Formal Languages and TQFTs with Defects

Luisa Boateng; Matilde Marcolli

REVIEW 4 major objections 5 minor 21 references

Formal Languages and TQFTs with Defects

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A functorial construction turns context-free grammars into 1D TQFTs with defects, reducing them to tree-contour grammars.

desk verdict A promising extension of the automata-to-TQFT program with a real, fixable bug in the modified pullback definition; the central functoriality claim does not currently hold as stated. read the letter →

arxiv 2412.09688 v1 pith:BQVXQFGG submitted 2024-12-12 math-ph cs.CLmath.MPmath.QA

Luisa Boateng , Matilde Marcolli This is my paper · ORCID

classification math-phcs.CLmath.MPmath.QA MSC 68Q4518M8581T45

keywords Boolean1DTQFTswithdefectsfinitestateautomatacontext-freegrammarsChomsky-Schützenbergertheoremcategoricaltransducerstree-contourcoloredoperadssubregularlanguages

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the recently introduced assignment of Boolean 1D TQFTs with defects to finite automata is not ad hoc: it is functorial under transducers, and it extends from regular to context-free grammars. The key move is to use a categorical version of the Chomsky–Schützenberger theorem, in which every context-free grammar is represented as a transducer image of a tree-contour grammar, the categorical analogue of the Dyck language. If correct, the TQFT associated to any context-free grammar is completely determined by the TQFTs of tree-contour grammars plus functoriality under transducers. The paper also shows that certain subregular language classes, such as strictly local and locally testable languages, leave cohomological or global traces on the TQFT side.

What carries the argument

The central objects are the colored operad of spliced arrows $\mathcal{W}_{\mathcal{C}}$, whose operations are words $w_0\square w_1\square\cdots\square w_n$ with square gaps as inputs, and the contour category $\mathrm{Cont}(\mathcal{O}_{\mathcal{S}})$, whose arrows trace the contours of trees in a free operad and whose associated tree-contour grammars are the categorical Dyck languages. A context-free grammar is encoded as a morphism of operads $P:\mathcal{O}_{\mathcal{S}}\to\mathcal{W}_{\mathcal{C}}$, and the categorical Chomsky–Schützenberger theorem factors any such $P$ through the universal tree-contour morphism $\mathcal{O}_{\mathcal{S}}\to\mathcal{W}_{\mathrm{Cont}(\mathcal{O}_{\mathcal{S}})}$ via a functor $\tau_G$. The TQFT side is carried by the colored operad $\mathcal{O}_{\mathrm{Cob},\mathcal{C}}$ of 1D cobordisms with defects and rectangular boxes cut out, composed by plugging matching boundary data; Proposition 4.9 converts a grammar into a morphism $\Xi_G:\mathcal{O}_{\mathrm{Cob},\mathcal{C}}\to\mathcal{O}_{\mathcal{B}\text{-Mod}}$, and Proposition 4.10 uses the contour factorization together with transducer naturality to reduce this morphism to tree contours.

What would settle it

Compute, for the grammar $S\to aSbS\mid \epsilon$, the operator $T_{a\square b}$ from equation (4.9) and compare it with the operator obtained by first mapping this grammar through its contour functor $\tau_G$ to a tree-contour grammar and then applying the transducer of Proposition 4.10; any difference on a basis state $\delta_C$ would falsify the main reduction. Alternatively, check that the natural transformation $\gamma_T$ of Theorem 4.6 commutes for a simple two-state transducer; a failure on a cup or half-line cobordism would falsify the claimed functoriality.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the passage from machines and grammars to 1D TQFTs with defects is a structural functor rather than a collection of examples. For finite automata, a transducer between automata induces a natural transformation between the associated Boolean 1D TQFTs with defects, after replacing the target category by spans of Boolean semimodules and using modified pullbacks. For categorical context-free grammars, seen as operad homomorphisms $P:\mathcal{O}_{\mathcal{S}}\to\mathcal{W}_{\mathcal{C}}$, the same functoriality holds; the associated TQFT becomes a morphism of colored operads, from an operad of 1D cobordisms with defects to an operad of Boolean semimodule operations. The categorical Chomsky–Schützenberger theorem then yields Proposition 4.10: every such TQFT is completely determined by its values on tree-contour grammars and by functoriality under transducers. Additionally, strictly local languages produce cohomological structures from forbidden factors, while locally testable languages require a global cobordism-level condition.

