REFLECTIONS on LOGIC PROGRAMMING and NONMONOTONIC REASONING by JACK MINKER UNIVERSITY OF MARYLAND

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REFLECTIONSonLOGIC PROGRAMMINGandNONMONOTONIC
REASONINGbyJACK MINKERUNIVERSITY OF MARYLAND
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INTRODUCTION

• BEGINNINGS
• LOGIC PROGRAMMING
• DISJUNCTIVE LOGIC PROGRAMMING
• NONMONOTONIC REASONING
• LP and NMR
• IMPLEMENTATIONS
• RECENT DEVELOPMENTS
• APPLICATIONS
• SUMMARY and CONCLUSIONS

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BEGINNINGS

• McCARTHY
• Common Sense Reasoning (1959)
• DEFINED OLDEST PROBLEM IN AI
• LIFSCHITZ, McCAIN, REMOLINA, TURNER (2000) CCALC
• Situations, Actions, Causal Laws (1963)
• GOLOG (LEVESQUE, REITER (1997))
• McCarthy and Hayes
• Philosophical Problems and Frame Axioms (1969)
• Seeming Need for Large Number of Axioms to
  Represent Changes
• REITER (1980, 1991), SHANAHAN (1997)
• Robinson (1965)
• Resolution Principle for Automated Theorem
  Proving
• Minsky Frame Paper and Critique of Logic in AI
  (1975)

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MINSKYS CRITIQUE OF LOGIC

• LOGICAL REASONING IS NOT FLEXIBLE FOR
  THINKING
• INCONSISTENT DATA CANNOT BE HANDLED
• FEASIBILITY OF REPRESENTING KNOWLEDGE BY SMALL
  TRUE PROPOSITIONS IS DOUBTFUL
• SEPARATION OF KNOWLEDGE AND RULES IS TOO RADICAL
• LOGIC IS MONOTONIC
• PROCEDURAL DESCRIPTIONS OVER DECLARATIVE
  DESCRIPTIONS

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LOGIC PROGRAMMING BEGINNINGS

• HODES (1966)
• GREEN (1969)
• HEWITT (1969)
• THNOT Operator PLANNER
• ELCOCK (1971)
• ABSYS and ABSET Declarative Languages
• HAYES (1973)
• Computation and Deduction
• COLMERAUER (1973)
• PROLOG NOT Operator
• Kowalski/Kuehner SLD for Horn Clauses
• WARREN, PEREIRA, PEREIRA (1977)
• EDINBURGH PROLOG
• Competitive With LISP

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HORN LOGIC PROGRAMMING FOUNDATIONS

• HORN CLAUSES
• p(t1, , tm) ? A1, , An
• KOWALSKI and KUEHNER SL Resolution (1971)
• Descendant of Model Elimination (Loveland 1969)
• LUSH/SLD (Hill 1974, Apt and Van Emden 1982)
• KOWALSKI (1974)
• Van EMDEN and KOWALSKI (1976)
• FIXPOINT SEMANTICS
• MODEL THEORY SEMANTICS
• OPERATIONAL SEMANTICS
• LOGIC and DATABASES (WORKSHOP 1977, BOOK 1978
  Gallaire, Minker)
• DEDUCTIVE DATABASES
• REITER (1978)
• NEGATION (REITER CLOSED WORLD ASSUMPTION)
• DOMAIN CLOSURE AXIOM
• UNIQUE NAME AXIOM
• CLARK (1978)
• NEGATION (CLARK COMPLETION THEORY COMP(P) IFF)
• p(t1, , tm) ? A1, , An p(x1, , xm)?? y1
  ? yp (x1 t1 ? ? xn tn ? A1 ? An)

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DISJUNCTIVE LOGIC PROGRAMMING FIRST STEP

• NON HORN CLAUSE (DISJUNCTIVE CLAUSE)
• P1, , Pn ? A1, , Am
• THEORY OF NEGATION
• REITERs CWA is INCONSISTENT for DISJUNCTION
• P v Q then by CWA, not P and not Q
• Minker (1982)
• GENERALIZED CLOSED WORLD ASSUMPTION
• MODEL THEORETIC - MINIMAL MODELS
• P, Q
• Positive Truths True in every minimal model
• Negative Truths Not True in any minimal
  model
• PROOF THEORETIC

