We review PLTLf and PLDLf, the pure-past versions of the well-known logics on finite traces LTLf and LDLf, respectively. PLTLf and PLDLf are logics about the past, and so scan the trace backwards from the end towards the beginning. Because of this, we can exploit a foundational result on reverse languages to get an exponential improvement, over LTLf /LDLf , for computing the corresponding DFA. This exponential improvement is reflected in several forms of sequential decision making involving temporal specifications, such as planning and decision problems in non-deterministic and non-Markovian domains. Interestingly, PLTLf (resp., PLDLf ) has the same expressive power as LTLf (resp., LDLf ), but transforming a PLTLf (resp., PLDLf ) formula into its equivalent LTLf (resp.,LDLf) is quite expensive. Hence, to take advantage of the exponential improvement, properties of interest must be directly expressed in PLTLf /PLDLf .
We investigate partial orders that are computable, in a precise sense, by finite automata. Our emphasis is on trees and linear orders. We study the relationship between automatic linear orders and trees in terms of rank functions that are related to Cantor-Bendixson rank. We prove that automatic linear orders and automatic trees have finite rank. As an application we provide a procedure for deciding the isomorphism problem for automatic ordinals. We also investigate the complexity and definability of infinite paths in automatic trees. In particular, we show that every infinite path in an automatic tree with countably many infinite paths is a regular language.
A structure has a (finite-string)automatic presentationif the elements of its domain can be named by finite strings in such a way that the coded domain and the coded atomic operations are recognised by synchronous multitape automata. Consequently, every structure with an automatic presentation has a decidable first-order theory. The problems surveyed here include the classification of classes of structures with automatic presentations, the complexity of the isomorphism problem, and the relationship between definability and recognisability.
We revisit the parameterized model checking problem for token-passing systems and specifications in indexed CTL * \X. Namjoshi (1995, 2003) have shown that parameterized model checking of indexed CTL * \X in uni-directional token rings can be reduced to checking rings up to some cutoff size. Clarke et al. (2004) have shown a similar result for general topologies and indexed LTL\X, provided processes cannot choose the directions for sending or receiving the token.We unify and substantially extend these results by systematically exploring fragments of indexed CTL * \X with respect to general topologies. For each fragment we establish whether a cutoff exists, and for some concrete topologies, such as rings, cliques and stars, we infer small cutoffs.Finally, we show that the problem becomes undecidable, and thus no cutoffs exist, if processes are allowed to choose the directions in which they send or from which they receive the token.
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