Autoepistemic logics as a unifying framework for the semantics of logic programs

In this paper, it is shown that a three-valued autoepistemic logic provides an elegant unifying framework for some of the major semantics of normal and disjunctive logic programs and logic programs with classical negation, namely, the stable semantics, the well-founded semantics, supported models, Fitting's semantics, Kunen's semantics, the stationary semantics, and answer sets. For the first time, so many semantics are embedded into one logic. The framework extends previous results—by Gelfond, Lifschitz, Marek, Subrahmanian, and Truszczynski —on the relationships between logic programming and Moore's autoepistemic logic. The framework suggests several new semantics for negation-as-failure. In particular, we will introduce the epistemic semantics for disjunctive logic programs. In order to motivate the epistemic semantics, an interesting class of applications called ignorance tests will be formalized; it will be proved that ignorance tests can be defined by means of the epistemic semantics, but not by means of the old semantics for disjunctive programs. The autoepistemic framework provides a formal foundation for an environment that integrates different forms of negation. The role of classical negation and various forms of negation-by-failure in logic programming will be briefly discussed.     

An approach to the DIN Kernel Lisp definition

A DIN Kernel LISP Draft (DKLisp) has been developed by DIN as Reaction to Action D1 (N79), short term goal, of ISO WG16. It defines a subset language, as compatible as possible with the ANSICommon-Lisp draft, but also with theEuLisp draft. It combines the most important LISP main stream features in a single, compact, but nevertheless complete language definition, which thereby could be well suited as basis for a short term InternationalLisp Standard. Besides the functional and knowledge processing features, the expressive power of the language is well comparable with contemporary procedural languages, as e.g. C++ (of course without libraries). Important features ofDKLisp are:

to be a “Lisp-1,” but allowing an easy “Lisp-2” transformation;

to be a simple, powerful and standardized educationalLisp;

to omit all features, which are unclean or in heavy discussion;

DKLisp programs run nearly unchanged inCommon-Lisp;

DKLisp contains a simple object and package system;

DKLisp contains those data classes and control structures also common to most modernLisp and non-Lisp languages;

DKLisp offers a simple stream I/O;

DKLisp contains a clean unified hierarchical class/type system;

DKLisp contains the typical “Lisp-features” in an orthogonal way;

DKLisp allows and encourages really small but powerful implementations;

DKLisp comes in levels, so allowing ANSICommon-Lisp to be an extension ofDKLisp level-1.

The present is the second version of the proposal, namely version 1.2, with slight differences with respect to the one sent to ISO. Sources of such changes were the remarks generously sent by many readers of the previous attempt.     

Thinking in parallel: four steps to Parlog

If you are familiar with Prolog but not with Parlog then this tutorial is aimed at you. In what follows I attempt to:

? explain the basics of Parlog

? demonstrate that Parlog programs can be powerful and elegant

? discuss the relationship of Parlog to Prolog, and

? identify some resources which will take you further.

These are what I call ‘four steps to Parlog’.     

Trial and error

A pac-learning algorithm isd-space bounded, if it stores at mostd examples from the sample at any time. We characterize thed-space learnable concept classes. For this purpose we introduce the compression parameter of a concept classb and design our Trial and Error Learning Algorithm. We show: b isd-space learnable if and only if the compression parameter ofb is at mostd. This learning algorithm does not produce a hypothesis consistent with the whole sample as previous approaches e.g. by Floyd, who presents consistent space bounded learning algorithms, but has to restrict herself to very special concept classes. On the other hand our algorithm needs large samples; the compression parameter appears as exponent in the sample size. We present several examples of polynomial time space bounded learnable concept classes:

- all intersection closed concept classes with finite VC-dimension.

- convexn-gons in ?².

- halfspaces in ?ⁿ.

- unions of triangles in ?².

We further relate the compression parameter to the VC-dimension, and discuss variants of this parameter.     

