Descriptive complexity theory is an important area in the study of computational complexity. In this direction, it is possible to describe combinatorial problems exclusively by logical methods, without resorting to the use of complicated algorithms. The first work in this direction was written in 1974 by the American mathematician Fagin. The article describes the development of methods of the theory of descriptive complexity.
We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the $\mathrm{AC}_{\mathbb{R}}$ and $\mathrm{NC}_{\mathbb{R}}$-classes for this setting. We give a theorem in the style of Immerman's theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer and we show …
We present several known formalizations of theorems from computational complexity in bounded arithmetic and formalize the PCP theorem in the theory PV1 (no formalization of this theorem was known). This includes a formalization of the existence and of some properties of the (n,d,λ)-graphs in PV1.
This paper investigates the relationship between the Logical Algorithms language (LA) of Ganzinger and McAllester and Constraint Handling Rules (CHR). We present a translation schema from LA to CHR-rp: CHR with rule priorities, and show that the meta-complexity theorem for LA can be applied to a subset of CHR-rp via inverse translation. Inspired by the high-level implementation proposal for Logical Algorithm by Ganzinger and McAllester and based on a new scheduling algorithm, we propose an alternative implementation for CHR-rp that gives strong complexity guarantees and results in a new and ac…
We prove that every triconnected planar graph is definable by a first order sentence that uses at most 15 variables and has quantifier depth at most $11\log_2 n+43$. As a consequence, a canonic form of such graphs is computable in $AC^1$ by the 14-dimensional Weisfeiler-Lehman algorithm. This provides another way to show that the planar graph isomorphism is solvable in $AC^1$.
We study the relations between the notions of highness, lowness and logical depth in the setting of complexity theory. We introduce a new notion of polylog depth based on time bounded Kolmogorov complexity. We show polylog depth satisfies all basic logical depth properties, namely sets in P are not polylog deep, sets with (time bounded)-Kolmogorov complexity greater than polylog are not polylog deep, and only polylog deep sets can polynomially Turing compute a polylog deep set. We prove that if NP does not have p-measure zero, then NP contains polylog deep sets. We show that every high set for…
In this paper two connected Szilard single molecule engines (with different temperature) model of Maxwell's demon are used to demonstrate and analysis the logical compare operation. The logical and physical complexity of compare operations are both showed to be kTln2. Then this limit was used to prove the time complexity lower bound of sorting problem. It confirmed the proposed way to measure the complexity of a problem, provided another evidence of the equivalence between information theoretical and thermodynamic entropies.
We review some semantic and syntactic complexity classes that were introduced to better understand the relationship between complexity classes P and NP. We also define several new complexity classes, some of which are associated with Mersenne numbers, and show their location in the complexity hierarchy.
We discuss the definability of finite graphs in first-order logic with two relation symbols for adjacency and equality of vertices. The logical depth $D(G)$ of a graph $G$ is equal to the minimum quantifier depth of a sentence defining $G$ up to isomorphism. The logical width $W(G)$ is the minimum number of variables occurring in such a sentence. The logical length $L(G)$ is the length of a shortest defining sentence. We survey known estimates for these graph parameters and discuss their relations to other topics (such as the efficiency of the Weisfeiler-Lehman algorithm in isomorphism testing…
Loveland complexity is a variant of Kolmogorov complexity, where it is asked to output separately the bits of the desired string, instead of the string itself. Similarly to the resource-bounded Kolmogorov sets we define Loveland sets. We highlight a structural connection between resource-bounded Loveland sets and some advice complexity classes. This structural connection enables us to map to advice complexity classes some properties of Kolmogorov sets first noticed by Hartmanis and thoroughly investigated in Longpré's thesis: 1. Non-inclusion properties of Loveland sets result in hierarchy pro…
This survey describes, at an introductory level, the algebraic complexity framework originally proposed by Leslie Valiant in 1979, and some of the insights that have been obtained more recently.
We provide a non-explicit separation of the number-on-forehead communication complexity classes RP and NP when the number of players is up to δlog(n) for any δ<1. Recent lower bounds on Set-Disjointness [LS08,CA08] provide an explicit separation between these classes when the number of players is only up to o(loglog(n)).
