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23.5.5 Belief Revision and Limitcycles Elimination The belief revision model we employed in criminal investigation is a simplified model we covered in chapter 10. In fact, we here used Looney’s model [12] with the additional considerations that the FPN may include cycles. Unlike multi-valued distributions as used in chapter 10, we here considered single valued tokens, representing the belief of the proposition / predicates associated with the places in the FPN. The belief revision cycle in the present context includes updating of the fuzzy truth token values associated with the transitions, followed by updating of fuzzy beliefs at all places. The following expressions were used for updating truth tokens at transitions and beliefs at places. For a given transition trq having input places {p1, p2, …, pv} with the corresponding single valued tokens {n1 (t), n2 (t), ….,nv (t)} at time t, let x= ∧ { nj(t) } 1 ≤j≤ v Now, if x exceeds the threshold marked with transition trq, the fuzzy truth token (FTT) at transition trq, denoted by tq, is computed as follows. tq(t+1) = x ∧ cfq =0, otherwise where cfq denotes the certainty factor of the rule, described by the transition. Further, the fuzzy belief ni (t+ 1) at time (t +1) is computed at place pi, the common output place of transitions tr1,tr2, …, tru by the following expression. ni (t +1) = ∨ {tk (t+1)} 1 ≤ k≤ u where tk(t +1) denotes the FTT of transition trk at time (t+1). If place pi is an axiom (i.e., it is not an output place of any transition), then ni (t+1) = 0. Let m be the number of transitions in the network. Now, it has been shown in [9] that there may be two situations after m number of belief revision cycles. First, the fuzzy beliefs at all places may reach an equilibrium (steady) state, following which there will be no more changes in beliefs. Alternatively, the beliefs at one or more places in the network may exhibit sustained oscillation.
We call this oscillation ‘limitcycles’. No inferences can be derived from an FPN exhibiting limitcycles. For elimination of limitcycles, either more information has to be added to the system, or one has to delink the cycles carefully, so that resulting inferences are minimally disturbed. We employed the second scheme here, assuming that we always will not get more information. The following definitions are useful to explain the scheme for limitcycles elimination. Definition 23.14: An arc represents a connectivity from a place to a transition and vice versa. Definition 23.15: An arc is called dominant, if it carries the highest FTT among all the arcs incoming to a place, or the least fuzzy belief (LFB) among all the arcs incoming to an enabled transition (i.e., a transition having tokens at all its input places) provided the LFB does not exceed the certainty factor of the transition. Definition 23.16: An arc is called permanently dominant, if it remains dominant for an infinite number of belief revision cycles. Definition 23.17: Fuzzy gain of an acyclic single reasoning path between two places is computed by taking the minimum of the fuzzy beliefs of the places and the certainty factors of the transitions on that path. In case there exists parallel reasoning paths between two places, then the overall fuzzy gain of the path is defined as the maximum of the individual fuzzy gains. Fuzzy gain of a reasoning path, starting from axioms up to a given transition, is defined as the maximum of the fuzzy gains from each axiom up to each of the input places of the transition. Limitcycles Elimination: It has already been discussed that existence of limitcycles in an FPN can be understood only after m number of belief revision cycles. An analysis [8] of FPN reveals that limitcycles occur because of permanent dominance of all the arcs on one or more cycles. The simplest way to eliminate limitcycles is to make at least one arc on those cycles non-dominant. Further, an input or output arc of a transition on a cycle can be non-dominant, if it has more than one input or output places. But how should one make an arc non-dominant? This calls for adjustment of threshold of some transitions on cycles, so that its output arc becomes permanently nondominant. But what should be the value of the threshold that would cause permanent non-dominance of its output arc? One choice of threshold is the largest fuzzy belief on the reasoning path from the axioms to the selected transition. A question that naturally arises: shall we not lose anything by adjusting thresholds of transitions? The answer to this is affirmative, if we can ensure
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Artificial Intelligence and Soft Co
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Library of Congress Cataloging-in-P
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understanding of the subject. The r
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4. Structural organization of the b
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ACKNOWLEDGMENT The author gratefull
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To my parents, Mr. Sailen Konar and
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Chapter 2: The Psychological Perspe
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5.4.2 Syntactic Methods for Theorem
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9.4.4 Realization of Fuzzy Inferenc
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Chapter 14: Machine Learning Using
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17.4.2.7 Fusing Multi-sensory Data
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22.4 Parallelism at Knowledge Repre
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1 Introduction to Artificial Intell
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instances of decision making [27],
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End. Begin state := new-states - ex
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(a) Generate and Test Approach: Thi
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this problem is useful for the foll
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disciplines. As a young discipline
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The bird and its attributes here ha
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[19] , Kosko [15] and Pedrycz [30]
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Reasoning in the presence of imprec
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where AI finds extensive applicatio
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M1 M2 M3 J2 J 1 8 5 J 2 J 1 2 8 J 2
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1.5.3 The Modern Period The modern
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A B AND C D E AND Y Fig. 1.12: A Pe
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improve reliability by realizing fr
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in AI problems, it supports massive
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[7] Dean, T., Allen, J. and Aloimon
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[34] Rajendra, C. and Chaudhury, D.
