29 research outputs found
A Fragment of Dependence Logic Capturing Polynomial Time
In this paper we study the expressive power of Horn-formulae in dependence
logic and show that they can express NP-complete problems. Therefore we define
an even smaller fragment D-Horn* and show that over finite successor structures
it captures the complexity class P of all sets decidable in polynomial time.
Furthermore we study the question which of our results can ge generalized to
the case of open formulae of D-Horn* and so-called downwards monotone
polynomial time properties of teams
Proof Complexity of Systems of (Non-Deterministic) Decision Trees and Branching Programs
This paper studies propositional proof systems in which lines are sequents of decision trees or branching programs, deterministic or non-deterministic. Decision trees (DTs) are represented by a natural term syntax, inducing the system LDT, and non-determinism is modelled by including disjunction, ?, as primitive (system LNDT). Branching programs generalise DTs to dag-like structures and are duly handled by extension variables in our setting, as is common in proof complexity (systems eLDT and eLNDT).
Deterministic and non-deterministic branching programs are natural nonuniform analogues of log-space (L) and nondeterministic log-space (NL), respectively. Thus eLDT and eLNDT serve as natural systems of reasoning corresponding to L and NL, respectively.
The main results of the paper are simulation and non-simulation results for tree-like and dag-like proofs in LDT, LNDT, eLDT and eLNDT. We also compare them with Frege systems, constant-depth Frege systems and extended Frege systems
Frameworks for logically classifying polynomial-time optimisation problems.
We show that a logical framework, based around a fragment of existential second-order logic formerly proposed by others so as to capture the class of polynomially-bounded P-optimisation problems, cannot hope to do so, under the assumption that P ≠NP. We do this by exhibiting polynomially-bounded maximisation and minimisation problems that can be expressed in the framework but whose decision versions are NP-complete. We propose an alternative logical framework, based around inflationary fixed-point logic, and show that we can capture the above classes of optimisation problems. We use the inductive depth of an inflationary fixed-point as a means to describe the objective functions of the instances of our optimisation problems
More on Descriptive Complexity of Second-Order HORN Logics
This paper concerns Gradel's question asked in 1992: whether all problems
which are in PTIME and closed under substructures are definable in second-order
HORN logic SO-HORN. We introduce revisions of SO-HORN and DATALOG by adding
first-order universal quantifiers over the second-order atoms in the bodies of
HORN clauses and DATALOG rules. We show that both logics are as expressive as
FO(LFP), the least fixed point logic. We also prove that FO(LFP) can not define
all of the problems that are in PTIME and closed under substructures. As a
corollary, we answer Gradel's question negatively
Universal First-Order Logic is Superfluous for NL, P, NP and coNP
In this work we continue the syntactic study of completeness that began with
the works of Immerman and Medina. In particular, we take a conjecture raised by
Medina in his dissertation that says if a conjunction of a second-order and a
first-order sentences defines an NP-complete problems via fops, then it must be
the case that the second-order conjoint alone also defines a NP-complete
problem. Although this claim looks very plausible and intuitive, currently we
cannot provide a definite answer for it. However, we can solve in the
affirmative a weaker claim that says that all ``consistent'' universal
first-order sentences can be safely eliminated without the fear of losing
completeness. Our methods are quite general and can be applied to complexity
classes other than NP (in this paper: to NLSPACE, PTIME, and coNP), provided
the class has a complete problem satisfying a certain combinatorial property
Capturing the polynomial hierarchy by second-order revised Krom logic
We study the expressive power and complexity of second-order revised Krom
logic (SO-KROM). On ordered finite structures, we show that its
existential fragment -KROM equals -KROM, and
captures NL. On all finite structures, for , we show that
equals -KROM if is even, and
equals -KROM if is odd. The result gives an alternative
logic to capture the polynomial hierarchy. We also introduce an extended
version of second-order Krom logic (SO-EKROM). On ordered finite structures, we
prove that SO-EKROM collapses to -EKROM and equals . Both
of SO-EKROM and -EKROM capture co-NP on ordered finite structures
Definability by Horn Formulas and Linear Time on Cellular Automata
We establish an exact logical characterization of linear time complexity of cellular automata of dimension d, for any fixed d: a set of pictures of dimension d belongs to this complexity class iff it is definable in existential second-order logic restricted to monotonic Horn formulas with built-in successor function and d+1 first-order variables. This logical characterization is optimal modulo an open problem in parallel complexity. Furthermore, its proof provides a systematic method for transforming an inductive formula defining some problem into a cellular automaton that computes it in linear time
On Constraint Satisfaction Problems below P
Symmetric Datalog, a fragment of the logic programming language Datalog, is conjectured to capture all constraint satisfaction problems (CSP) in L. Therefore developing tools that help us understand whether or not a CSP can be defined in symmetric Datalog is an important task. It is widely known that a CSP is definable in Datalog and linear Datalog iff that CSP has bounded treewidth and bounded pathwidth duality, respectively. In the case of symmetric Datalog, Bulatov, Krokhin and Larose ask for such a duality [2008]. We provide two such dualities, and give applications. In particular, we give a short and simple new proof of the result of Dalmau and Larose that "Maltsev + Datalog -> symmetric Datalog" [2008].
In the second part of the paper, we provide some evidence for the conjecture of Dalmau [2002] that every CSP in NL is definable in linear Datalog. Our results also show that a wide class of CSPs ---CSPs which do not have bounded pathwidth duality (e.g. the P-complete Horn-3Sat problem)--- cannot be defined by any polynomial size family of monotone read-once nondeterministic branching programs.
We consider the following restrictions of the previous models: read-once linDat(suc) (1-linDat(suc)), and monotone readonce nondeterministic branching programs (mnBP1). Although restricted, these models can still define NL-complete problems such as directed st-Connectivity, and also nontrivial problems in NL which are not definable in linear Datalog. We show that any CSP definable by a 1-linDat(suc) program or by a poly-size family of mnBP1s can also be defined by a linear Datalog program. It also follows that a wide class of CSPs ---CSPs which do not have bounded pathwidth duality (e.g. the P-complete Horn-3Sat problem)--- cannot be defined by any 1-linDat(suc) program or by any poly-size family of mnBP1s