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STRATIFIED AND NORMAL LOGIC PROGRAMMING

• P ? A1, A2, An, not B1,, not Bm
• p? not q, q? p (not stratified)
• p? not q, r , q? q, not r rewritten for
  stratification as r, q? q, not r, p? not q
• STRATIFIED LP
• APT, BLAIR and WALKER (1988)
• VAN GELDER (1988)
• PRZYMUSINSKI (1988)
• PERFECT MODELS
• NORMAL LP
• VAN GELDER, ROSS and SCHLIPF (1988)
• WELL FOUNDED SEMANTICS (WFS)
• p? not q, q? not p WFS p and q are unknown
• GELFOND and LIFSCHITZ (1989)
• STABLE MODEL SEMANTICS
• Stable models p, q

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STABLE MODELS

• GELFOND, LIFSCHITZ (1991)
• REDUCT PI of P w.r.t Interpretation I
• Delete all rules with a negative false literal
  (w.r.t. I)
• Delete the negative literals from the bodies of
  the remaining literals
• A Stable Model of a program P is an
  interpretation I such that I is an answer set of
  PI

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DISJUNCTIVE LOGIC PROGRAMMING THEORY

• P1, P2, , Pm ? A1, A2, , An
• MINKER and RAJASEKAR (1987)
• FIXPOINT OPERATOR
• MODEL THEORY
• PROOF THEORY
• LUST/SLI (MINKER, ZANON 1982, LOBO,
  MINKER,RAJASEKAR 1992)
• EXTENDED DLP (with Baral, Lobo, Ruiz, Seipel)
  (Gelfond and Lifschitz 1991)
• P1, P2, , Pm ? A1, A2, , An, not B1, not B2,
  , not Bk
• Negation in body of clauses
• SLINF (MINKER, RAJASEKAR 1990)
• LOBO, MINKER, RAJESEKAR (1992)
• FOUNDATIONS of DISJUNCTIVE LOGIC PROGRAMMING
• GELFOND and LIFSCHITZ
• Classical Negation (1991)
• Answer Set Semantics (1999)

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APPLICATIONS DISJUNCTIVE LP

• KNOWLEDGE REPRESENTATION
• BARAL, GELFOND (1995)
• BARAL (2002)
• Knowledge Representation, Reasoning and
  Declarative Problem Solving
• OTHER APPLICATIONS
• 3 Color Problem
• Hamiltonian Path
• See Problems in LPNMR07 ASP Contest

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ABDUCTIVE LOGIC PROGRAMMING

• ABDUCTION INTRODUCED BY PHILOSOPHER C.S, PIERCE
  (1955)
• An Inference Process of Forming a hypothesis
  that explains given observed phenomena
• Study of Abduction in LP Introduced in Late 1990s
• Eshgi, Kowalski, Denecker, Kakas, Mancarella
  early workers in field
• Kowalski, Kakas and Toni (1993) Abductive Logic
  Programming
• Answer Set Programming used as basis for some
  implementations
• Performing Abduction in Disjunctive Logic
  Programming Studied by Eiter, Leone, Mateis,
  Pfeifer, Scarcello (1998) and by Sato and Inoue
  who discussed abduction and DLP
• Mancarella, Sadri, Terreni and Toni (2007 at
  LPNMR07), discuss the use of CIFF for abductive
  reasoning with constraints and show that their
  system compares favorably with A-System, DLV and
  Smodels

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NONMONOTONIC THEORIES

• CIRCUMSCRIPTION (McCARTHY 1980)
• DEFAULT REASONING (REITER 1980)
• AUTOEPISTEMIC REASONING (MOORE 1985)

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CIRCUMSCRIPTION

• Let A be a sentence of FOL containing
  predicate symbol P(x1,,xn) written as P(x). We
  write A(Ø) as result for replacing all predicates
  P in A by the predicate expression Ø.
• The CIRCUMSCRIPTION OF P IN A(P) is the
  sentence schema
• A(Ø) ? ?x(Ø(x) ? P(x)) ? ?x(P(x) ? Ø(x))
  (1)
• LIFSCHITZ POINTWISE, PRIORITIZED, PARALLEL,
  INTROSPECTIVE