Load-bearing premise

The load-bearing premise is the categorical Chomsky–Schützenberger theorem itself: every categorical context-free grammar must factor through a tree-contour grammar by a functor and be realizable as a transducer image of a color-injective tree-contour grammar; if that companion theorem fails, the paper's reduction to tree-contour TQFTs collapses.

Editorial extensions

If this is right

For every categorical context-free grammar, the associated Boolean 1D TQFT with defects is determined by the tree-contour grammar of its species and the transducer realizing the grammar.
Strictly local languages of order 2 give cohomological structures $H_{ab}=\ker(T_a)/\operatorname{im}(T_b)$ for non-admissible factors, and higher-order strictly local languages give analogous structures for forbidden $k$-factors.
Locally testable languages cannot be captured by local composition of defects; they require the global set of cobordisms containing required factors, with commuting actions of the two endomorphism monoids.
Transducer morphisms between automata become natural transformations between TQFTs, so the TQFT construction is a functor from automata and cobordisms to spans of Boolean semimodules.
The operadic reformulation allows composition of defect TQFTs by plugging cobordisms into boxes, yielding a compositional calculus for context-free language TQFTs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the reduction is made constructive, it suggests an algorithm for computing a context-free TQFT by first computing the tree-contour TQFT and then applying the transducer, potentially simplifying semiring parsing of context-free languages.
The cohomological structures attached to strictly local languages may be invariants of the language rather than the automaton; checking whether two minimal automata for the same language yield isomorphic cohomology would be a direct test.
The same operadic machinery may extend to higher-dimensional automata and 2D TQFTs with defects, where Frobenius algebra descriptions could give algebraic invariants; the paper itself lists this as future work.
Locally testable languages' global cobordism condition resembles a cohomology of the endomorphism monoid action, and making that precise could give a model-theoretic-to-TQFT dictionary for subregular classes beyond SL and LT.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper does something genuinely new: it makes the automaton-to-Boolean-1D-TQFT construction functorial with respect to transducers, and then extends the whole picture to context-free grammars via the Melliès–Zeilberger categorical Chomsky–Schützenberger theorem. The operadic formalism in Sections 4.5–4.6 is a real step beyond the existing literature, and the writing is clear about what is being assumed.

The soft spots are serious, though. Definition 3.2 defines the modified pullback α*Φ as the largest subspace invariant under the defect operators. As the stress-test note observes, this does not match Lemma 3.2's identification with Φ_{α^{-1}(M)}. With a sink state u, the largest invariant subspace is the whole space, because T_A(δ_u)=0, so every basis vector is invariant; the intended subspace consists only of states that occur as endpoints of α-transitions. That discrepancy breaks the natural transformations in Theorem 3.3 and Theorem 4.6, and those natural transformations are exactly what Proposition 4.10 relies on to reduce all context-free TQFTs to tree-contour grammars. The fix is straightforward—define the pullback as the submodule generated by the images of the pulled-back defect operators—but as written the central functoriality claim is not established.

Section 3.3 overreaches in a different way. The claim that a non-admissible 2-factor implies T_a T_{a'} = 0 is not a property of the language; it depends on the chosen automaton. A non-minimal automaton for (AB)^n can have extra states that make the product nonzero. The cohomological structure is a feature of the specific TQFT, not of the SL2 language, and the section needs to say so.

Also, the context-free reduction itself is largely imported from [17]. That is fine, but the authors should be clearer about which statements are new and which are directly taken from Melliès–Zeilberger.

Despite these problems, the paper deserves a serious referee. The topic is timely, the categorical framework is appropriate, and the main bug is genuinely repairable. With the pullback fixed and the subregular section corrected, this could become a useful organizing contribution. My own verdict would be revise-and-resubmit, not accept.

I would bring it to reading group only if people want to wrestle with the pullback issue. I would not cite it in its current form.

Referee Report

4 major / 5 minor

Summary. This paper extends the Boolean 1D TQFT-with-defects construction of Gustafson, Im, Kaldawy, Khovanov, and Lihn from finite state automata to a category whose morphisms are transducers, then to categorical finite state automata and to categorical context-free grammars viewed as morphisms of colored operads P: O_S → W_C. The advertised results are functoriality under transducers (Theorems 3.3, 4.3, 4.6), a cohomological interpretation of certain subregular language classes (Section 3.3), and a reduction of all categorical context-free grammar TQFTs to tree-contour grammars via the Melliès–Zeilberger version of the Chomsky–Schützenberger theorem (Proposition 4.10). The paper is exploratory and contains a number of local gaps and typos, but its intended categorical/operadic bridge between automata theory and defect TQFTs is clear.