We present a method for estimating the complexity of an image based on Bennett's concept of logical depth. Bennett identified logical depth as the appropriate measure of organized complexity, and hence as being better suited to the evaluation of the complexity of objects in the physical world. Its use results in a different, and in some sense a finer characterization than is obtained through the application of the concept of Kolmogorov complexity alone. We use this measure to classify images by their information content. The method provides a means for classifying and evaluating the complexity…
We study the problem of generating monomials of a polynomial in the context of enumeration complexity. In this setting, the complexity measure is the delay between two solutions and the total time. We present two new algorithms for restricted classes of polynomials, which have a good delay and the same global running time as the classical ones. Moreover they are simple to describe, use little evaluation points and one of them is parallelizable. We introduce three new complexity classes, TotalPP, IncPP and DelayPP, which are probabilistic counterparts of the most common classes for enumeration …
Complexity theory provides a wealth of complexity classes for analyzing the complexity of decision and counting problems. Despite the practical relevance of enumeration problems, the tools provided by complexity theory for this important class of problems are very limited. In particular, complexity classes analogous to the polynomial hierarchy and an appropriate notion of problem reduction are missing. In this work, we lay the foundations for a complexity theory of hard enumeration problems by proposing a hierarchy of complexity classes and by investigating notions of reductions for enumeratio…
The Descriptive Complexity of Relation Modification Problems
Institute for Theoretical Computer Science, Universität zu Lübeck（吕贝克大学理论计算机科学研究所）；Algorithms and Complexity Group, TU Wien（维也纳工业大学算法与复杂性组）；Institute of Computer Science, Friedrich Schiller University Jena（耶拿弗里德里希·席勒大学计算机科学研究所）
This volume contains the proceedings of the International Workshop on Developments in Implicit Computational complExity (DICE 2010), which took place on March 27-28 2010 in Paphos, Cyprus, as a satellite event of the Joint European Conference on Theory and Practice of Software, ETAPS 2010. Implicit Computational Complexity aims at studying computational complexity without referring to external measuring conditions or particular machine models, but instead by considering restrictions on programming languages or logical principles implying complexity properties. The aim of this workshop was to b…
This paper deals with descriptive complexity of picture languages of any dimension by syntactical fragments of existential second-order logic. - We uniformly generalize to any dimension the characterization by Giammarresi et al. \cite{GRST96} of the class of \emph{recognizable} picture languages in existential monadic second-order logic. - We state several logical characterizations of the class of picture languages recognized in linear time on nondeterministic cellular automata of any dimension. They are the first machine-independent characterizations of complexity classes of cellular automata…
The work is devoted to Computability Logic (CoL) -- the philosophical/mathematical platform and long-term project for redeveloping classical logic after replacing truth} by computability in its underlying semantics (see http://www.cis.upenn.edu/~giorgi/cl.html). This article elaborates some basic complexity theory for the CoL framework. Then it proves soundness and completeness for the deductive system CL12 with respect to the semantics of CoL, including the version of the latter based on polynomial time computability instead of computability-in-principle. CL12 is a sequent calculus system, wh…
We study the computational complexity of sequences of projective varieties. We define analogues of the complexity classes P and NP for these and prove the NP-completeness of a sequence called the universal circuit resultant. This is the first family of compact spaces shown to be NP-complete in a geometric setting.
Treating a conjecture, P^#P != NP, on the separation of complexity classes as an axiom, an implication is found in three manifold topology with little obvious connection to complexity theory. This is reminiscent of Harvey Friedman's work on finitistic interpretations of large cardinal axioms.
