Abstract
The paper presents how formation trees, a type of syntax trees for a formula, can be used to represent semantic information about formulas of classical propositional logic in form of graphs, permutations and tables. The three representation types are discussed in terms of the construction process as well as their cognitive potentials and observational advantages.
My work was supported financially by National Science Centre in Poland, grant no 2017/26/E/HS1/00127.
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Notes
- 1.
Proof of a formula, in this context, consists in finding a particular homomorphism between graphs, and so has been characterised by Hughes as a “graphical” [6].
- 2.
In discrete mathematics, cographs are often represented by cotrees and parse trees, the latter corresponding to formation trees. Parse trees are binary trees, with inner nodes of types 1 and 0, corresponding to \(\alpha \)– and \(\beta \)–nodes of the formation tree. Their structure mirrors the recursive procedure of construction of cographs, with 1-nodes corresponding to join operation and 0-nodes to disjoint union. See [3, 7] for other properties of cographs.
- 3.
A clique in a graph G is a complete induced subgraph H of G. A maximal clique is a clique that cannot be extended to a greater clique by adding new vertices. An independent set in a graph G is a subgraph H of G such that no \(v_{1}, v_{2} \in H\) are adjacent. Similarly, a maximal independent set in G is one that cannot be extended by adding new vertices to it.
- 4.
Note that each horizontal path crosses each vertical path in exactly one point (occurrence of a literal). This is consistent with an interesting property of cographs: any maximal clique of a cograph intersects any maximal independent set in a single vertex.
References
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Sochański, M. (2022). Representing Formulas of Propositional Logic by Cographs, Permutations and Tables. In: Giardino, V., Linker, S., Burns, R., Bellucci, F., Boucheix, JM., Viana, P. (eds) Diagrammatic Representation and Inference. Diagrams 2022. Lecture Notes in Computer Science(), vol 13462. Springer, Cham. https://doi.org/10.1007/978-3-031-15146-0_26
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