Significance. If the main theorems were correct, the paper would give a useful functorial and operadic framework for the language–TQFT correspondence: transducer morphisms would lift to natural transformations, context-free grammars would be encoded as operad maps, and the Chomsky–Schützenberger theorem would reduce the whole construction to tree-contour grammars. The paper is explicit, has no fitted parameters, and transparently relies on the external theorem [17]. However, as written, the central functoriality statements are not established because of an incorrect modified pullback definition, and the subregular cohomology claim is not valid for arbitrary automata computing a language. These are local, repairable defects, so the framework remains promising, but the current version cannot serve as a rigorous foundation for the context-free results.

major comments (4)

[§3.2, Definition 3.2 and Lemma 3.2] Definition 3.2 defines the modified pullback α^*Φ(+) as the largest subspace invariant under all operators Φ(c_{α(a)}), but Lemma 3.2 and the proof of Theorem 3.3 require this subspace to be B^{Q_{M,α}}, the span of states occurring in α-transitions. These are not equal: if a state is not the source of any α(a)-transition, its basis vector is annihilated by all T_{α(a)} and hence lies in every invariant subspace. Concretely, let M have states {s,t,u}, transitions s--A-->t, t--B-->s, u--B-->u, and let α(a)=A. Then Q_{M,α}={s,t}, so Lemma 3.2 identifies α^*Φ_M(+) with B^{{s,t}}, but the largest T_A-invariant subspace of B^{{s,t,u}} is the whole space because T_A(δ_u)=0. The proof of Theorem 3.3 explicitly uses the equality α^*Φ_M = Φ_{α^{-1}(M)}, so Theorem 3.3 is not established as stated. The repair is local: replace 'largest invariant subspace' by the Boolean submodule spanned by all states that occur as sources or targets of transitions labelled by α(a), and then re-check the naturality argument.
[§3.3] The inference from inadmissible k-factors to vanishing of products of defect operators is not valid for arbitrary FSAs computing the given language. Earlier in §3.2 the paper correctly stresses that the TQFT depends on the automaton, not only the language, yet §3.3 asserts for (AB)^n that 'TB is a coboundary operator (T_B^2=0) on any of the spaces Φ_M(ϵ), and so is TA.' This is false in general. Add to the standard two-state automaton for (AB)^n an extra state r with a transition r--A-->r (and, if desired, make r reachable by adding q0--A-->r and keeping q0 final); the recognized language is unchanged, but T_A^2(δ_r)=δ_r, so T_A is not square-zero. The cohomological structure of SL2 languages therefore needs an explicit hypothesis on the automaton, such as a canonical minimal automaton, together with a proof, rather than being presented as a language-level consequence.
[§4.5–4.6, Theorem 4.6 and Proposition 4.10] The same defect propagates to the context-free setting. Theorem 4.6 is stated using 'the modified pullbacks as in Definition 3.2', and its proof identifies Π_{Wα}Ψ_G(+) with B(X_{C,S}∩Obj(Q_T)). This is exactly the equality that fails for the largest-invariant-subspace definition. Consequently the natural transformations γ_T : Wα^*Ψ_G → Wβ^*Ψ_{G'} are not established, and Proposition 4.10, which concludes that all context-free TQFTs are determined by tree-contour grammars and functoriality under transducers, has no valid proof at present. The authors should repair Definition 3.2, or avoid modified pullbacks in this theorem, and then verify that the naturality diagram in Theorem 4.6 commutes with the corrected pullback.
[§4.6, Proposition 4.10] Proposition 4.10 makes essential use of the Melliès–Zeilberger theorem [17], which is legitimate, but the proof as written does not verify that the hypotheses of Proposition 4.9 and Theorem 4.6 are met after the reduction. In particular, Theorem 4.6 assumes that α is injective on objects and that the operad map is C-chromatic, while the factorization step introduces the color-surjective morphism φCol: O_S→O_{φColS}; the argument that the original, possibly non-C-chromatic, grammar G is a transducer image of a C-chromatic tree-contour grammar is compressed into a diagram and is not self-contained. Please state the precise theorem from [17] that is being used and check the two hypotheses explicitly; otherwise Proposition 4.10 is conditional on an unstated strengthening of [17].

minor comments (5)

