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Definability of some $k$-ary Relations Over Second Order kinds of Logics | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 04 اسفند 1404 اصل مقاله (495.07 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.31006.2705 | ||
| نویسندگان | ||
| Simone Costa1؛ Marco Dalai2؛ Stefano Della Fiore2؛ Anita Pasotti* 1 | ||
| 1DICATAM - Sez. Matematica, Universit`a degli Studi di Brescia, Via Branze 43, I-25123 Brescia, Italy | ||
| 2DII, Università degli Studi di Brescia, Via Branze 38, I-25123 Brescia, Italy | ||
| چکیده | ||
| We consider the exprissibility in monadic second order logic of certain relations of importance in computer science. For integers $n\geq 1$ and $k\leq b$, a $k$-tuple of sequences in $\{0,1,\ldots, b-1\}^n$ are said to be $k$-hashed if there is a coordinate where they all differ. A set $\mathcal{C}$ of sequences is said to be a $k$-hash code if any $k$ distinct elements are $k$-hashed. Testing whether a code is $k$-hashing and determining the largest size of $k$-hash codes is an important problem in computer science. The use of general purpose solvers for this problem leads to question what minimal logic is needed to represent the problem. In this paper, we prove that the $k$-hashing relation on $k$-tuples is not definable in Monadic Second Order Logic (MSO), highlighting its limitations for this problem. Instead, the property can be expressed in extensions of the MSO that add the equi-cardinality relation. Finally, since to the best of our knowledge there are no available solvers for MSO with such global cardinality constraints, we provide a practical SMT encoding in Z3 for fixed parameters. In addition, we computationally recover the non-existence of a trifferent code of size $11$ and length $5$. | ||
| کلیدواژهها | ||
| k-hash code؛ inexpressibility؛ monadic second order logic | ||
| مراجع | ||
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