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2 The Psychological Perspective of
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The scope of realization of the pro
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2.2.3 Feature-based Approach for Pa
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From Sensory Registers Echoic Reg.
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cognitive memory and its utilizatio
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Tulving’s model bridges a potenti
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A " (a) (b) Fig. 2.6: The character
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his friend s facial image in memory
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sum need not be 180 degrees. Thus a
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same evening. Then he started medit
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construction of knowledge and its o
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saved in a magnetic media, which he
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set-point + error Controller Plant
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Cognitive science has emerged as a
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Further show that the Euclidean lea
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[4] Biswas, B., Mukherjee, A. K. an
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[28] Peterson, M. A., Kihlstrom, J.
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Production Systems, Logical Calculu
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against the data items of the WM. I
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Suppose the WM contains the data Bi
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To demonstrate the working principl
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i) Prefer rules for firing, where u
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In fig. 3.3, the antecedent clauses
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possible offspring states are gener
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Backward Reasoning: The backward re
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p q r s denotes joint premises t w
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3.11.1 Isolation of Knowledge and C
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state, as the predecessor states of
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[3] Buchanan, B. G. and Shortliffe,
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issues: first ‘what to search’
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We will now present two typical alg
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Further, the first state within the
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n1 n2 n3 n8 (a) (b) (c) (d) (e) (f
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The iterative deepening search thus
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legal next states will also lie on
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End. Then push those children into
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Procedure Best-First-Search Begin 1
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End. have multiple parents; Under t
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node. Under this circumstance, the
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6. Pn- Γ the set of paths from nod
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Therefore, f * (n’) = g* (n’) +
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0 +10 B 1+3 C +4 D +5 E F G 2+2 3+1
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The above algorithm considers two d
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Step 1: X (7) Step 2: Step 3: U X (
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(iv) If all of the nodes connected
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If the values are propagated up to
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ii) the alpha value of any MAX node
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αmin =2 βmax =1 C αmin =1 e(n) =
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eductions. Constraint satisfaction
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5 The Logic of Propositions and Pre
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5.2 Formal Definitions The followin
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Definition 5.7: A propositional for
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5.4.1 Semantic Method for Theorem P
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8. p \/ (q Λ ¬ q) ⇔ p 9. p Λ (
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) AND, OR Removal: If the L.H.S. co
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Since all the terminals of the tree
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p v q ¬ q v r p v r ¬ p v s r v s
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Let AS = {A1, A2,, …..,An} be the
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4. If P is a WFF and X is not a qua
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original expression: ( ¬ P11 ∨
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S: = S ∪ {variable or constant /
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¬ Loves (john, mary) ¬Child (X)
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which implies that if q is true the
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ii) C is a ground instance of C [9]
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) p→ (q→ r) ⇔ (p Λ q) → r
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[6] Leinweber, D., “Knowledge-bas
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of a ‘classroom scene’ interpre
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Sits-on (Y, bench, classhour), Pers
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Sits-on (mita, bench, classhour). 3
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Teacher (X) Sub-goal fails Person (
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Definition 6.5: A definite goal is
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Example 6.5, presented below, illus
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6.6.1 Risk of Using CUT It is to be
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In the SLD tree for the Taxpayer pr
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6.9 Fixed Points in Non-Horn Clause
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6.11 Conclusions The main advantage
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preferred X50 >= 2, default X100
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7.1 Introduction Predicate Logic is
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proposed this logic and called it n
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We now present a few modal properti
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Expert Reasoning System New data Dy
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PR5: ¬Visits (Wife, market, Monday
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IN (n1) + IN (n2) + (n11) retracts
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may hold (and not must hold). An ex
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The resulting stable consequences t
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In this example, we call ψ = Bird
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important factor in non-monotonic r
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a) Boy(X) ∧ ¬ L ¬ Likes (X, cho
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8 Structured Approach to Knowledge
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predicate boy (jim). This can be re
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Consequently, we represent the abov
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Is-a Is-a Is-a Protozoa Bacteria Is
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Example 8.2: This example illustrat
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A common question that may be raise
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∴ q is true. 2) Defeasible deriva
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Examination Hall Seat Arrangement I
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Petri nets [5]. It, however, does n
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Since p1 and p 2 contain tokens and
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For building dependency structures,
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The most significant drawback of CD
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Once a script structure for a given
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[6] Nilsson, N. J., Principles of A
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techniques, we first introduce the
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of precision of data and certainty
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where D and S stand for disease and
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To illustrate the reasoning process
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2. It might happen that none of the
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One interesting property of Bayesia
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We now compute the messages that no
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The main steps [8], [12] of the bel
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Updating a node X thus involves upd
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j ↓ i→ round oval-shaped and MB
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Formally, the set of all possible o
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The orthogonal summation of belief
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The orthogonal summation operations
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integers, then we can definitely sa
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where a / b in fast-runner subset r
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where the “o” denotes a fuzzy A