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DEFAULT REASONING

• DEFAULT REASONING
• DEFAULT RULES ??
• 
  -----
• 
  ?
• If ? is true and ? is
  consistent with a set of beliefs, then ? is
  believed
• EXTENSIONS TO DEFAULT REASONING
• DISJUNCTIVE DEFAULTS (GELFOND,LIFSCHITZ,
  PRZYMUSINSKA, TRUSZCZYNSKI (1991))
• ??1, , ?m
• 
  ------------------
• 
  ?1 ?n
• Generalizes the semantics of disjunctive and
  extended disjunctive databases
• CONSTRAINED (DELGRAND, SCHAUB, JACKSON (1999))
• CUMULATIVE DEFAULT LOGIC (BREWKA (1991))
• JUSTIFIED DEFAULT LOGIC (LUKASZIEWICZ (1988))
• RATIONAL DEFAULT LOGIC (MIKITIUK, TRUSZCZYNSKI
  (1988))
• DEFAULTS WITH PREFERENCES AND INHERITANCE
  (DELGRANDE, SCHAUB (2002))

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MODAL THEORIES

• AUTOEPISTEMIC LOGIC
• Modal Logic augments FOL by operators
• such as B (believes), K (knows) that take
  sentences as arguments rather than terms.
• Invented by Hintikka (1962). Kripke (1963)
  defined semantics of modal logic of knowledge in
  terms of possible worlds.
• Moore related modal logic of knowledge to
  reasoning about knowledge which refers directly
  to possible worlds in FOL.

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RELATIONSHIPSAE/DEFAULT/CIRCUMSCRIPTION

• PERLIS (1988)and LIFSCHITZ (1989)
• VARIANTS OF CIRCUMSCRIPTION ANALOGOUS TO AEL
• KONOLIGE (1987)
• STRENGTHENS AEL TO BE EQUIVALENT TO PROPOSITIONAL
  FORM OF DEFAULT LOGIC
• MAREK/TRUSZCZYNSKI (1989)
• EXTEND WORK OF KONOLIGE
• MAREK/SUBRAHMANIAN (1989)
• RELATE FORMAL MODELS OF NORMAL PROGRAMS AND
  EXPANSIONS OF AE THEORIES

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ADDITIONAL RELATIONSHIPS AE/CIRCUMSCRIPTION/DEFAUL
T/LP

• REITER (1982)
• FIRST TO RELATE CIRCUMSCRIPTION TO LOGIC
  PROGRAMMING
• Marek and Truszczynski (1989)
• Stable Models for Default Logic
• GELFOND (1987)
• GENERAL LOGIC PROGRAMS TRANSLATE TO AEL
• GELFOND/LIFSCHITZ (1988)
• STABLE MODEL SEMATICS EQUIVALENT TO TRANSLATION
  OF LOGIC PROGRAMS TO AEL
• LIFSCHITZ (1989)
• AEL, STABLE MODELS AND INTROSPECTIVE
  CIRCUMSCRIPTION PROVIDE 3 EQUIVALENT DESCRIPTIONS
  OF PROPOSITIONAL LOGIC PROGRAMS
• PRZYMUSINSKI (1988)
• RELATIONSHIPS BETWEEN LP AND NMR
• EXTENDS AEL TO GENERALIZED AEL AND RELATES
• AEL TO REITERS CWA
• GAEL TO MINKERS GCWA

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ADDITIONAL RELATIONS

• Bonatti (1993)
• AEL Programs Generalize Ideas in LP
• Stable, Supported WFS, Fittings and Kunens
  Semantics and Abduction can be Captured by AEL
  Translations
• Generalized SLDNF and a Generate and Test Method
  To Provide Sound and Complete Methods for AE
  Programs
• Lin, ZHOU (2007)
• Answer Sets and Circumscription
• Map Pearce Equilibrium Logic (2001) and
  Ferrariss General Logic Programs (2005) to Lin
  and Shohams Knowledge of Justified Assumptions
  (1992) (a nonmonotonic modal logic that includes
  as special cases Reiters default logic in
  propositional case and Moores AEL).
• Allows a Mapping from general logic programming
  to propositional circumscription.

