Guarded Kleene Algebra with Tests: Coequations, Coinduction, and Completeness

@inproceedings{Schmid2021GuardedKA,
  title={Guarded Kleene Algebra with Tests: Coequations, Coinduction, and Completeness},
  author={Todd J. Schmid and Tobias Kapp'e and Dexter Kozen and Alexandra Silva},
  booktitle={ICALP},
  year={2021}
}
Guarded Kleene Algebra with Tests (GKAT) is an efficient fragment of KAT, as it allows for almost linear decidability of equivalence. In this paper, we study the (co)algebraic properties of GKAT. Our initial focus is on the fragment that can distinguish between unsuccessful programs performing different actions, by omitting the so-called early termination axiom. We develop an operational (coalgebraic) and denotational (algebraic) semantics and show that they coincide. We then characterize the… 

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References

SHOWING 1-10 OF 37 REFERENCES
Nonlocal Flow of Control and Kleene Algebra with Tests
  • D. Kozen
  • Mathematics
    2008 23rd Annual IEEE Symposium on Logic in Computer Science
  • 2008
TLDR
This paper shows how KAT can be used to give a rigorous equational treatment of control constructs involving nonlocal transfer of control such as unconditional jumps, loop statements with multi-level breaks, and exception handlers, and develops a compositional semantics and a complete equational axiomatization.
Guarded Kleene Algebra with Tests: Verification of Uninterpreted Programs in Nearly Linear Time
TLDR
This work develops the (co)algebraic theory of GKAT and shows how it can be efficiently used to reason about imperative programs and provides a full Kleene theorem and proves completeness for an analogue of Salomaa's axiomatization of Kleene Algebra.
On the Coalgebraic Theory of Kleene Algebra with Tests
  • D. Kozen
  • Mathematics, Computer Science
  • 2017
TLDR
It follows that coinductive equivalence proofs can be generated automatically in PSPACE, and this matches the bound of Worthington (2008) for the automatic generation of equational proofs in \(\mathsf {KAT}\).
A Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity
TLDR
It is proved that the adaptation of Milner's system over the subclass of regular expressions that arises by dropping the constant 1, and by changing to binary Kleene star iteration is complete.
The Böhm-Jacopini Theorem Is False, Propositionally
TLDR
A purely propositional account of the Bohm---Jacopini theorem, which states that any deterministic flowchart program is equivalent to a while program, and reformulate the problems at the propositional level in terms of automata on guarded strings, the automata-theoretic counterpart to Kleene algebra with tests.
Universal coalgebra: a theory of systems
Equational Theories of Abnormal Termination Based on Kleene Algebra
TLDR
This work investigates two equational theories in the abstract framework of Kleene algebra, and proposes two simple and intuitive equational axiomatizations, and proves very general conservativity results.
Kleene Algebra with Tests: Completeness and Decidability
TLDR
The completeness of the equational theory of Kleene algebras with tests and *-continuous Kleene algebra with tests over language-theoretic and relational models is proved.
The Complexity of Kleene Algebra with Tests
TLDR
The equational theory of Kleene algebras with tests has been shown to be decidable in at most exponential time by an efficient reduction to Propositional Dynamic Logic.
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