Static semantics for normal and disjunctive logic programs

In this paper, we propose a newsemantic framework for disjunctive logic programming by introducingstatic expansions of disjunctive programs. The class of static expansions extends both the classes of stable, well-founded and stationary models of normal programs and the class of minimal models of positive disjunctive programs. Any static expansion of a programP provides the corresponding semantics forP consisting of the set of all sentences logically implied by the expansion. We show that among all static expansions of a disjunctive programP there is always theleast static expansion, which we call thestatic completion ¯P ofP. The static completion¯P can be defined as the least fixed point of a naturalminimal model operator and can be constructed by means of a simpleiterative procedure. The semantics defined by the static completion¯P is called thestatic semantics ofP. It coincides with the set of sentences that are true inall static expansions ofP. For normal programs, it coincides with the well-founded semantics. The class of static expansions represents a semantic framework which differs significantly from the other semantics proposed recently for disjunctive programs and databases. It is also defined for a much broader class of programs.Dedicated to Jack MinkerPartially supported by the National Science Foundation grant # IRI-9313061.     

On the degree of boolean functions as real polynomials

Every Boolean function may be represented as a real polynomial. In this paper, we characterize the degree of this polynomial in terms of certain combinatorial properties of the Boolean function. Our first result is a tight lower bound of Ω(logn) on the degree needed to represent any Boolean function that depends onn variables. Our second result states that for every Boolean functionf, the following measures are all polynomially related:

o The decision tree complexity off.

o The degree of the polynomial representingf.

o The smallest degree of a polynomialapproximating f in theL _max norm.

     

Solving visibility and separability problems on a mesh-of-processors

In this paper we study parallel algorithms for the Mesh-of-Processors architecture to solve visibility and related separability problems for sets of simple polygons in the plane. In particular, we present the following algorithms:

- AnO( \(\sqrt N\) time algorithm for computing on a Mesh-of-Processors of size N the visibility polygon from a point located in anN-vertex polygon, possibly with holes.

-O( \(\sqrt N\) ) time algorithms for computing on a Mesh-of-Processors of sizeN the set of all points on the boundary of anN-vertex polygonP which are visible in a given directiond as well as the visibility hull ofP for a given directiond.

- AnO( \(\sqrt N\) ) time algorithm for detecting on a Mesh-of-Processors of size 2N whether twoN-vertex polygons are separable in a given direction and anO( \(\sqrt {MN}\) ) time algorithm for detecting on a Mesh-of-Processors of sizeMN whetherM N-vertex polygons are sequentially separable in a given direction.

All proposed algorithms are asymptotically optimal (for the Mesh-of-Processors) with respect to time and number of processors.     

On the power of unambiguity in log-space

We report progress on the NL versus UL problem.

We show that counting the number of s-t paths in graphs where the number of s-v paths for any v is bounded by a polynomial can be done in FUL: the unambiguous log-space function class. Several new upper bounds follow from this including ${{{ReachFewL} \subseteq {UL}}}$ and ${{{LFew} \subseteq {UL}^{FewL}}}$

We investigate the complexity of min-uniqueness—a central notion in studying the NL versus UL problem. In this regard we revisit the class OptL[log n] and introduce UOptL[log n], an unambiguous version of OptL[log n]. We investigate the relation between UOptL[log n] and other existing complexity classes.

We consider the unambiguous hierarchies over UL and UOptL[log n]. We show that the hierarchy over UOptL[log n] collapses. This implies that ${{{ULH} \subseteq {L}^{{promiseUL}}}}$ thus collapsing the UL hierarchy.

We show that the reachability problem over graphs embedded on 3 pages is complete for NL. This contrasts with the reachability problem over graphs embedded on 2 pages, which is log-space equivalent to the reachability problem in planar graphs and hence is in UL.