[Introduction] In the first paragraph, 'and and additional operadic structure' contains a duplicated word that should be removed.
[§3.1] The text repeats 'FM(+) = BQ∨' after defining FM(+)=BQ; the second occurrence should be FM(−)=BQ∨.
[Throughout] There are numerous typos, including 'wth', 'inital', and inconsistent spelling of Melliès; a thorough proofreading pass is needed before publication.
[§3.3] The definition of H_{ab}(Φ_M(ϵ),T) := Ker(T_a)/Image(T_b) uses quotients and kernels of Boolean semimodules without specifying the relevant universal properties; this should be clarified, especially since the paper otherwise works with spans of B-modules.
[§2.2] The proof of Proposition 2.4 is very terse; in particular, the transition from the automaton for α^{-1}(L)∩L_reg to the automaton for β(α^{-1}(L)∩L_reg) is described only informally via 'subdivisions' and should be formalized with a precise definition of the intermediate state set Q'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the context-free reduction rests on the external Melliès–Zeilberger theorem, not on the paper's own construction.

full rationale

The paper's central derivation, Proposition 4.10, claims that the assignment of 1D Boolean TQFTs with defects to categorical context-free grammars is determined by tree-contour grammars and by functoriality under transducers. The proof begins by citing [17] for the factorization of any categorical context-free grammar through a tree-contour grammar; this is an external theorem by Melliès and Zeilberger, not a result derived from the TQFT construction in this paper. The TQFT-to-transducer natural transformations in Theorems 3.3 and 4.6 are established by explicit span and pullback constructions, not by assuming the conclusion. No equation is shown to be equal to its own input by construction, and no fitted parameter is later renamed as a prediction. The only self-citation is [7], coauthored by Marcolli, used to recall the category of formal languages with rational transductions and to justify associativity of transducer composition; this is background structure rather than the source of the Chomsky–Schützenberger reduction. Even if the skeptical objection about the modified pullback in Definition 3.2 were correct, it would be a mathematical correctness issue in Theorem 3.3, not a circularity: the theorem would be false or unproved, not true by definition. The central claim therefore has independent grounding in [17].

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted constants or empirical entities appear. The paper's novel definitions, such as operadic Boolean TQFTs and spliced defects, are mathematical structures and are not listed as invented entities. The main non-local input is the Mellies-Zeilberger theorem [17].

assumptions (4)

domain assumption The category F L of formal languages with rational transductions and the category A of automata with transducers have associative composition as defined in [7].
Used in Section 2.1.2 and Proposition 2.4 to define all morphisms; the paper cites [7] for associativity rather than proving it.
domain assumption The construction of Boolean 1D TQFTs with defects from finite automata in [10] is valid as recalled in Section 3.1.
Theorems 3.3, 4.3, and 4.5 build directly on this construction; if it failed, there would be no base TQFT to make functorial.
domain assumption The Mellies-Zeilberger categorical Chomsky-Schützenberger theorem [17] holds, including the tree-contour grammar factorization and the preservation of categorical context-free languages under transducer action.
This external theorem is the load-bearing input for Proposition 4.10 and Section 4.4; the paper applies it without restating the proof.
domain assumption The category B-Mod has pullbacks, and the splice operads WC and tree-contour categories Cont(OS) have the properties stated in [16,17,21].
Needed for spans in Span(B-Mod) and for the operadic constructions in Sections 4.2 through 4.6.

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Cite this review

Pith. "Pith review of Formal Languages and TQFTs with Defects." pith.science (2026). https://pith.science/paper/BQVXQFGG

@misc{pith2026241209688,
  author       = {Pith},
  title        = {Pith review of: Formal Languages and TQFTs with Defects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BQVXQFGG}},
  note         = {Machine review of arXiv:2412.09688}
}

read the original abstract

A construction that assigns a Boolean 1D TQFT with defects to a finite state automaton was recently developed by Gustafson, Im, Kaldawy, Khovanov, and Lihn. We show that the construction is functorial with respect to the category of finite state automata with transducers as morphisms. Certain classes of subregular languages correspond to additional cohomological structures on the associated TQFTs. We also show that the construction generalizes to context-free grammars through a categorical version of the Chomsky-Sch\"utzenberger representation theorem, due to Melli\`es and Zeilberger. The corresponding TQFTs are then described as morphisms of colored operads on an operad of cobordisms with defects.

Figures

Figures reproduced from arXiv: 2412.09688 by the authors.

