Jerzy Marcinkowski

The Undecidability of the Logic of Subintervals

The Halpern--Shoham logic is a modal logic of time intervals. Some effort has been put in last ten years to classify fragments of this beautiful logic with respect to decidability of its satisfiability problem. We complete this classification by showing ---what we believe is quite an unexpected result---that the logic of subintervals, the fragment of the Halpern--Shoham logic where only the operator “during”, or D, is allowed, is undecidable over discrete structures. This is surprising as this, apparently very simple, logic is decidable over dense orders and its reflexive variant is known to be decidable over discrete structures. Our result subsumes a lot of previous undecidability results of fragments that include D.

A Toolkit for First Order Extensions of Monadic Games

In 1974 R. Fagin proved that properties of structures which are in NP are exactly the same as those expressible by existential second order sentences, that is sentences of the form: there exist P→ such that ϕ, where P→ is a tuple of relation symbols and ϕ is a first order formula. Fagin was also the first to study monadic NP: the class of properties expressible by existential second order sentences where all the quantified relations are unary. In [AFS00] Ajtai, Fagin and Stockmeyer introduce closed monadic NP: the class of properties which can be expressed by a kind of monadic second order existential formula, where the second order quantifiers can interleave with first order quantifiers. In order to prove that such alternation of quantifiers gives substantial additional expressive power they construct graph properties P1 and P2: P1 is expressible by a sentence with the quantifier prefix in the class (∃∀)* ∃*(∃∀)* 1 but not by a boolean combination of sentences from monadic NP (i.e with the prefix of the form ∃*(∃∀)*) and P2 is expressible by a sentence ∃*(∃∀)* ∃*(∃∀)* but not by a Boolean combination of sentences of the form (∃∀)* ∃*(∃∀)*. A natural question arises here whether the hierarchy inside closed monadic NP, defined by the number of blocks of second order existential quantifiers, is strict. In this paper we present a technology for proving some non expressibility results for monadic second order logic. As a corollary we get a new, easy, proof of the two results from [AFS00] mentioned above. With our technology we can also make a first small step towards an answer to the hierarchy question by showing that the hierarchy inside closed monadic NP does not collapse on a first order level. The monadic complexity of properties definable in Kozens mu-calculus is also considered as our technology also applies to the mu-calculus itself.

The 3 Frenchmen Method Proves Undecidability of the Uniform Boundedness for Single Recursive Rule Ternary DATALOG Programs

DATALOG is the language of logic programs without function symbols. It is considered to be the paradigmatic database query language. If it is possible to eliminate the recursion from the program then it is uniformly bounded. We show that the uniform boundedness is undecidable for ternary DATALOG programs containing only one recursive rule, and for linear programs of arity 3. The proof is based on the discovery of, how we call it, Achilles-Turtle machine. It computes the subsequent iterations of a Conway function and is, up to our knowledge, the simplest known universal machine.

Directed Reachability: From Ajtai-Fagin to Ehrenfeucht-Fraïssé Games

In 1974 Ronald Fagin proved that properties of structures which are in NP are exactly the same as those expressible by existential second order sentences, that is sentences of the form ∃P→Φ, where P→is a tuple of relation symbols. and Φ is a first order formula. Fagin was also the first to study monadic NP: the class of properties expressible by existential second order sentences where all quantified relations are unary. In their very difficult paper [AF90] Ajtai and Fagin show that directed reachability is not in monadic NP. In [AFS97] Ajtai, Fagin and Stockmeyer introduce closed monadic NP: the class of properties which can be expressed by a kind of monadic second order existential formula, where the second order quantifiers can interleave with first order quantifiers. Among other results they show that directed reachability is expressible by a formula of the form ∃P¬x∃P1 Φ, where P and P1 are unary relation symbols and Φ is first order. They state the question if this property is in the positive first order closure of monadic NP, that is if it is expressible by a sentence of the form Q→x∃P→Φ, where Q→x is a tuple of first order quantifiers and PΦis a tuple of unary relation symbols. In this paper we give a negative solution to the problem.

DATALOG SIRUPs uniform boundedness is undecidable

DATALOG is the paradigmatic database query language. If it is possible to eliminate recursion from a DATALOG program then it is uniformly bounded. Since uniformly bounded programs can be executed in parallel constant time, the possibility of automated boundedness detection is an important issue, and has been studied in many papers. In this paper we solve one of the most famous open problems in the theory of deductive databases (see e.g. P.C. Kanellakis, Elements of Relational Database Theory in Handbook of Theoretical Computer Science) showing that uniform boundedness is undecidable for single rule programs (called also sirups).

A Horn Clause that Implies and Undecidable Set of Horn Clauses

In this paper we prove that there exists a Horn clauseHsuch that the problem: given a Horn clause G. Is G a consequence of H? is not recursive. Equivalently, there exists a one-clause PROLOG program such that there is no PROLOG implementation answering TRUE if the program implies a given goal and FALSE otherwise. We give a short survey of earlier results concerning clause implication and prove a classical Linial-Post theorem as a consequence of one of them.

On the BDD/FC conjecture

Bounded Derivation Depth property (BDD) and Finite Controllability (FC) are two properties of sets of datalog rules and tuple generating dependencies (known as Datalog3 programs), which recently attracted some attention. We conjecture that the first of these properties implies the second, and support this conjecture by some evidence proving, among other results, that it holds true for all theories over binary signature.

On the expressive power of graph logic

Graph Logic, a query language being a sublogic of Monadic Second Order Logic is studied in [CGG02]. In the paper [DGG04] the expressiveness power of Graph Logic is examined, and it is shown, for many MSO properties, how to express them in Graph Logic. But despite of the positive examples, it is conjectured there that Graph Logic is strictly less expressive than MSO Logic. Here we give a proof of this conjecture.

Converging to the chase – A tool for finite controllability

Abstract We solve a problem, stated in [5] , showing that Sticky Datalog∃, defined in the cited paper as an element of the Datalog± project, has the Finite Controllability property, which means that whenever a query Ψ is not logically implied by a set of atoms D and a Sticky Datalog∃ theory T a finite structure M can be found such that M ⊨ D , T , but M ⊭ Ψ . In order to do that, we develop a technique, which we believe can have further applications, of approximating C h a s e ( T , D ) , for a database instance D and a set of tuple generating dependencies and Datalog rules T , by an infinite sequence of finite structures, all of them being models of T and D .

On a semantic subsumption test

We observe, that subsumption of clauses (in the language of first order logic), so far understood as a syntactic notion, can also be defined by semantical means. Subsumption is NP-complete and testing subsumption takes roughly half of the running time of a typical first order resolution-based theorem prover. We also give some experimental evidence, that replacing syntactic indexing for subsumption by our semantic Monte Carlo technique can, in some situations, significantly decrease the cost of subsumption testing.