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This may be formally written as µ
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cumulative maximum of the two succe
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“If you can manage to evaluate th
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[7] Shenoy, P. P. and Shafer, G.,
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10.1 Introduction Databases associa
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a finite set of places, D= {d1, d2
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µfast runner(x) 0.9 0.6 0.2 0.1 5
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present in the consequent part; Aug
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Procedure Cycle-detection (P, Q, m,
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Procedure Put-transition (Newlistk,
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The trace of the algorithm cycle- d
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p1 p2 pm n1 n2 nm Fig. 10.4: A tran
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where the elements qw / ∈ {0, 1}
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Q' m and Rm matrices. It may be not
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Definition 10.9: An arc is called p
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The procedure forward reasoning is
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In the above example, A, B, and C a
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n V (q k j ∧r j k ) to be close t
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Let us assume for the sake of simpl
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It is evident from the above distri
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= R b m o ((P ' f m) T o Nf c (t+1)
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Since such choice of Rfm and Rbm sa
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one step of belief revision in the
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10.9 Conclusions The chapter presen
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References [1] Bugarin, A. J. and B
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[24] Looney, C. G., "Fuzzy Petri ne
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11 Reasoning with Space and Time Th
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The other one is an extension by ne
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touches another, the common points
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Procedure Move-optimal (R , Startin
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at Jadavpur University verified thi
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µ above(θ) =sin 2 θ , when -∏
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c c a b q r a b (b) (a) d d p q r p
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11.5 Temporal Reasoning by Situatio
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Holds( now, rain-ceased) Axiom (3)
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Interpretation I Interpretation II
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Example 11.3: Prove that Ā(p) ≡
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11.6.3 Some Elementary Proofs in PT
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The second rule is generally called
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11.9 Conclusions The chapter demons
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12 Intelligent Planning This chapte
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sets opposite to the TV. So they oc
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where On (X, Y) means the object X
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this, let us consider the last prob
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(B), we generate the old state 3, w
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12.3 Least Commitment Planning The
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To start solving the problem, we fi
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We now have three partially ordered
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12.4 Hierarchical Task Network Plan
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d=0 d=1 d=2 d=3 Fig. 12.16: A hiera
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J1 J2 M1 M2 M3 or Job schedules J2
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job k from time 0. For example, the
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3. Design a hierarchical plan for t
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trainer, who supplies the input-out
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Definition 13.1: An object is an en
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The restriction of hypotheses can b
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Positive instances: 1. Object (larg
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To illustrate how LEX performs inte
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alive dead Z=A/W 2 1 0 0 1 2 Living
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where Pi denotes the proportion of
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3. Apply Mapping Function: Apply th
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determine the animals because it do
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simplest form of reinforcement lear
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Let us now assume that the game is
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How can we compute the utility valu
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13.5 Learning by Inductive Logic Pr
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know f ? These questions can be ans
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|H| (1-ε) m ≤ δ ⇒ m ≥ (1/ε
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[7] Michalski, R. S., “A theory a
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However, most biologists are of the
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The most common non-linear function
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(d) b1 a1 (e) c1 cj cq bi ah Fig.14
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transverse diameter, etc. as the in
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x 1 x 2 x n Penultimate layer ∑+F
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Drawbacks of back-propagation algor
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x1 x2 xn 14.6 Widrow-Hoff's Multi-l
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x 1 x 2 x 1 x 2 The training proces
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• If not, select another ADALINE,
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ni 1 = '1' output ( firing ). Each
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designed the weights (W) of a singl
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AND z1 s11 OR w11 w12 w 1n v 11 v12
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− ~ w X k, l ∼ = ⎡ ⎤ ⎢
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V x ∑ w O j = Sgn ( j+ ji i + A
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neural nets in the fields mentioned
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where the first summation is over a
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circuit,” IEEE Trans. on Circuits
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their joint usage in many problems,
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1 0 1 0 11 0 1 0 1 0 1 1 0 10 1 1 1
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15.2 Deterministic Explanation of H
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Thus, in a total of N selections wi
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and population size = 2, which mean
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The behavior of GA without mutation
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a = ∑ Lt (π 0 P n ) i i=1 n→
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Eθ GA Program Z1, Z2, Z3, Z4, X1,
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e [W1 W2 W3 W4 W5 W6] T and [ G1 G2
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function. So, GA in each evolution
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f(x) = Sin (x) + [x * x + y] 1/2 ca
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(EQ (DU(MT CS) (NOT CS)) (DU(MS NN)
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[4] Chakraborty, U. K. and Muehlenb
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16 Realizing Cognition Using Fuzzy
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Cognitive maps are generally used f
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Theorem 16.1: With initial values o
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Here W T is a transposed weight mat
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example, for moving one’s arm, th
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and the weight adaptation equation
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Xk+1 ∆Wk = α Ek οXk+1 T / ((Xk+
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d5 d9 d1 d6 d7 d8 d10 d11 d1=Front
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16.10 Conclusions and Future Direct
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[6] Kosko, B., “Fuzzy cognitive m
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however, have a narrow spectrum of
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contaminated with it. However, for
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1. Move the mask over the image, so
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∇ 2 (g * f ) = ( ∇ 2 g ) * f. W
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It may be noted that segmenting an
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machine has to narrate the game onl
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schematic diagram, briefly outlinin
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same process for all clusters, we g
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Level 1 Level 2 Level 0 Neuron Fig.
