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arxiv: 1906.10494 · v1 · pith:TXGOMQBAnew · submitted 2019-06-25 · 💻 cs.FL

New Pumping Technique for 2-dimensional VASS

Wojciech Czerwi\'nski , S{\l}awomir Lasota , Christof L\"oding , Rados{\l}aw Pi\'orkowski This is my paper

Pith reviewed 2026-05-25 16:00 UTC · model grok-4.3

classification 💻 cs.FL
keywords 2-VASSvector addition systemspumping lemmageometric propertiesexponential boundregular separabilityformal languages
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The pith

A new geometric pumping technique for 2-VASS yields an exponential bound on shortest accepting runs and a pumping lemma for their languages.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a pumping technique for two-dimensional vector addition systems with states that draws directly from the geometric properties of runs. The authors apply it to recover the known exponential bound on the shortest accepting run and to establish a fresh pumping lemma for 2-VASS languages. A sympathetic reader would care because the method replaces intricate combinatorial case analysis with geometric reasoning and is presented as a tool for resolving further questions about these languages.

Core claim

The central claim is that natural geometric properties of runs in 2-VASS suffice to define a pumping technique that simultaneously recovers the exponential length bound on shortest accepting runs and produces a pumping lemma for the accepted languages.

What carries the argument

The new pumping technique built on natural geometric properties of runs in 2-VASS

Load-bearing premise

Natural geometric properties of runs suffice to obtain both the exponential bound and the pumping lemma without hidden combinatorial restrictions.

What would settle it

A concrete 2-VASS whose shortest accepting run exceeds exponential length in the system size, or whose runs cannot be pumped according to the geometric rule, would show the technique does not hold in general.

Figures

Figures reproduced from arXiv: 1906.10494 by Christof L\"oding, Rados{\l}aw Pi\'orkowski, S{\l}awomir Lasota, Wojciech Czerwi\'nski.

Figure 1
Figure 1. Figure 1: Thin (above) and thick run (below). Points correspond to counter values, and control states along a run are ignored. Contribution. The above-mentioned techniques are mostly oriented towards reachable sets, and henceforth may ignore certain runs as long as the reachable set is preserved. In consequence, they are not very helpful in solving decision problems for￾mulated in terms of the whole language ac￾cept… view at source ↗
Figure 2
Figure 2. Figure 2: Above u1  u2  . . .  u11  u1. Also, u4  u9, but u4 6 u11 and u11  u4. Pairs of vectors ui , ui+6 are contralinear, for i = 1, . . . , 5. Sequential cones. For a vec￾tor v ∈ Z 2 , define the half-line in￾duced by v as ℓv := Q≥0 · v = {αv | α ∈ Q≥0}. We call two vec￾tors v, w colinear if ℓv = ℓw, and contralinear if ℓv = ℓ−w. For two vectors u, v ∈ Z 2 \ {(0, 0)}, define the angle ∡[u, v] ⊆ Q2 as the … view at source ↗
Figure 3
Figure 3. Figure 3: Thin run within belts Bv,W . The main result of this section (cf. theorem 1 below) classifies (0, 0)-runs in a 2-VASS into thin and thick ones. Throughout this section we consider an arbi￾trary fixed 2-VASS V = (Q, T ). Let n = |Q| and M = kV k. Thin runs. The belt of di￾rection v ∈ N 2 and width W is the set Bv,W = {u ∈ N 2 | dist(u, ℓv) ≤ W}, where dist(u, ℓv) denotes the Euclidean distance between the p… view at source ↗
Figure 4
Figure 4. Figure 4: Thick run. Blue angles denote sequential [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Contracted paths ρ, e ρe ′ (left) and reconstructed (0, 0)-run ¯τ = ¯ρ ρ¯ ′ (right). Proof of theorem 3. We use m from the last claim to modify all factors of τ ex￾cept for ρ1 and ρ ′ 1 , in order to reduce their lengths to at most n·m2 . W.l.o.g. as￾sume m to be larger than A (take a sufficient multiplicity of m otherwise); this assumption allows us to proceed uniformly, irrespectively whether v1 is posit… view at source ↗
Figure 6
Figure 6. Figure 6: A-belts intersect only within square S. Proof. Assume that w ∈ Bu,A for some u (kuk ≤ A). We will show w ′ ∈ Bu,A. Notice that w ′ ∈ Bu,W (since kw √ ′ − wk2 ≤ 2kw ′ − wk ≤ √ 2M). Towards contradic￾tion assume that v ′ also belongs to some A-belt Bv,A 6= Bu,A (i.e. v and u non￾colinear). Then of course w ′ ∈ Bv,W too. We will show that this implies w ′ ∈ S. W.l.o.g. assume that u  v. Let I = (1, 1). Notic… view at source ↗
read the original abstract

We propose a new pumping technique for 2-dimensional vector addition systems with states (2-VASS) building on natural geometric properties of runs. We illustrate its applicability by reproving an exponential bound on the length of the shortest accepting run, and by proving a new pumping lemma for languages of 2-VASS. The technique is expected to be useful for settling questions concerning languages of 2-VASS, e.g., for establishing decidability status of the regular separability problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces a new pumping technique for 2-dimensional vector addition systems with states (2-VASS) that rests on natural geometric properties of runs. It applies the technique to reprove the known exponential bound on the length of the shortest accepting run and to establish a new pumping lemma for 2-VASS languages. The authors indicate that the approach should be useful for open questions such as the decidability of regular separability for 2-VASS.

Significance. The geometric construction supplies an independent, self-contained argument that simultaneously recovers the exponential bound and yields a new pumping lemma. Because the derivations are presented explicitly without hidden combinatorial restrictions or unjustified case distinctions, the technique constitutes a reusable tool for the theory of 2-VASS languages. If the geometric arguments hold, the work strengthens the toolkit available for settling decidability questions in this class.

minor comments (2)
  1. [Abstract] The abstract states that the technique 'reproves an exponential bound' but does not name the precise bound (e.g., the constant or the base of the exponential); adding this reference would improve readability.
  2. [Introduction] The manuscript would benefit from an explicit statement, early in the introduction, of the precise geometric property (e.g., the relevant invariant on the direction or the convex hull) that the pumping construction exploits.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript and the recommendation to accept. The report correctly identifies the self-contained geometric approach as the main contribution and notes its potential applicability to open problems such as regular separability.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via geometric arguments

full rationale

The paper introduces a pumping technique derived directly from geometric properties of runs in 2-VASS, using explicit constructions to reprove the exponential bound on shortest accepting runs and to establish a new pumping lemma. All load-bearing steps rely on independent geometric observations and case analysis presented within the manuscript itself, without reduction to fitted parameters, self-definitional loops, or load-bearing self-citations whose validity depends on the current results. The central claims are supported by the paper's own derivations rather than by renaming known results or smuggling ansatzes via prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no explicit free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5617 in / 939 out tokens · 40833 ms · 2026-05-25T16:00:37.608752+00:00 · methodology

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Reference graph

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