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IMPLEMENTATIONS at LBAI 2000

• Niemela, Simon (1997)
• SMODELS
• Marek and Truszczynski
• DeReS
• Warren, et al. (1999)
• XSB (Well Founded Models)
• Eiter, Leone, Mateis, Pfeifer, Scarcello (1997)
• DLV (Disjunctive Theories)
• Zaniolo, Arni, Ong (1993)
• LDL

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IMPLEMENTATIONS at LBAI 2000 (CONT)

• PLANNING
• TLPlan (Bacchus et al.)
• GPT (Bonet/Geffner)
• Blackbox (Kautz/Selman/Huang)
• CCALC (Lifschitz/McCain/Turner)
• Golog (Levesque et al.)
• INDUCTIVE LOGIC PROGRAMMING
• CPROLOG (Muggleton/Srinivasan)
• MULTIAGENT APPLICATIONS
• IMPACT (Subrahmanian et al.)

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NONMONOTONIC REASONING PARADIGM

• Use any NMR Theory to Define your Problem
• Translate the Theory to LP/DLP system
• Depending upon your translation and whether or
  not the translation has recursion through
  negation, select an existing system that best
  meets your needs
• Dominant semantics is Answer Set Semantics
• Implement and Test your System
• Build Capabilities Using Existing Systems
• A-Prolog Implemented on Top of Smodels (Gelfond
  et al.) (2002)
• GnT Built on Top of Smodels to achieve disjunction

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IMPLEMENTATION REPOSITORY

• DAGSTUHL INITIATIVE PROPOSAL (1996)
• Minker Proposed Developing a Database of
  Information about LP System Implementations and
  Applications.
• University of Koblenz developed web site listing
  systems and applications. (Furbach)
• 32 SYSTEMS LISTED (Last updated 2000)
• Applications Page Inaccessible
• DAGSTUHL INITIATIVE PROPOSAL (2002)
• Develop infrastructure for benchmarking ASP
  solvers
• Environment for submitting and archiving
  benchmarking problems and instances in which ASP
  systems can be benchmarked under equal and
  reproducible conditions, leading to independent
  results.
• Asparagus Web Site http//asparagus.cs.uni-potsda
  m.de/
• International Board
• Assure Continuation and Generate Continued
  Interest
• Consider Broadening the Material in the Asparagus
  Web Site, not necessarily for the contest
• Information about other nonmonotonic systems
  (WFS), Successful Real Applications, Cognotive
  Robotics, Logic Planning Programs,
• FIRST INTERNATIONAL CONTEST ASP SYSTEMS LPNMR 07
• Evaluation Committee GEBSER, LIU, NAMASIVAYAN,
  NEUMANN, TRAUB, TRUSZCZYNSKI
• SYSTEMS Asper, Angers Assat, Hong Kong Clasp
  Potsdam Cmodels, Texas dlv, Vienna/Rende gnt,
  Helsinki lp2sat, Helsinki nomore, Potsdam
  pbmodels, Kentucky Smodels, Helsinki
• 37 problems listed for First Answer Set
  Programming System Contest
• THE COMPETITION COMMITTEE HAS AUTHORIZED ME TO
  ANNOUNCE THE WINNER IS

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• TO BE
• ANNOUNCED
• BY THE
• First Answer Set Programming System
  Competition Committee

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SIGNIFICANT DEVELOPMENTS -1

• IMPRESSSED BY WORK THAT HAS COMBINED THEORY,
  COMPLEXITY, IMPLEMENTATION AND EXPERIMENTAL WORK,
  PRIMARILY ON ANSWER SET PROGRAMMING
• EXTENSIONS TO ANSWER SET PROGRAMMING - SMODELS
• Choice Rules, Cardinality and Weight Constraints
  (NIEMELA, SIMONS 2000)
• Cardinality Constraint La1, , an, not b1, ,
  bmU
• Cardinality and Weight Constraints are form of
  AGGREGATES that correspond to COUNT and SUM
  (first to introduce into non stratified programs)
• Disjunction capability, GnT, Built on Top of
  Smodels (2000)
• Unfolding Partiality and Disjunctions in Stable
  Model Semantics (Janhusen, Niemela, Seipel,
  Simons, You 2006)
• Develop Implementation methodology for partial
  disjunctive stable models where partiality and
  disjunctions are unfolded
• Implementation of stable models of normal
  (disjunction-free) logic programs can be used to
  compute stable models for disjunctive logic
  programs
• They show partial stable models can be captured
  by total stable models using a simple linear
  modular program transformation.
• Experiments on several classes of problems
  compares favorably with DLV

