2 research outputs found
Non-definability of languages by generalized first-order formulas over (N,+)
We consider first-order logic with monoidal quantifiers over words. We show
that all languages with a neutral letter, definable using the addition
numerical predicate are also definable with the order predicate as the only
numerical predicate. Let S be a subset of monoids.
Let LS be the logic closed under quantification over the monoids in S and N
be the class of neutral letter languages. Then we show that: LS[<,+] cap N =
LS[<] Our result can be interpreted as the Crane Beach conjecture to hold for
the logic LS[<,+]. As a corollary of our result we get the result of Roy and
Straubing that FO+MOD[<,+] collapses to FO+MOD[<].
For cyclic groups, we answer an open question of Roy and Straubing, proving
that MOD[<,+] collapses to MOD[<]. Our result also shows that multiplication is
necessary for Barrington's theorem to hold.
All these results can be viewed as separation results for very uniform
circuit classes. For example we separate FO[<,+]-uniform CC0 from
FO[<,+]-uniform ACC0
"Most of" leads to undecidability: Failure of adding frequencies to LTL
Linear Temporal Logic (LTL) interpreted on finite traces is a robust
specification framework popular in formal verification. However, despite the
high interest in the logic in recent years, the topic of their quantitative
extensions is not yet fully explored. The main goal of this work is to study
the effect of adding weak forms of percentage constraints (e.g. that most of
the positions in the past satisfy a given condition, or that sigma is the
most-frequent letter occurring in the past) to fragments of LTL. Such
extensions could potentially be used for the verification of influence networks
or statistical reasoning. Unfortunately, as we prove in the paper, it turns out
that percentage extensions of even tiny fragments of LTL have undecidable
satisfiability and model-checking problems. Our undecidability proofs not only
sharpen most of the undecidability results on logics with arithmetics
interpreted on words known from the literature, but also are fairly simple. We
also show that the undecidability can be avoided by restricting the allowed
usage of the negation, and briefly discuss how the undecidability results
transfer to first-order logic on words.Comment: Full version of FOSSACS 2021 pape