When Input Integers are Given in the Unary Numeral Representation

Many NP-complete problems take integers as part of their input instances. These input integers are generally binarized, that is, provided in the form of the "binary" numeral representation, and the lengths of such binary forms are used as a basis unit to measure the computational complexity of the problems. In sharp contrast, the "unarization" (or the "unary" numeral representation) of numbers has been known to bring a remarkably different effect onto the computational complexity of the problems. When no computational-complexity difference is observed between binarization and unarization of instances, on the contrary, the problems are said to be strong NP-complete. This work attempts to spotlight an issue of how the unarization of instances affects the computational complexity of various combinatorial problems. We present numerous NP-complete (or even NP-hard) problems, which turn out to be easily solvable when input integers are represented in unary. We then discuss the computational complexities of such problems when taking unary-form integer inputs. We hope that a list of such problems signifies the structural differences between strong NP-completeness and non-strong NP-completeness.Comment: (A4, 10pt, 12 pages, 1 figure) This is a preliminary report of the current work, which has appeared in the Proceedings of the 24th Italian Conference on Theoretical Computer Science (ICTCS 2023), Palermo, Italy, September 13--15, 2023, CEUR Workshop Proceedings (CEUR-WS.org

Program schemes with deep pushdown storage.

Inspired by recent work of Meduna on deep pushdown automata, we consider the computational power of a class of basic program schemes, TeX, based around assignments, while-loops and non- deterministic guessing but with access to a deep pushdown stack which, apart from having the usual push and pop instructions, also has deep-push instructions which allow elements to be pushed to stack locations deep within the stack. We syntactically define sub-classes of TeX by restricting the occurrences of pops, pushes and deep-pushes and capture the complexity classes NP and PSPACE. Furthermore, we show that all problems accepted by program schemes of TeX are in EXPTIME

Extended Computation Tree Logic

We introduce a generic extension of the popular branching-time logic CTL which refines the temporal until and release operators with formal languages. For instance, a language may determine the moments along a path that an until property may be fulfilled. We consider several classes of languages leading to logics with different expressive power and complexity, whose importance is motivated by their use in model checking, synthesis, abstract interpretation, etc. We show that even with context-free languages on the until operator the logic still allows for polynomial time model-checking despite the significant increase in expressive power. This makes the logic a promising candidate for applications in verification. In addition, we analyse the complexity of satisfiability and compare the expressive power of these logics to CTL* and extensions of PDL

Turing machines with access to history

AbstractWe study remembering Turing machines, that is Turing machines with the capability to access freely the history of their computations. These devices can detect in one step via the oracle mechanism whether the storage tapes have exactly the same contents at the moment of inquiry as at some past moment in the computation. The s(n)-space-bounded remembering Turing machines are shown to be able to recognize exactly the languages in the time-complexity class determined by bounds exponential in s(n). This is proved for deterministic, non-deterministic, and alternating Turing machines

Unboundedness Problems for Machines with Reversal-Bounded Counters

We consider a general class of decision problems concerning formal languages, called (one-dimensional) unboundedness predicates, for automata that feature reversal-bounded counters (RBCA). We show that each problem in this class reduces-non-deterministically in polynomial time to the same problem for just nite automata. We also show an analogous reduction for automata that have access to both a push- down stack and reversal-bounded counters (PRBCA). This allows us to answer several open questions: For example, we settle the complexity of deciding whether a given (P)RBCA language L is bounded, meaning whether there exist words w1, . . . , wn with L ⊆ w1∗ · · · wn∗ . For PRBCA, even decidability was open. Our methods also show that there is no language of a (P)RBCA of intermediate growth. Part of our proof is likely of independent interest: We show that one can translate an RBCA into a machine with Z-counters in logarithmic space

Automata theory and formal languages

These lecture notes present some basic notions and results on Automata Theory, Formal Languages Theory, Computability Theory, and Parsing Theory. I prepared these notes for a course on Automata, Languages, and Translators which I am teaching at the University of Roma Tor Vergata. More material on these topics and on parsing techniques for context-free languages can be found in standard textbooks such as [1, 8, 9]. The reader is encouraged to look at those books. A theorem denoted by the triple k.m.n is in Chapter k and Section m, and within that section it is identified by the number n. Analogous numbering system is used for algorithms, corollaries, definitions, examples, exercises, figures, and remarks. We use ‘iff’ to mean ‘if and only if’. Many thanks to my colleagues of the Department of Informatics, Systems, and Production of the University of Roma Tor Vergata. I am also grateful to my stu- dents and co-workers and, in particular, to Lorenzo Clemente, Corrado Di Pietro, Fulvio Forni, Fabio Lecca, Maurizio Proietti, and Valerio Senni for their help and encouragement. Finally, I am grateful to Francesca Di Benedetto, Alessandro Colombo, Donato Corvaglia, Gioacchino Onorati, and Leonardo Rinaldi of the Aracne Publishing Com- pany for their kind cooperation

Reasoning about reversal-bounded counter machines

International audienceIn this paper, we present a short survey on reversal-bounded counter machines. It focuses on the main techniques for model-checking such counter machines with specifications expressed with formulae from some linear-time temporal logic. All the decision procedures are designed by translation into Presburger arithmetic. We provide a proof that is alternative to Ibarra's original one for showing that reachability sets are effectively definable in Presburger arithmetic. Extensions to repeated control state reachability and to additional temporal properties are discussed in the paper. The article is written to the honor of Professor Ewa Orłowska and focuses on several topics that are developped in her works

Real-time multipushdown and multicounter automata networks and hierarchies

Ph.D.William I. Grosk