Diagonalization in the spirit of Cantor's diagonal arguments is a widely used tool in theoretical computer sciences to obtain structural results about computational problems and complexity classes by indirect proofs. The Uniform Diagonalization Theorem allows the construction of problems outside complexity classes while still being reducible to a specific decision problem. This paper provides a generalization of the Uniform Diagonalization Theorem by extending it to promise problems and the complexity classes they form, e.g. randomized and quantum complexity classes. The theorem requires from …
This volume contains the proceedings of the Second International Workshop on Developments in Implicit Computational complExity (DICE 2011), which took place on April 2-3 2011 in Saarbruecken, Germany, as a satellite event of the Joint European Conference on Theory and Practice of Software, ETAPS 2011. Implicit Computational Complexity aims at studying computational complexity without referring to external measuring conditions or particular machine models, but instead by considering restrictions on programming languages or logical principles implying complexity properties. The aim of this works…
We prove two sets of results concerning computational complexity classes. The first concerns a variation of the random oracle hypothesis posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, P not equal NP with probability 1. This hypothesis was quickly disproven in several ways, most famously in 1992 with the result that IP equals PSPACE, in spite of the classes being shown unequal with probability 1. Here we propose a variation of what it means to be ``large'' using the Ellentuck topology. In this new context, we demonstrate that the set of oracles separating…
We present an approach to non-uniform complexity in which single-pass instruction sequences play a key part, and answer various questions that arise from this approach. We introduce several kinds of non-uniform complexity classes. One kind includes a counterpart of the well-known non-uniform complexity class P/poly and another kind includes a counterpart of the well-known non-uniform complexity class NP/poly. Moreover, we introduce a general notion of completeness for the non-uniform complexity classes of the latter kind. We also formulate a counterpart of the well-known complexity theoretic c…
The second author previously discussed how classical complexity separation conjectures, we call them "axioms", have implications in three manifold topology: polynomial length stings of operations which preserve certain Jones polynomial evaluations cannot produce exponential simplifications of link diagrams. In this paper, we continue this theme, exploring now more subtle separation axioms for quantum complexity classes. Surprisingly, we now find that similar strings are unable to effect even linear simplifications of the diagrams.
In our thesis, we try to shed more light onto the complexity of quantum complexity classes by refining the related part of the hierarchy. First, we review the basic concepts of quantum computing in general. Then, inspired by BQP, we define new complexity classes. They are placed between BPP and PSPACE. We show that they incorporate the current important quantum algorithms. Furthermore, the importance of the unitarity constraint given by quantum mechanics is revealed. Without this requirement, we naturally arrive at the class AWPP, which was up to now thought to be just an artificially defined …
A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed in order to define such problems. One of these forms allows us to define complete problems on ordered structures, and another form to define them on unordered non-Aristotelian structures. Using the canonical forms, logics are developed for complete problems in various complexity classes. Evidence is shown that there cannot be any complete problem on Aristotelian structures for several complexity classes. Our approach is extended beyond complete problems. Using a simil…
Proving lower bounds remains the most difficult of tasks in computational complexity theory. In this paper, we show that whereas most natural NP-complete problems belong to NLIN (linear time on nondeterministic RAMs), some of them, typically the planar versions of many NP-complete problems are recognized by nondeterministic RAMs in linear time and sublinear space. The main results of this paper are the following: as the second author did for NLIN, we give exact logical characterizations of nondeterministic polynomial time-space complexity classes; we derive from them a class of problems, which…
In a recent paper, the author has shown how Interaction Graphs models for linear logic can be used to obtain implicit characterisations of non-deterministic complexity classes. In this paper, we show how this semantic approach to Implicit Complexity Theory (ICC) can be used to characterise deterministic and probabilistic models of computation. In doing so, we obtain correspondences between group actions and both deterministic and probabilistic hierarchies of complexity classes. As a particular case, we provide the first implicit characterisations of the classes PLogspace (un-bounded error prob…
Reversible computing can reduce the energy dissipation of computation, which can improve cost-efficiency in some contexts. But the practical applicability of this method depends sensitively on the space and time overhead required by reversible algorithms. Time and space complexity classes for reversible machines match conventional ones, but we conjecture that the joint space-time complexity classes are different, and that a particular reduction by Bennett minimizes the space-time product complexity of general reversible computations. We provide an oracle-relativized proof of the separation, an…
Over the past few decades, non-monotonic reasoning has developed to be one of the most important topics in computational logic and artificial intelligence. Different ways to introduce non-monotonic aspects to classical logic have been considered, e.g., extension with default rules, extension with modal belief operators, or modification of the semantics. In this survey we consider a logical formalism from each of the above possibilities, namely Reiter's default logic, Moore's autoepistemic logic and McCarthy's circumscription. Additionally, we consider abduction, where one is not interested in …
Quantum information scrambling has attracted much attention amid the effort to reconcile the conflict between quantum-mechanical unitarity and the thermalizaiton-irreversibility in many-body systems. Here we propose an unconventional mechanism to generate quantum information scrambling through a high-complexity mapping from logical to physical degrees-of-freedom that hides the logical information into non-separable many-body-correlations. Corresponding to this mapping, we develop an algorithm to efficiently sample a Slater-determinant wavefunction and compute all physical observables in dynami…