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SIGNIFICANT DEVELOPMENTS -2

• DLV
• Generate Test Paradigm (Eiter, Leone 2002)
• Disjunctive Rule Guesses Solution Candidate S
• Integrity constraints which check admissibility
  of S
• Recursive Aggregates in Disjunctive Logic
  Programming Semantics and Complexity (Faber,
  Leone, Pfeifer 2004) (Faber and Leone )
• Enhancing Magic Sets for Disjunctive Datalog
  (Cumbo, Faber, Greco, Leone)
• Magic Sets and Data Integration (Faber, Greco,
  Leone 2007)
• INFOMIX (Calabria, Roma, Vienna, Warsaw Groups
  2005)
• Data Integration
• Integrity Constraints over global schema
• Sound and complete logic-based methods for query
  answering
• Deal with incomplete and inconsistent data
• DLV and disjunctive data

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SIGNIFICANT DEVELOPMENTS - 3

• Extensions to Handle Ordered Disjunctions and
  Inconsistencies, CR-PROLOG2 (Consistency
  Restoring ) (BALDUCCINI, MELLARKOD 2004)
• r A1, , Ak ? l1, , lm, not lm1, , not
  ln
• r. A1 x x Ak ? l1, , lm, not lm1, ,
  not ln (introduced by Brewka, Niemela, Syrajnen
  2003)
• cr. H ? l1, , lm, not lm1, , not ln
  
• may possibly believe
  one of the elements of the head if agent has no
  way to obtain a consistent set of beliefs using
  regular rules only.
• Extend ASP to Include Probabilities - Allows
  Probabilistic Causal Reasoning (BARAL, GELFOND,
  RUSHTON 2007)
• Combines ASP with ideas of Judea Pearl
• Allows reasoning with causal probabilities and
  probabilistic updates
• AI_at_50 Debated whether AI should be logic-based or
  probability based. This work indicates that there
  need not be a dichotomy.

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SIGNIFICANT DEVELOPMENTS - 4

• Loop Formulas (Lin, Zhao 2002)
• Relationship Between Clarks Completion and
  Stable Models
• Loop formulas are those needed to be added to the
  Clark completion of the Program to get exact
  characterization of its stable models
• Loop p?q, q?p program has a unique answer set
• comp p?q, q?p has 2 models p, q
• Loop formula (p ? q) ? false none of them can
  be in answer set
• Serves as new basis to implement stable model
  semantics (ASSAT)
• Complete the program
• Conjoin with loop formulas
• Invoke SAT solver to find satisfying truth
  assignments
• Output truth assignments as stable models of
  program

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APPLICATIONS

• ACADEMIC APPLICATIONS USEFUL FOR TESTING AND
  INTRODUCING NEW FEATURES (3-COLOR, HAMILTONIAN
  CIRCUIT, )
• NON-ACADEMIC REALISTIC APPLICATIONS NEEDED
• DEMONSTRATE UTILITY OF LPNMR
• HANDLE LARGE APPLICATIONS (E.G. INTERFACE WITH
  SQL SYSTEM)
• HANDLE PROBLEMS NEEDED by USERS, EFFECTIVE
  INTERFACES, DEBUGGERS, OPTIMIZERS, HEURISTICS,
• TRANSFER TECHNOLOGY TO USER

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NON-ACADEMIC APPLICATIONS -1

• XSB (Warren)
• Ontology Management Work from textual database
  fields and technical drawings
• Extracted and inferred attributes of parts from
  textual database fields so organization could
  better understand what they had how many parts
  used, or how many parts included a strategic
  material such as titanium.
• Written in XSB with SQL server as a backing
  store, and included some parsing, a bit of
  ontological reasoning and a little bit of NMR --
  in parts using a WFS preference logic for
  parsing.
• Deductive Spread Sheet
• Implemented as add-in to MS Excel. Allows users
  to create deductive systems in a spreadsheet
  environment. XSB is backend computation engine
  and spreadsheet can be viewed as showing base
  data and the results of tabled computations.
  Whenever the user changes a spreadsheet cell that
  other cells depend on, those other cells are
  immediately updated.
• This is implemented using the new XSB incremental
  table maintenance facility.

