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Register a new userLeafy Automata for Higher-Order Concurrency
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Added by
Alex Dixon
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Finitary Idealized Concurrent Algol (FICA) is a prototypical programming language combining functional, imperative, and concurrent computation. There exists a fully abstract game model of FICA, which in principle can be used to prove equivalence and safety of FICA programs. Unfortunately, the problems are undecidable for the whole language, and only very rudimentary decidable sub-languages are known. We propose leafy automata as a dedicated automata-theoretic formalism for representing the game semantics of FICA. The automata use an infinite alphabet with a tree structure. We show that the game semantics of any FICA term can be represented by traces of a leafy automaton. Conversely, the traces of any leafy automaton can be represented by a FICA term. Because of the close match with FICA, we view leafy automata as a promising starting point for finding decidable subclasses of the language and, more generally, to provide a new perspective on models of higher-order concurrent computation. Moreover, we identify a fragment of FICA that is amenable to verification by translation into a particular class of leafy automata. Using a locality property of the latter class, where communication between levels is restricted and every other level is bounded, we show that their emptiness problem is decidable by reduction to Petri net reachability.
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Higher-order constrained Horn clauses (HoCHC) are a semantically-invariant system of higher-order logic modulo theories. With semi-decidable unsolvability over a semi-decidable background theory, HoCHC is suitable for safety verification. Less is known about its relation to larger classes of higher-order verification problems. Motivated by program equivalence, we introduce a coinductive version of HoCHC that enjoys a greatest model property. We define an encoding of higher-order recursion schemes (HoRS) into HoCHC logic programs. Correctness of this encoding reduces decidability of the open HoRS equivalence problem -- and, thus, the LambdaY-calculus Bohm tree equivalence problem -- to semi-decidability of coinductive HoCHC over a complete and decidable theory of trees.
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Automata for unordered unranked trees are relevant for defining schemas and queries for data trees in Json or Xml format. While the existing notions are well-investigated concerning expressiveness, they all lack a proper notion of determinism, which makes it difficult to distinguish subclasses of automata for which problems such as inclusion, equivalence, and minimization can be solved efficiently. In this paper, we propose and investigate different notions of horizontal determinism, starting from automata for unranked trees in which the horizontal evaluation is performed by finite state automata. We show that a restriction to confluent horizontal evaluation leads to polynomial-time emptiness and universality, but still suffers from coNP-completeness of the emptiness of binary intersections. Finally, efficient algorithms can be obtained by imposing an order of horizontal evaluation globally for all automata in the class. Depending on the choice of the order, we obtain different classes of automata, each of which has the same expressiveness as CMso.
A weight normalization procedure, commonly called pushing, is introduced for weighted tree automata (wta) over commutative semifields. The normalization preserves the recognized weighted tree language even for nondeterministic wta, but it is most useful for bottom-up deterministic wta, where it can be used for minimization and equivalence testing. In both applications a careful selection of the weights to be redistributed followed by normalization allows a reduction of the general problem to the corresponding problem for bottom-up deterministic unweighted tree automata. This approach was already successfully used by Mohri and Eisner for the minimization of deterministic weighted string automata. Moreover, the new equivalence test for two wta $M$ and $M$ runs in time $mathcal O((lvert M rvert + lvert Mrvert) cdot log {(lvert Qrvert + lvert Qrvert)})$, where $Q$ and $Q$ are the states of $M$ and $M$, respectively, which improves the previously best run-time $mathcal O(lvert M rvert cdot lvert Mrvert)$.
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