**Figure 1.** The generating morphisms of the category of one-dimensional cobordisms. The cup and cap morphisms and permutations which compose with them to get the opposite cup and cap, respectively [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** All four types of morphisms in the category of 1-dimensional cobordisms with defects, Cob1, P, with P = {a, b, c}. these half-edges cobordisms to either a morphism m : M → P (a choice of an element of P) or a morphism m∨ : M → R (a choice of an element in M∨ ). The category Cob can also be extended to 1D cobordisms with defects, namely marked points inserted in the one-dimensional cobordisms and marked by the elemen… view at source ↗

**Figure 3.** Two finite state automata which compute the same regular language (AB) n . Ta : BQ → BQ defined by the transition functions (q, a, q′ ) ∈ τ of the automaton M by Ta : q 7→ X (q,a,q′)∈τ q ′ , and its dual T ∨ a for the case with the opposite orientation. 3.2. Transducers and functorial properties. The TQFT construction in [10] is based on the automaton M not on the regular language it computes, in the sense that diff… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Operad composition f ◦1 g of the sequence g = u0□u1□u2 into the sequence f = w0□w1□w2□w3, from [17]. 4.2. Operads of spliced arrows and species. We recall the following setting on species, colores operads, and operads of spliced words in a category from [16], [17]. Let…

**Figure 5.** Figure 5: Operations in the operad OCob,C of 1D cobordisms with defects. with defects in an auxiliary category C. An operadic Boolean 1D TQFT with defects in C is a morphism of operads Ψ : OCob,C → OBMod . We can visualize graphically the spliced colored operad of cobordisms WCo…

**Figure 6.** Figure 6: 1D cobordisms with defects in the operad WCob1,MorC and the operad composition [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

**Figure 7.** Figure 7: 1D cobordisms with defects in the WC and the composition of defects. Theorem 4.5. The construction M 7→ ΦM of a 1D Boolean TQFT with defects from a categorical FSA, as in Theorem 4.1, determines a construction M 7→ ΨM of an operadic Boolean 1D TQFT with defects ΞM : OC…

**Figure 8.** Figure 8: The operad composition in WCob1,WC. we can also associate a corresponding morphism of colored operads WΦM : WCob1,MorC → WB-Mod between the respective colored operads of spliced arrows. Given M = (Q, C, τ : Q → C,(q0, qf )) and ΦM constructed as in Theorem 4.1, the ass…

**Figure 9.** Figure 9: Decomposition of cobordisms with defect. refinement of the decomposition of c in the form above, and the corresponding image of the pieces of the decomposition then corresponds to the operad composition in ORMod. □ We can also rewrite the action (4.6) of the operators …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 17 canonical work pages

[17]

Melli` es, N

P.A. Melli` es, N. Zeilberger,The categorical contours of the Chomsky–Sch¨ utzenberger representation theorem, arXiv:2405.14703

arXiv
[1]

Jean Berstel, Transductions and Context-Free Languages, Teubner Verlag, 1979

work page 1979
[2]

Carqueville, Lecture notes on 2-dimensional defect TQFT , arXiv:1607.05747

N. Carqueville, Lecture notes on 2-dimensional defect TQFT , arXiv:1607.05747

arXiv
[3]

N. Chomsky. Three models for the description of language . IRE Transactions on Information Theory. 2 (1956) N.3, 113–124

work page 1956
[4]

Chomsky, On certain formal properties of grammars , Information and Control, 2 (1959) 137–167

N. Chomsky, On certain formal properties of grammars , Information and Control, 2 (1959) 137–167

work page 1959
[5]

Computer program- ming and formal systems

N. Chomsky and M. P. Sch¨ utzenberger,The algebraic theory of context-free languages, in “Computer program- ming and formal systems”, pp. 118–161. North Holland, 1963

work page 1963
[6]

Connes, D

A. Connes, D. Kreimer, Hopf algebras, Renormalization and Noncommutative geometry , Comm. Math. Phys 199 (1998) 203–242

work page 1998
[7]

Formal languages, spin systems, and quasicrystals

F. Fernandes, M. Marcolli, Formal languages, spin systems, and quasicrystals , arXiv:2405.12485

work page Pith review arXiv

Show all 21 references

[8]

Algebra and Coalgebra in Computer Science

N. Ghani, A. Kurz, higher dimensional trees, algebraically , in “Algebra and Coalgebra in Computer Science”, Lecture Notes in Computer Science (LNTCS), Vol. 4624, pp. 226–241
[9]

