Lindstrom theorems for fragments of first-order logic

Lindstr\"om theorems characterize logics in terms of model-theoretic conditions such as Compactness and the L\"owenheim-Skolem property. Most existing characterizations of this kind concern extensions of first-order logic. But on the other hand, many logics relevant to computer science are fragments or extensions of fragments of first-order logic, e.g., k-variable logics and various modal logics. Finding Lindstr\"om theorems for these languages can be challenging, as most known techniques rely on coding arguments that seem to require the full expressive power of first-order logic. In this paper, we provide Lindstr\"om theorems for several fragments of first-order logic, including the k-variable fragments for k>2, Tarski's relation algebra, graded modal logic, and the binary guarded fragment. We use two different proof techniques. One is a modification of the original Lindstr\"om proof. The other involves the modal concepts of bisimulation, tree unraveling, and finite depth. Our results also imply semantic preservation theorems.Comment: Appears in Logical Methods in Computer Science (LMCS

Investigating Logics for Feasible Computation

The most celebrated open problem in theoretical computer science is, undoubtedly, the problem of whether P = NP. This is actually one instance of the many unresolved questions in the area of computational complexity. Many different classes of decision problems have been defined in terms of the resources needed to recognize them on various models of computation, such as deterministic or non-deterministic Turing machines, parallel machines and randomized machines. Most of the non-trivial questions concerning the inter-relationship between these classes remain unresolved. On the other hand, these classes have proved to be robustly defined, not only in that they are closed under natural transformations, but many different characterizations have independently defined the same classes. One such alternative approach is that of descriptive complexity, which seeks to define the complexity, not of computing a problem, but of describing it in a language such as the Predicate Calculus. It is particularly interesting that this approach yields a surprisingly close correspondence to computational complexity classes. This provides a natural characterization of many complexity classes that is not tied to a particular machine model of computation

Living without Beth and Craig: Definitions and Interpolants in the Guarded and Two-Variable Fragments

In logics with the Craig interpolation property (CIP) the existence of an interpolant for an implication follows from the validity of the implication. In logics with the projective Beth definability property (PBDP), the existence of an explicit definition of a relation follows from the validity of a formula expressing its implicit definability. The two-variable fragment, FO2, and the guarded fragment, GF, of first-order logic both fail to have the CIP and the PBDP. We show that nevertheless in both fragments the existence of interpolants and explicit definitions is decidable. In GF, both problems are 3ExpTime-complete in general, and 2ExpTime-complete if the arity of relation symbols is bounded by a constant c not smaller than 3. In FO2, we prove a coN2ExpTime upper bound and a 2ExpTime lower bound for both problems. Thus, both for GF and FO2 existence of interpolants and explicit definitions are decidable but harder than validity (in case of FO2 under standard complexity assumptions)

Coalgebraic Automata Theory: Basic Results

We generalize some of the central results in automata theory to the abstraction level of coalgebras and thus lay out the foundations of a universal theory of automata operating on infinite objects. Let F be any set functor that preserves weak pullbacks. We show that the class of recognizable languages of F-coalgebras is closed under taking unions, intersections, and projections. We also prove that if a nondeterministic F-automaton accepts some coalgebra it accepts a finite one of the size of the automaton. Our main technical result concerns an explicit construction which transforms a given alternating F-automaton into an equivalent nondeterministic one, whose size is exponentially bound by the size of the original automaton.Comment: 43 page