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NON-ACADEMIC APPLICATIONS -2

• SPACE SHUTTLE REACTION CONTROL SYSTEM (GELFOND ET
  AL. 2001)
• Primary responsibility - maneuver aircraft
  while in space.
• Consists of fuel and oxidizer tanks, valves and
  other plumbing needed to provide propellant to
  shuttles maneuvering jets. Includes electronic
  circuitry both to control valves in fuel lines
  and to prepare jets to receive firing commands.
• During normal shuttle operations, pre-scripted
  plans tell astronauts what to do to achieve
  certain goals. System failures change situation.
  The number of possible sets of failures is too
  large to pre-plan for all of them. Continued
  correct operation of the RCS is then needed to
  allow mission completion of the mission and
  ensure crew safety. An intelligent system to
  verify and generate plans was needed.
• RCS/USA-Advisor is part of a decision support
  system for shuttle controllers. It is based on a
  reasoning system and a user interface. The
  reasoning system is capable of checking
  correctness of plans and finding plans for the
  operation of the RCS.
• Employs a programming methodology based on
  A-Prolog, algorithms for computing answer sets of
  programs of A-Prolog, and programming systems
  implementing these algorithms.
• User interface written in Java. Allows the user
  to specify the reasoning task to be performed,
  and then assembles into a program various
  A-Prolog modules, chosen according the components
  of the RCS that are involved in the task.
  Finally, the interface invokes program smodels to
  compute the answer sets of the A-Prolog program,
  and presents the results to the user.

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SPACE SHUTTLE (CONT)

• Large Practical System written in A-Prolog
• Importance of Careful Initial Design Simplified
  the Program
• Java Interface to Select Modules to Solve a
  Problem and Integrate Modules into Final A-Prolog
  Worked Well
• Structuring Problems as LP modules Useful for
  Reusability and Proving Correctness of
  Integration.
• System of Substantial Size Used for Planning
  Built on Theory of Action and Changes
• A-Prolog Allowed Use of Recursive Causal Laws
• System Tested and Worked. Not yet Used on a Space
  Mission.
• Demonstrates Practical Use of LPNMR
• Important to Collect and Publicize Successes in
  LPNMR

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LPNMR COMPANIES

• XSB, INC. (Warren, XSB)
• Advanced Techniques To Transform Unstructured
  Data
• NEOTIDE (Simon, SMODELS)
• License SMODELS
• HERZUM (COLLABORATION with EXECURA SPIN-OFF,
  CALABRIA, DLV)
• Market OLEX (Semantic Categorizer) and
• HiLeX Advanced Semantic Information Extractor

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SUMMARY AND CONCLUSIONS

• SIGNIFICANT DEVELOPMENTS/RELATIONSHIPS IN LPNMR
• LPNMR IS MATURE DISCIPLINE THEORY/IMPLEMENTATIONS
  
• BASED ON LOGICAL FOUNDATIONS NOT AD-HOCKERY
• SIGNIFICANT IMPLEMENTATIONS
• TOOLS AVAILABLE FOR REAL WORLD APPLICATIONS
• SEVERAL SYSTEMS SCALE TO LARGE PROBLEMS
• ADDITIONAL TOOLS NEEDED FOR USERS
• FUTURE DIRECTIONS
• ASP and Grounding Extend to Variables Without
  Grounding
• SIGNIFICANT REALISTIC APPLICATION NEEDED
• EXPAND IMPLEMENTATION REPOSITORY
• EXPAND WORK TO LOGIC-BASED AI (and PROBABILISTIC
  METHODS)
• AGENTS AND BELIEFS, LOGIC AND LANGUAGE,
  MECHANICAL CHECKING, LOGIC FOR CAUSATION AND
  ACTIONS, COGNITIVE ROBOTICS, BIOLOGIC MODELS,
• SEMANTIC WEB