Agi Kurucz

On the computational complexity of decidable fragments of first-order linear temporal logics

We study the complexity of some fragments of first-order temporal logic over natural numbers time. The one-variable fragment of linear first-order temporal logic even with sole temporal operator /spl square/ is EXPSPACE-complete (this solves an open problem of J. Halpern and M. Vardi (1989)). So are the one-variable, two-variable and monadic monodic fragments with Until and Since. If we add the operators O/sup n/, with n given in binary, the fragment becomes 2EXPSPACE-complete. The packed monodic fragment has the same complexity as its pure first-order part - 2EXPTIME-complete. Over any class of flows of time containing one with an infinite ascending sequence - e.g., rationals and real numbers time, and arbitrary strict linear orders - we obtain EXPSPACE lower bounds (which solves an open problem of M. Reynolds (1997)). Our results continue to hold if we restrict to models with finite first-order domains.

Representable Cylindric Algebras and Many-Dimensional Modal Logics

The equationally expressible properties of the cylindrifications and the diagonals in finite-dimensional representable cylindric algebras can be divided into two groups: (i) ‘One-dimensional’ properties describing individual cylindrifications. These can be fully characterised by finitely many equations saying that each c i , for i < n, is a normal (c i 0 = 0), additive (c i (x+y) = c i x+c i y) and complemented closure operator:

Undecidable Propositional Bimodal Logics and One-Variable First-Order Linear Temporal Logics with Counting

x \leqslant C_i x\quad \quad C_i c_i x \leqslant C_i x\quad \quad C_i \left( { - C_i x} \right) \leqslant - C_i x.

Towards a natural language semantics without functors and operands

(2.0.1) (ii) ‘Dimension-connecting’ properties, that is, equations describing the diagonals and interaction between different cylindrifications and/or diagonals. These properties are much harder to describe completely, and there are many results in the literature on their complexity.

A Note on Axiomatisations of Two-Dimensional Modal Logics

We investigate the satisfiability problem for Horn fragments of the Halpern-Shoham interval temporal logic depending on the type (box or diamond) of the interval modal operators, the type of the underlying linear order (discrete or dense), and the type of semantics for the interval relations (reflexive or irreflexive). For example, we show that satisfiability of Horn formulas with diamonds is undecidable for any type of linear orders and semantics. On the contrary, satisfiability of Horn formulas with boxes is tractable over both discrete and dense orders under the reflexive semantics and over dense orders under the irreflexive semantics but becomes undecidable over discrete orders under the irreflexive semantics. Satisfiability of binary Horn formulas with both boxes and diamonds is always undecidable under the irreflexive semantics.

Many-dimensional modal logics: theory and applications

First-order temporal logics are notorious for their bad computational behavior. It is known that even the two-variable monadic fragment is highly undecidable over various linear timelines, and over branching time even one-variable fragments might be undecidable. However, there have been several attempts at finding well-behaved fragments of first-order temporal logics and related temporal description logics, mostly either by restricting the available quantifier patterns or by considering sub-Boolean languages. Here we analyze seemingly “mild” extensions of decidable one-variable fragments with counting capabilities, interpreted in models with constant, decreasing, and expanding first-order domains. We show that over most classes of linear orders, these logics are (sometimes highly) undecidable, even without constant and function symbols, and with the sole temporal operator “eventually.” We establish connections with bimodal logics over 2D product structures having linear and “difference” (inequality) component relations and prove our results in this bimodal setting. We show a general result saying that satisfiability over many classes of bimodal models with commuting “unbounded” linear and difference relations is undecidable. As a byproduct, we also obtain new examples of finitely axiomatizable but Kripke incomplete bimodal logics. Our results generalize similar lower bounds on bimodal logics over products of two linear relations, and our proof methods are quite different from the known proofs of these results. Unlike previous proofs that first “diagonally encode” an infinite grid and then use reductions of tiling or Turing machine problems, here we make direct use of the grid-like structure of product frames and obtain lower-complexity bounds by reductions of counter (Minsky) machine problems. Representing counter machine runs apparently requires less control over neighboring grid points than tilings or Turing machine runs, and so this technique is possibly more versatile, even if one component of the underlying product structures is “close to” being the universal relation.

Combining modal logics

Built on the foundations laid by Peirce, Schroder, and others in the 19th century, the modern development of relation algebras started with the work of Tarski and his colleagues [21, 22]. They showed that relation algebras can capture strong first-order theories like ZFC, and so their equational theory is undecidable. The less expressive class WA of weakly associative relation algebras was introduced by Maddux [7]. Nemeti [16] showed that WAs have a decidable universal theory. There has been extensive research on increasing the expressive power of WA by adding new operations [1, 4, 11, 13, 20]. Extensions of this kind usually also have decidable universal theories. Here we give an example – extending WAs with set-theoretic projection elements – where this is not the case. These “logical” connectives are set-theoretic counterparts of the axiomatic quasi-projections that have been investigated in the representation theory of relation algebras [22, 6, 19]. We prove that the quasi-equational theory of the extended class PWA is not recursively enumerable. By adding the difference operator D one can turn WA and PWA to discriminator classes where each universal formula is equivalent to some equation. Hence our result implies that the projections turn the decidable equational theory of “WA + D ” to non-recursively enumerable (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Spatial Logic + Temporal Logic = ?

The paper sets out to offer an alternative to the function/argument approach to the most essential aspects of natural language meanings. That is, we question the assumption that semantic completeness (of, e.g., propositions) or incompleteness (of, e.g., predicates) exactly replicate the corresponding grammatical concepts (of, e.g., sentences and verbs, respectively). We argue that even if one gives up this assumption, it is still possible to keep the compositionality of the semantic interpretation of simple predicate/argument structures. In our opinion, compositionality presupposes that we are able to compare arbitrary meanings in term of information content. This is why our proposal relies on an ‘intrinsically’ type free algebraic semantic theory. The basic entities in our models are neither individuals, nor eventualities, nor their properties, but ‘pieces of evidence’ for believing in the ‘truth’ or ‘existence’ or ‘identity’ of any kind of phenomenon. Our formal language contains a single binary non-associative constructor used for creating structured complex terms representing arbitrary phenomena. We give a finite Hilbert-style axiomatisation and a decision algorithm for the entailment problem of the suggested system.