Deciding the Word Problem for Ground and Strongly Shallow Identities w.r.t. Extensional Symbols

• Published in: Journal of automated reasoning Vol. 66; no. 3; pp. 301 - 329
• Main Authors: Baader, Franz, Kapur, Deepak
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-08-2022; Springer Nature B.V
• Subjects: Algorithms; Artificial Intelligence; Computer Science; Mathematical Logic and Formal Languages; Mathematical Logic and Foundations; Polynomials; Semantics; Symbolic and Algebraic Manipulation; Symbols; Ordered Rewriting; Extensionality; Shallow identities; Semantic congruence closure; Description Logics; Interpreted function symbols; Word problem for ground identities
• ISSN: 0168-7433; 1573-0670
• DOI: 10.1007/s10817-022-09624-4

The word problem for a finite set of ground identities is known to be decidable in polynomial time using congruence closure, and this is also the case if some of the function symbols are assumed to be commutative or defined by certain shallow identities, called strongly shallow. We show that decidability in P is preserved if we add the assumption that certain function symbols
f
are
extensional
in the sense that
f
(
s
1
,
…
,
s
n
)
≈
f
(
t
1
,
…
,
t
n
)
implies
s
1
≈
t
1
,
…
,
s
n
≈
t
n
. In addition, we investigate a variant of extensionality that is more appropriate for commutative function symbols, but which raises the complexity of the word problem to coNP.