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REFLECTIONSon

LOGIC PROGRAMMINGand

NONMONOTONIC REASONING

byJACK MINKER

UNIVERSITY 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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MINSKY’S 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 Λ … Λ xn = tn Λ A1 … Λ An)• Clark Equational Theory (CET)

• LLOYD (1984, 1987) FOUNDATIONS OF LOGIC PROGRAMMING • SURVEYS ON NEGATION

– SHEPHERDSON (1988, 1998)

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DISJUNCTIVE LOGIC PROGRAMMING – FIRST STEP

• NON HORN CLAUSE (DISJUNCTIVE CLAUSE)• P1, …, Pn A1, …, Am

• THEORY OF NEGATION– REITER’s 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 modelNegative 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) Λ 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/DEFAULT/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 REITER’S CWA• GAEL TO MINKER’S GCWA

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ADDITIONAL RELATIONS• Bonatti (1993)

– AEL Programs Generalize Ideas in LP– Stable, Supported WFS, Fitting’s and Kunen’s 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 Ferraris’s General Logic

Programs (2005) to Lin and Shoham’s Knowledge of Justified Assumptions (1992) (a nonmonotonic modal logic that includes as special cases Reiter’s default logic in propositional case and Moore’s 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 (CON’T)

• 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-potsdam.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 L{a1, …, an, not b1, …, bm}U• 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 lm+1, …, not ln• r. A1 x … x Ak l1, …, lm, not lm+1, …, not ln (introduced by Brewka,

Niemela, Syrajnen 2003)• cr. H + l1, …, lm, not lm+1, …, 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@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 Clark’s 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 {pq, qp} program has a unique answer set– comp: {pq, qp} 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 shuttle’s 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 (CON’T)• 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