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: while n < 10: 4 n = n + 1 5 n = n + 100 6 # make progress 7 8 else: assert(False) 9 Guillermo A. P´ erez 3/27
One-Counter Automata 1 n = 5 2 n = max(0, n - 5) 3 if n = 0: − 5 ≥ 10 +100 5 = 0 while n < 10: 4 ≥ 1 n = n + 1 5 +1 n = n + 100 6 # make progress false 7 8 else: assert(False) 9 ◮ Configurations: ( q , c ) , c ≥ 0; Guillermo A. P´ erez 3/27
One-Counter Automata 1 def foobar(x): n = 5 2 n = max(0, n - x) 3 if n = 0: 4 while n < 10: 5 n = n + 1 6 n = n + 100 7 # make progress 8 else: 9 assert(False) 10 Guillermo A. P´ erez 4/27
One-Counter Automata 1 def foobar(x): n = 5 2 n = max(0, n - x) 3 − x ≥ 10 5 = 0 +100 if n = 0: 4 while n < 10: 5 ≥ 1 n = n + 1 +1 6 n = n + 100 7 false # make progress 8 else: 9 assert(False) 10 Guillermo A. P´ erez 4/27
Parametric One-Counter Automata X = { x 1 , . . . , x n } − x 1 ≥ x 2 + x 3 5 = 0 ≥ 1 +1 false Guillermo A. P´ erez 5/27
Parametric One-Counter Automata X = { x 1 , . . . , x n } − x 1 ≥ x 2 + x 3 5 = 0 ≥ 1 +1 false Definition (Succinct OCA with Parameters) A = ( Q , q in , T , δ, X ) δ : T → Op with Op the union of Guillermo A. P´ erez 5/27
Parametric One-Counter Automata X = { x 1 , . . . , x n } − x 1 ≥ x 2 + x 3 5 = 0 ≥ 1 +1 false Definition (Succinct OCA with Parameters) A = ( Q , q in , 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 X = { x 1 , . . . , x n } − x 1 ≥ x 2 + x 3 5 = 0 ≥ 1 +1 false Definition (Succinct OCA with Parameters) A = ( Q , q in , 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 X = { x 1 , . . . , x n } − x 1 ≥ x 2 + x 3 5 = 0 ≥ 1 +1 false Definition (Succinct OCA with Parameters) A = ( Q , q in , 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 ✓ ✘ ✓ ✘ OCAPT {− 1 , 0 , 1 } ✘ ✓ ✓ Parametric SOCAP ✓ ✓ ✓ ✓ Guillermo A. P´ erez 6/27
Decision Problems Non-Parametric: Parametric: Guillermo A. P´ erez 7/27
Decision Problems Non-Parametric: Parametric: Reach ∃ ρ such that ρ ( q in , 0) − → q f Guillermo A. P´ erez 7/27
Decision Problems Non-Parametric: Parametric: Reach ∃ ρ such that ρ ( q in , 0) − → q f [NP-complete (HKOW’09)] Guillermo A. P´ erez 7/27
Decision Problems Non-Parametric: Parametric: Reach Par-Reach ∃ ρ such that ∃ V : X → N s.t. ∃ ρ , ρ ρ ( q in , 0) − → q f ( q in , 0) − → V q f [NP-complete (HKOW’09)] Guillermo A. P´ erez 7/27
Decision Problems Non-Parametric: Parametric: Reach Par-Reach ∃ ρ such that ∃ V : X → N s.t. ∃ ρ , ρ ρ ( q in , 0) → q f − ( q in , 0) − → V q f [NP-complete (HKOW’09)] [in NEXP (HKOW’09,LOW’15)] Guillermo A. P´ erez 7/27
Decision Problems Non-Parametric: Parametric: Reach Par-Reach ∃ ρ such that ∃ V : X → N s.t. ∃ ρ , ρ ρ ( q in , 0) − → q f ( q in , 0) − → V q f [NP-complete (HKOW’09)] [in NEXP (HKOW’09,LOW’15)] UnivReach For all infinite ρ , ρ ( q in , 0) → q f − Guillermo A. P´ erez 7/27
