Dependency Tree Automata

Stirling, Colin

doi:10.1007/978-3-642-00596-1_8

Cited by 12 publications

(11 citation statements)

References 14 publications

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“…A run is accepting if it ends in a final state. One can think of these automata as a deterministic version of Stirling's dependency tree automata [19] restricted to words.…”

Section: Regular Sets Of Collapsible Stacksmentioning

confidence: 99%

Recursion Schemes and Logical Reflection

Broadbent

Carayol

Ong

et al. 2010

2010 25th Annual IEEE Symposium on Logic in Computer Science

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Let R be a class of generators of node-labelled infinite trees, and L be a logical language for describing correctness properties of these trees. Given R ∈ R and ϕ ∈ L, we say that Rϕ is a ϕ-reflection of R just if (i) R and Rϕ generate the same underlying tree, and (ii) suppose a node u of the tree [[R]] generated by R has label f , then the label of the node u of] is the computation tree of a program R, we may regard Rϕ as a transform of R that can internally observe its behaviour against a specification ϕ. We say that R is (constructively) reflective w.r.t. L just if there is an algorithm that transforms a given pair (R, ϕ) to Rϕ. In this paper, we prove that higher-order recursion schemes are reflective w.r.t. both modal µ-calculus and monadic second order (MSO) logic. To obtain this result, we give the first characterisation of the winning regions of parity games over the transition graphs of collapsible pushdown automata (CPDA): they are regular sets defined by a new class of automata. (Order-n recursion schemes are equi-expressive with order-n CPDA for generating trees.) As a corollary, we show that these schemes are closed under the operation of MSO-interpretation followed by tree unfoldingà la Caucal.

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“…A run is accepting if it ends in a final state. One can think of these automata as a deterministic version of Stirling's dependency tree automata [19] restricted to words.…”

Section: Regular Sets Of Collapsible Stacksmentioning

confidence: 99%

Recursion Schemes and Logical Reflection

Broadbent

Carayol

Ong

et al. 2010

2010 25th Annual IEEE Symposium on Logic in Computer Science

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“…On the one hand, type-checking games generalise the property-checking games (such as parity games) which are played over trees, to games which are played over trees with binders; on the other, they may be viewed as an extension of Stirling's dependency tree automata [21] to infinite (binding) trees with ω-regular conditions. The table below summarises the various aspects of types as a higher-order analogue of alternating parity tree automata (APT).…”

Section: Contributions Of the Papermentioning

confidence: 99%

“…Remark 7. The type-checking games are closely related to (alternating) dependency tree automata [21], automata for (finite) trees with binders like the λ-abstraction of Böhm trees. As recogniser of well-kinded finite Böhm trees (i.e.…”

Section: Type-checking Games Over Böhm Treesmentioning

confidence: 99%

Compositional higher-order model checking via ω -regular games over Böhm trees

Tsukada

Ong

2014

Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Nin

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We introduce type-checking games, which are ω-regular games over Böhm trees, determined by a type of the Kobayashi-Ong intersection type system. These games are a higher-type extension of parity games over trees, determined by an alternating parity tree automaton. However, in contrast to these games over trees, the "game boards" of our type-checking games are composable, using the composition of Böhm trees. Moreover the winner (and winning strategies) of a composite game is completely determined by the respective winners (and winning strategies) of the component games.To our knowledge, type-checking games give the first compositional analysis of higher-order model checking, or the model checking of trees generated by recursion schemes. We study a highertype analogue of higher-order model checking, namely, the problem to decide the winner of a type-checking game over the Böhm tree generated by an arbitrary λY-term. We introduce a new typeassignment system and use it to prove that the problem is decidable.On the semantic side, we develop a novel (two-level) arena game model for type-checking games, which is a cartesian closed category equipped with parametric monad and comonad that themselves form a parametrised adjunction.

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“…Terms are represented as special kinds of tree (that we call binding trees in [12,14]) with dummy lambdas and an explicit binding relation. A term of the form y a is represented as a tree with a single node labelled y a .…”

Section: Preliminariesmentioning

confidence: 99%

“…If n is a leaf node (of D(z) or C(x)) then the position is final and play ends; otherwise, player ∀ chooses a successor ni of n and the next position is ni θ. (15) or (17), is associated with (2), the successor of (1), in D(z 1 , z 2 ) as shown by the entry for (12) in the current look-up table. The next position is at the observable node (13).…”

Section: Definition 12mentioning

confidence: 99%