Goodman, Semiring parsing, Computational Linguistics, Vol.25 (1999) N.4, 573–605

J. Goodman, Semiring parsing, Computational Linguistics, Vol.25 (1999) N.4, 573–605

1999
[10]

Gustafson, M.S

P. Gustafson, M.S. Im, R. Kaldawy, M. Khovanov, Z. Lihn, Automata and one-dimensional TQFTs with defects, arXiv:2301.00700

arXiv
[11]

Gustafson, M.S

P. Gustafson, M.S. Im, M. Khovanov, Boolean TQFTs with accumulating defects, sofic systems, and automata for infinite words , arXiv:2312.17033

arXiv
[12]

M.S. Im, M. Khovanov, Topological theories and automata, arXiv:2202.13398

arXiv
[13]

M.S. Im, M. Khovanov, From finite state automata to tangle cobordisms: a TQFT journey from on to four dimensions, arXiv:2309.00708

arXiv
[14]

J¨ ager, J

G. J¨ ager, J. Rogers, Formal language theory: refining the Chomsky hierarchy , Phil. Trans. R. Soc. B (2012) 367, 1956–1970

2012
[15]

Kambites, Formal languages and groups as memory , arXiv:math/060161

M. Kambites, Formal languages and groups as memory , arXiv:math/060161. 28 LUISA BOATENG AND MATILDE MARCOLLI
[16]

Proceedings of the 38th Conference on Mathematical Foundations of Programming Semantics (MFPS 2022)

P.A. Melli` es, N. Zeilberger,Parsing as a lifting problem and the Chomsky–Sch¨ utzenberger representation theo- rem, in “Proceedings of the 38th Conference on Mathematical Foundations of Programming Semantics (MFPS 2022)”, arXiv:2212.09060

2022 arXiv
[18]

Rawski, Tensor Product Representations of Subregular Formal Languages , arXiv:1908.08132

J. Rawski, Tensor Product Representations of Subregular Formal Languages , arXiv:1908.08132

1908 arXiv
[19]

Rogers, Syntactic structures as multi-dimensional trees , Research on Language and Computation 1(3–4), 265–305 (2003)

J. Rogers, Syntactic structures as multi-dimensional trees , Research on Language and Computation 1(3–4), 265–305 (2003)

2003
[20]

Spivak, P

D.I. Spivak, P. Schultz, D. Rupel, String diagrams for traced and compact categories are oriented 1-cobordisms, J. Pure Appl. Algebra 221 (2017), no. 8, 2064–2110

2017
[21]

Takahashi, Extensions of Semimodules I , Math

M. Takahashi, Extensions of Semimodules I , Math. Sem. Notes Kobe Univ. 10 (1982), 563–592. Department of Mathematics, Stanford University, Stanford, CA 94305, USA Email address : luisa.mboateng@gmail.com Department of Mathematics and Department of Computing and Mathematical S...

1982

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [17]

Melli` es, N

P.A. Melli` es, N. Zeilberger,The categorical contours of the Chomsky–Sch¨ utzenberger representation theorem, arXiv:2405.14703

arXiv

[2] [1]

Jean Berstel, Transductions and Context-Free Languages, Teubner Verlag, 1979

work page 1979

[3] [2]

Carqueville, Lecture notes on 2-dimensional defect TQFT , arXiv:1607.05747

N. Carqueville, Lecture notes on 2-dimensional defect TQFT , arXiv:1607.05747

arXiv

[4] [3]

N. Chomsky. Three models for the description of language . IRE Transactions on Information Theory. 2 (1956) N.3, 113–124

work page 1956

[5] [4]

Chomsky, On certain formal properties of grammars , Information and Control, 2 (1959) 137–167

N. Chomsky, On certain formal properties of grammars , Information and Control, 2 (1959) 137–167

work page 1959

[6] [5]

Computer program- ming and formal systems

N. Chomsky and M. P. Sch¨ utzenberger,The algebraic theory of context-free languages, in “Computer program- ming and formal systems”, pp. 118–161. North Holland, 1963

work page 1963

[7] [6]

Connes, D

A. Connes, D. Kreimer, Hopf algebras, Renormalization and Noncommutative geometry , Comm. Math. Phys 199 (1998) 203–242

work page 1998

[8] [7]

Formal languages, spin systems, and quasicrystals

F. Fernandes, M. Marcolli, Formal languages, spin systems, and quasicrystals , arXiv:2405.12485

work page Pith review arXiv