Decision Problems Non-Parametric: Parametric: Reach Par-Reach ∃ ρ such that ∃ V : X → N s.t. ∃ ρ , ρ ρ ( q in , 0) → q f − ( q in , 0) → V q f − [NP-complete (HKOW’09)] [in NEXP (HKOW’09,LOW’15)] SynthReach UnivReach ∃ V s.t. for all infinite For all infinite ρ , ρ ρ ρ , ( q in , 0) − → V q f ( q in , 0) → q f − Guillermo A. P´ erez 7/27
Decision Problems Non-Parametric: Parametric: Reach Par-Reach ∃ ρ such that ∃ V : X → N s.t. ∃ ρ , ρ ρ ( q in , 0) → q f − ( q in , 0) → V q f − [NP-complete (HKOW’09)] [in NEXP (HKOW’09,LOW’15)] SynthReach UnivReach ∃ V s.t. for all infinite For all infinite ρ , ρ ρ ρ , ( q in , 0) − → V q f ( q in , 0) → q f − Guillermo A. P´ erez 7/27
UnivReach for SOCA Model and problem ◮ Op = CU ∪ ZT ◮ UnivReach : all infinite paths reach q f Guillermo A. P´ erez 8/27
UnivReach for SOCA Model and problem ◮ Op = CU ∪ ZT ◮ UnivReach : all infinite paths reach q f ∃ infinite path avoiding q f = two Reach queries Guillermo A. P´ erez 8/27
UnivReach for SOCA Model and problem ◮ Op = CU ∪ ZT ◮ UnivReach : all infinite paths reach q f ∃ infinite path avoiding q f = two Reach queries (coNP) Guillermo A. P´ erez 8/27
UnivReach for SOCA Model and problem ◮ Op = CU ∪ ZT ◮ UnivReach : all infinite paths reach q f ∃ infinite path avoiding q f = 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: Parametric: Reach Par-Reach ∃ ρ such that ∃ V : X N s.t. ∃ ρ , ρ ρ ( q in , 0) → q f − → V q f − ( q in , 0) [NP-complete (HKOW’09)] [in NEXP (HKOW’09,LOW’15)] UnivReach SynthReach For all infinite ρ , ∃ V s.t. for all infinite ρ ρ ( q in , 0) → q f − − → V q f ρ , ( q in , 0) Guillermo A. P´ erez 9/27
Decision Problems Non-Parametric: Parametric: Reach Par-Reach ∃ ρ such that ∃ V : X N s.t. ∃ ρ , ρ ρ ( q in , 0) → q f − → V q f − ( q in , 0) [NP-complete (HKOW’09)] [in NEXP (HKOW’09,LOW’15)] ? UnivReach SynthReach For all infinite ρ , ∃ V s.t. for all infinite ρ ρ ( q in , 0) − → q f − → V q f ρ , ( q in , 0) [coNP-complete] 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
∀∃ R PAD & Undecidability ◮ ∀∃ R PAD := ∀ z 1 . . . ∀ z n ∃ x 1 . . . ∃ x m . ϕ ( x , z ) ◮ divisibilities of the form f ( z ) | g ( x , z ) Guillermo A. P´ erez 12/27
∀∃ R PAD & Undecidability ◮ ∀∃ R PAD := ∀ z 1 . . . ∀ z n ∃ x 1 . . . ∃ x m . ϕ ( x , z ) ◮ divisibilities of the form f ( z ) | g ( x , z ) ◮ ∀∃ R PAD + := ∀∃ R PAD with ¬ not allowed before divisibility. Guillermo A. P´ erez 12/27
∀∃ R PAD & Undecidability ◮ ∀∃ R PAD := ∀ z 1 . . . ∀ z n ∃ x 1 . . . ∃ x m . ϕ ( x , z ) ◮ divisibilities of the form f ( z ) | g ( x , z ) ◮ ∀∃ R PAD + := ∀∃ R PAD with ¬ not allowed before divisibility. Claim (Bozga-Iosif’05, Lechner’15) SynthReach for SOCAP is decidable by a reduction to ∀∃ R PAD + . Guillermo A. P´ erez 12/27
∀∃ R PAD & Undecidability ◮ ∀∃ R PAD := ∀ z 1 . . . ∀ z n ∃ x 1 . . . ∃ x m . ϕ ( x , z ) ◮ divisibilities of the form f ( z ) | g ( x , z ) ◮ ∀∃ R PAD + := ∀∃ R PAD with ¬ not allowed before divisibility. Claim (Bozga-Iosif’05, Lechner’15) SynthReach for SOCAP is decidable by a reduction to ∀∃ R PAD + . ∀∃ R PAD ≡ ∀∃ R PAD + Guillermo A. P´ erez 12/27
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