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Combining Equational Reasoning

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNAI,volume 5749)

Abstract

Given a theory \(\mathbb{T}\), a set of equations E, and a single equation e, the uniform word problem (UWP) is to determine if \(E\Rightarrow e\) in the theory \(\mathbb{T}\). We recall the classic Nelson-Oppen combination result for solving the UWP over combinations of theories and then present a constructive version of this result for equational theories. We present three applications of this constructive variant. First, we use it on the pure theory of equality (\(\mathbb{T}_{EQ}\)) and arrive at an algorithm for computing congruence closure of a set of ground term equations. Second, we use it on the theory of associativity and commutativity (\(\mathbb{T}_{AC}\)) and obtain a procedure for computing congruence closure modulo AC. Finally, we use it on the combination theory \(\mathbb{T}_{EQ}\cup\mathbb{T}_{AC}\cup\mathbb{T}_{PR}\), where \(\mathbb{T}_{PR}\) is the theory of polynomial rings, to present a decision procedure for solving the UWP for this combination.

Keywords

  • Inference Rule
  • Decision Procedure
  • Polynomial Ring
  • Function Symbol
  • Equational Theory

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Research supported in part by NSF grants CNS-0720721 and CNS-0834810 and NASA grant NNX08AB95A.

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Tiwari, A. (2009). Combining Equational Reasoning. In: Ghilardi, S., Sebastiani, R. (eds) Frontiers of Combining Systems. FroCoS 2009. Lecture Notes in Computer Science(), vol 5749. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04222-5_4

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  • DOI: https://doi.org/10.1007/978-3-642-04222-5_4

  • Publisher Name: Springer, Berlin, Heidelberg

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