     

Association Fields via Cuspless Sub-Riemannian Geodesics in SE(2)

To model association fields that underly perceptional organization (gestalt) in psychophysics we consider the problem P _curve of minimizing $\int _{0}^{\ell} \sqrt{\xi^{2} +\kappa^{2}(s)} {\rm d}s $ for a planar curve having fixed initial and final positions and directions. Here κ(s) is the curvature of the curve with free total length ?. This problem comes from a model of geometry of vision due to Petitot (in J. Physiol. Paris 97:265–309, 2003; Math. Inf. Sci. Humaines 145:5–101, 1999), and Citti & Sarti (in J. Math. Imaging Vis. 24(3):307–326, 2006). In previous work we proved that the range $\mathcal{R} \subset\mathrm{SE}(2)$ of the exponential map of the underlying geometric problem formulated on SE(2) consists of precisely those end-conditions (x _fin,y _fin,θ _fin) that can be connected by a globally minimizing geodesic starting at the origin (x _in,y _in,θ _in)=(0,0,0). From the applied imaging point of view it is relevant to analyze the sub-Riemannian geodesics and $\mathcal{R}$ in detail. In this article we

• show that $\mathcal{R}$ is contained in half space x≥0 and (0,y _fin)≠(0,0) is reached with angle π,
• show that the boundary $\partial\mathcal{R}$ consists of endpoints of minimizers either starting or ending in a cusp,
• analyze and plot the cones of reachable angles θ _fin per spatial endpoint (x _fin,y _fin),
• relate the endings of association fields to $\partial\mathcal {R}$ and compute the length towards a cusp,
• analyze the exponential map both with the common arc-length parametrization t in the sub-Riemannian manifold $(\mathrm{SE}(2),\mathrm{Ker}(-\sin\theta{\rm d}x +\cos\theta {\rm d}y), \mathcal{G}_{\xi}:=\xi^{2}(\cos\theta{\rm d}x+ \sin\theta {\rm d}y) \otimes(\cos\theta{\rm d}x+ \sin\theta{\rm d}y) + {\rm d}\theta \otimes{\rm d}\theta)$ and with spatial arc-length parametrization s in the plane $\mathbb{R}^{2}$ . Surprisingly, s-parametrization simplifies the exponential map, the curvature formulas, the cusp-surface, and the boundary value problem,
• present a novel efficient algorithm solving the boundary value problem,
• show that sub-Riemannian geodesics solve Petitot’s circle bundle model (cf. Petitot in J. Physiol. Paris 97:265–309, [2003]),
• show a clear similarity with association field lines and sub-Riemannian geodesics.

     

A connection between the Cantor–Bendixson derivative and the well-founded semantics of finite logic programs

Results of Schlipf (J Comput Syst Sci 51:64?C86, 1995) and Fitting (Theor Comput Sci 278:25?C51, 2001) show that the well-founded semantics of a finite predicate logic program can be quite complex. In this paper, we show that there is a close connection between the construction of the perfect kernel of a $\Pi^0_1$ class via the iteration of the Cantor?CBendixson derivative through the ordinals and the construction of the well-founded semantics for finite predicate logic programs via Van Gelder??s alternating fixpoint construction. This connection allows us to transfer known complexity results for the perfect kernel of $\Pi^0_1$ classes to give new complexity results for various questions about the well-founded semantics ${\mathit{wfs}}(P)$ of a finite predicate logic program P.     

The addition of bounded quantification and partial functions to a computational logic and its theorem prover

We describe an extension to our quantifier-free computational logic to provide the expressive power and convenience of bounded quantifiers and partial functions. By quantifier we mean a formal construct which introduces a bound or indicial variable whose scope is some subexpression of the quantifier expression. A familiar quantifier is the Σ operator which sums the values of an expression over some range of values on the bound variable. Our method is to represent expressions of the logic as objects in the logic, to define an interpreter for such expressions as a function in the logic, and then define quantifiers as ‘mapping functions’. The novelty of our approach lies in the formalization of the interpreter and its interaction with the underlying logic. Our method has several advantages over other formal systems that provide quantifiers and partial functions in a logical setting. The most important advantage is that proofs not involving quantification or partial recursive functions are not complicated by such notions as ‘capturing’, ‘bottom’, or ‘continuity’. Naturally enough, our formalization of the partial functions is nonconstructive. The theorem prover for the logic has been modified to support these new features. We describe the modifications. The system has proved many theorems that could not previously be stated in our logic. Among them are:

? classic quantifier manipulation theorems, such as $$\sum\limits_{{\text{l}} = 0}^{\text{n}} {{\text{g}}({\text{l}}) + {\text{h(l) = }}} \sum\limits_{{\text{l = }}0}^{\text{n}} {{\text{g}}({\text{l}})} + \sum\limits_{{\text{l = }}0}^{\text{n}} {{\text{h(l)}};} $$

? elementary theorems involving quantifiers, such as the Binomial Theorem: $$(a + b)^{\text{n}} = \sum\limits_{{\text{l = }}0}^{\text{n}} {\left( {_{\text{i}}^{\text{n}} } \right)} \user2{ }{\text{a}}^{\text{l}} {\text{b}}^{{\text{n - l}}} ;$$

? elementary theorems about ‘mapping functions’ such as: $$(FOLDR\user2{ }'PLUS\user2{ O L) = }\sum\limits_{{\text{i}} \in {\text{L}}}^{} {{\text{i}};} $$

? termination properties of many partial recursive functions such as the fact that an application of the partial function described by $$\begin{gathered} (LEN X) \hfill \\ \Leftarrow \hfill \\ ({\rm I}F ({\rm E}QUAL X NIL) \hfill \\ {\rm O} \hfill \\ (ADD1 (LEN (CDR X)))) \hfill \\ \end{gathered} $$ terminates if and only if the argument ends in NIL;

? theorems about functions satisfying unusual recurrence equations such as the 91-function and the following list reverse function: $$\begin{gathered} (RV X) \hfill \\ \Leftarrow \hfill \\ ({\rm I}F (AND (LISTP X) (LISTP (CDR X))) \hfill \\ (CONS (CAR (RV (CDR X))) \hfill \\ (RV (CONS (CAR X) \hfill \\ (RV (CDR (RV (CDR X))))))) \hfill \\ X). \hfill \\ \end{gathered} $$

     

Parameterized Domination in Circle Graphs

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Appl. Math., 42(1):51–63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction:

• Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution.
• Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs.
• If T is a given tree, deciding whether a circle graph G has a dominating set inducing a graph isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by t=|V(T)|. We prove that the FPT algorithm runs in subexponential time, namely $2^{\mathcal{O}(t \cdot\frac{\log\log t}{\log t})} \cdot n^{\mathcal{O}(1)}$ , where n=|V(G)|.

     

Well-founded and stationary models of logic programs

We investigate the class ofstationary or partial stable models of normal logic programs. This important class of models includes all (total)stable models, and, moreover, thewell-founded model is always its smallest member. Stationary models have several natural fixed-point definitions and can be equivalently obtained as expansions or extensions of suitable autoepistemic or default theories. By taking a particular subclass of this class of models one can obtain different semantics of logic programs, including the stable semantics and the well-founded semantics. Stationary models can be also naturally extended to the class of all disjunctive logic programs. These features of stationary models designate them as an important class of models with applications reaching far beyond the realm of logic programming.Partially supported by the National Science Foundation grant #IRI-9313061.     

Two-Way Automata Versus Logarithmic Space

We strengthen a previously known connection between the size complexity of two-way finite automata ( ) and the space complexity of Turing machines (tms). Specifically, we prove that

every s-state has a poly(s)-state that agrees with it on all inputs of length ≤s if and only if NL?L/poly, and

every s-state has a poly(s)-state that agrees with it on all inputs of length ≤2 ^s if and only if NLL?LL/polylog.

Here, and are the deterministic and nondeterministic , NL and L/poly are the standard classes of languages recognizable in logarithmic space by nondeterministic tms and by deterministic tms with access to polynomially long advice, and NLL and LL/polylog are the corresponding complexity classes for space O(loglogn) and advice length poly(logn). Our arguments strengthen and extend an old theorem by Berman and Lingas and can be used to obtain variants of the above statements for other modes of computation or other combinations of bounds for the input length, the space usage, and the length of advice.