach for OCAPT is in PSPACE.

Guillermo A. P´ erez 22/27

OCAPT to A2A

Proposition

SynthReach for OCAPT is in PSPACE. ◮ For every A2A T there is an NBA B of exponential size accepting same language ◮ Non-emptiness witnesses for NBAs are simple “lassos” ◮ = ⇒ SynthReach admits exponential (w.r.t. the OCAPT) witnesses and thus polynomial in binary encoding

Guillermo A. P´ erez 22/27

OCAPT to A2A

Proposition

SynthReach for OCAPT is in PSPACE. ◮ For every A2A T there is an NBA B of exponential size accepting same language ◮ Non-emptiness witnesses for NBAs are simple “lassos” ◮ = ⇒ SynthReach admits exponential (w.r.t. the OCAPT) witnesses and thus polynomial in binary encoding ◮ Guess a valuation and check UnivReach for resulting SOCA

Theorem

SynthReach for OCAPT is in NPcoNP.

Guillermo A. P´ erez 22/27

Outline

• 1. One-Counter Automata (Parametric) and Synthesis Problem
• 2. Previous Approach
• 3. Approach with Alternating Two-Way Automata for a subclass
• 4. Approach with Partial Observation Games

Guillermo A. P´ erez 23/27

Partial Observation Energy Games

a, 4 a, 1

b, −5 a, −2 a, −3 b, 0 Σ, 0 Σ, 0 Σ, 0

Guillermo A. P´ erez 24/27

Partial Observation Energy Games

a, 4 a, 1

b, −5 a, −2 a, −3 b, 0 Σ, 0 Σ, 0 Σ, 0

Guillermo A. P´ erez 24/27

Partial Observation Energy Games

a, 4 a, 1

b, −5 a, −2 a, −3 b, 0 Σ, 0 Σ, 0 Σ, 0

◮ Chooses an action ◮ Resolves non-determinism

Guillermo A. P´ erez 24/27

Partial Observation Energy Games

a, 4 a, 1

b, −5 a, −2 a, −3 b, 0 Σ, 0 Σ, 0 Σ, 0

◮ Chooses an action ◮ Keeps the energy level positive ◮ Resolves non-determinism ◮ Wants it eventually negative

Guillermo A. P´ erez 24/27

Partial Observation Energy Games

a, 4 a, 1

b, −5 a, −2 a, −3 b, 0 Σ, 0 Σ, 0 Σ, 0

◮ Chooses an action ◮ Keeps the energy level positive ◮ Resolves non-determinism ◮ Wants it eventually negative

Guillermo A. P´ erez 24/27

SOCAP to POEG

Claim

For every SOCAP A we construct a POEG which Eve wins iff for all valuations V there exists a reaching run of A.

Guillermo A. P´ erez 25/27

SOCAP to POEG

Claim

For every SOCAP A we construct a POEG which Eve wins iff for all valuations V there exists a reaching run of A. Idea: ◮ Adam chooses valuations, Eve simulates a reaching run

Guillermo A. P´ erez 25/27

SOCAP to POEG

Claim

For every SOCAP A we construct a POEG which Eve wins iff for all valuations V there exists a reaching run of A. Idea: ◮ Adam chooses valuations, Eve simulates a reaching run ◮ We create gadgets in the game simulating Op

Guillermo A. P´ erez 25/27

SOCAP to POEG

Claim

For every SOCAP A we construct a POEG which Eve wins iff for all valuations V there exists a reaching run of A. Idea: ◮ Adam chooses valuations, Eve simulates a reaching run ◮ We create gadgets in the game simulating Op ◮ The PO helps Adam force Eve to simulate runs faithfully

Guillermo A. P´ erez 25/27

SOCAP to POEG

Claim

For every SOCAP A we construct a POEG which Eve wins iff for all valuations V there exists a reaching run of A. Idea: ◮ Adam chooses valuations, Eve simulates a reaching run ◮ We create gadgets in the game simulating Op ◮ The PO helps Adam force Eve to simulate runs faithfully

Conjecture

SynthReach problem for SOCAP is decidable.

Guillermo A. P´ erez 25/27

Decision Problems

Non-Parametric: ∃ρ such that (qin, 0)

ρ

− → qf Reach [NP-complete] For all infinite ρ, (qin, 0)

ρ

− → qf UnivReach [coNP-complete] Parametric: ∃V : X → N s.t. ∃ρ, (qin, 0)

ρ

− →V qf Par-Reach [in NEXP] ∃V s.t. for all infinite ρ, (qin, 0)

ρ

− →V qf SynthReach [Decidable?]

Guillermo A. P´ erez 26/27

Conclusion

◮ If parameters are only allowed on tests, the problem is in NPNP ◮ In full generality, for SOCAP it is still open

Guillermo A. P´ erez 27/27