Infinitary Logic and Inductive Definability Over Finite Structures

The extensions of first-order logic with a least fixed point operators (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [AV91b] investigated the relation of these two logics in the absence of an ordering, using a mchine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables, Lω∞ω (see, for instance, [KV90]). We present a treatment of the results in [AV91b] from this point of view. In particular, we show that we can write a formula of FO + LFP and P from ordered structures to classes of structures where every element is definable. We also settle a conjecture mentioned in [AV91b] by showing that FO + LFP in properly contained in the polynomial time computable fragment of Lω∞ω, raising the question of whether the latter fragment is a recursively enumerable class

Towards Automated Reasoning in Herbrand Structures

Herbrand structures have the advantage, computationally speaking, of being guided by the definability of all elements in them. A salient feature of the logics induced by them is that they internally exhibit the induction scheme, thus providing a congenial, computationally-oriented framework for formal inductive reasoning. Nonetheless, their enhanced expressivity renders any effective proof system for them incomplete. Furthermore, the fact that they are not compact poses yet another prooftheoretic challenge. This paper offers several layers for coping with the inherent incompleteness and non-compactness of these logics. First, two types of infinitary proof system are introduced—one of infinite width and one of infinite height—which manipulate infinite sequents and are sound and complete for the intended semantics. The restriction of these systems to finite sequents induces a completeness result for finite entailments. Then, in search of effectiveness, two finite approximations of these systems are presented and explored. Interestingly, the approximation of the infinite-width system via an explicit induction scheme turns out to be weaker than the effective cyclic fragment of the infinite-height system

Enhancing Fixed Point Logic with Cardinality Quantifiers

Let Q IPP be any quantifier such that FO(QIFP), first-order logic enhanced with Q IPP and its vectorizations, equals inductive fixed point logic, IFP in expressive power. It is known that for certain quantifiers Q, the equivalence FO(QIFP) ≡ IFP is no longer true if Q is added on both sides. Rather, we have FO (QIFP, Q) < IFP(Q) in such cases. We extend these results to a great variety of quantifiers, namely all unbounded simple cardinality quantifiers. Our argument also applies to partial fixed point logic, PFP. In order to establish an analogous result for least fixed point logic, LFP, we exhibit a general method to pass from arbitrary quantifiers to monotone quantifiers. Our proof shows that the three isomorphism problem is not definable in, infinitary logic extended with all monadic quantifiers and their vectorizations, where a finite bound is imposed to the number of variables as well as to the number of nested quantifiers in Q1. This strengthens a result of Etessami and Immerman by which tree isomorphism is not definable in TC + COUNTIN

Linear Datalog and Bounded Path Duality of Relational Structures

In this paper we systematically investigate the connections between logics with a finite number of variables, structures of bounded pathwidth, and linear Datalog Programs. We prove that, in the context of Constraint Satisfaction Problems, all these concepts correspond to different mathematical embodiments of a unique robust notion that we call bounded path duality. We also study the computational complexity implications of the notion of bounded path duality. We show that every constraint satisfaction problem \csp(\best) with bounded path duality is solvable in NL and that this notion explains in a uniform way all families of CSPs known to be in NL. Finally, we use the results developed in the paper to identify new problems in NL