By Gabriele Puppis

Since 2002, FoLLI awards an annual prize for a great dissertation within the fields of good judgment, Language, and data. This publication relies at the Ph.D. thesis of Gabriele Puppis, who used to be the winner of the E.W. Beth dissertation award for 2007.

Puppis' thesis makes a speciality of common sense and Computation and, extra in particular, on automata-based decidability innovations for time granularity and on a brand new strategy for figuring out Monadic moment Order theories of bushes. the consequences provided characterize an important step in the direction of a greater knowing of the adjustments in granularity degrees that people make so simply in cognition of time, house, and different phenomena, while their logical and computational constitution poses tough conceptual and computational challenges.

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**Example text**

3 Compact and Tractable Representations 41 unique (up to isomorphism) among all equivalent NCSSA, as it happens, for instance, with the case of deterministic ﬁnite automata. Hereafter, we denote by n(A) the number of states of an NCCSA A. , by partitioning the state space or by exploiting characterizations of recognizable words in terms of suitable expressions. In the following, we tackle both the complexity optimization problem and the state optimization problem by using dynamic programming, namely, by computing optimal NCSSA starting from smaller (optimal) ones in a bottom-up fashion.

As a matter of fact, this also proves that any RCSSA can be eﬀectively translated into an equivalent SSA. For the sake of completeness, below we explicitly describe the translation from RCSSA to SSA. From such a result and from the fact that the equivalence problem for SSA is solvable in linear time, we have that the equivalence problem for RCSSA is in EXPTIME. Proposition 2. Any RCSSA A can be eﬀectively translated into an equivalent SSA A having size at most exponential in the size of A (note that the size of A comprises the size of the binary expansion of every constant that appears inside a guard of a secondary transition of A).

The (unique) inﬁnite word recognized by A is given by the sequence ↓3 ρ = ↓3 ρ(1) ↓3 ρ(2) ↓3 ρ(3) .... 5 depicts a CSSA representing the time granularity BusinessWeek. Such an automaton uses one counter i, which is initialized to 0. , i ← i + 1); the guards of the secondary transitions are speciﬁed as additional annotations, having the form of constraints like i = 4, of the corresponding arrows. 6, which depicts a CSSA representing the time granularity Month over the temporal domain of days. 22 2 Word Automata and Time Granularities j mod 12 = 0 i ← 0; j ← 0; k ← k + 1 s0 j ← i mod 26 = 0 s1 k mod 400 = 200 ∧ k mod 400 = 300 s2 s6 j mod 12 = 10 0; k mod 4 = 0 ∧ k mod 400 = 100 ∧ j mod 12 = 8 ∧ ← 1 s8 j mod 12 = 5 ∧ i + j mod 12 = 1 s5 j j mod 12 = 3 ∧ s7 i ← 0; j ← j + 1 s3 s4 i←i+1 i ← 0; j ← j + 1 i ← 0; j ← j + 1 Fig.