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I (u, v) YI Fig. 17.12: Camera mode
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epipolar planes meet the image plan
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(a, b, 1) Fig. 17.15: A 3-D represe
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case III: Planes not parallel to th
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yi = -fi (xi,* ai - 1*) + ( fi / a
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Sample Execution: This is a program
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U t11 t12 t13 t14 x V = t21 t22 t23
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obtained after linearizing measurem
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co-ordinate systems. The correspond
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18 Linguistic Perception Building p
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psychological aspects of understand
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S NP VP ART N VP the man VP V NP ki
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( S ( NP ( ( ART the ) (N man ) ) (
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Sentence: NP VP 1 6 NP: ART N 2 ART
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18.2.3.1 Learning For adaptation of
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context sensitive grammar is thus m
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Generally, we start the sentence S
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VP: V NP S0 S1 Tagged procedure V:=
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Laugh agent animal enjoyer entity p
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18.5 Discourse and Pragmatic Analys
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Thus the elements of communicative
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19 Problem Solving by Constraint Sa
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The recognition problem is the pres
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CSP deals with two classical proble
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= 10 10 = 10 = * Fig. 19.2 (b): Pro
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+ _ V1 R1 V 2 V3 I1 I2 I3 V R2 R3 S
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V = Now assuming the values of V, R
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On (A, Ta) ∧ On (B, Ta) ∧On (C,
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We now represent the neighboring re
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Step 1: (x
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such variable xj , xk, etc., attach
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of C3. If the constraints are liste
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The possible labels of the junction
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The main part of CSP is the formula
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20 Acquisition of Knowledge Acquisi
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invite the experts to attend a meet
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B E A= test ?, B = test results, C
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The database in fig. 20.2 is extrac
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20.5 Knowledge Refinement by Hebbia
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* * * n i ( t + 1) = n i ( t ) at t
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L(r,s) L(s,r) Y(r) Y(s) OS(r,s) n1
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The FPN of fig. 20.5 is formed with
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L(a,l) L(l,a) Y(a) Y(l) OS(a,l) n1
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[3] Cooke, N. and Macdonald, J.,
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21 Validation, Verification and Mai
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Here the problem characteristic is
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The chapter will cover various issu
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21.2.1 Qualitative Methods for Perf
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Paired t-test: Suppose xi ∈ X and
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Ancestor P = {p1,p2} Tr = {tr1,tr2,
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21.3.2 Inconsistencies in Knowledge
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q tr1 tr2 p1 p2 …………… pn
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A rule may also include a number of
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21.4 Maintenance of Knowledge Based
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control of data and knowledge is un
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8. Stefik, M., Introduction to Know
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are not suitable for AI application
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Because of non-determinism in the s
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Procedure Partial-Expansion Begin W
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Step: Fig.22.4: A tree expanded by
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Procedure Generous-Sharing Begin Fl
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To evaluate the performance of the
Page 656 and 657:
Definition 22.2: If the antecedents
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a) AND -Parallelism Consider a logi
Page 660 and 661: However, given sufficient computing
Page 662 and 663: EPN = { P, Tr, D, f, m, A, a, I, o
Page 664 and 665: It is to be noted that if- then ope
Page 666 and 667: After the bindings of the variables
Page 668 and 669: F(x) ← A(X) , B(Y) (1) A(1) ← (
Page 670 and 671: the result produced by clause (2) m
Page 672 and 673: Else Flag:= true; End; Until no-of
Page 674 and 675: transition for the PTVVM. The PTVVM
Page 676 and 677: For the sake of convenience, the pi
Page 678 and 679: 22.6 Conclusions The chapter starte
Page 680 and 681: Rule 3: S(X, Y) → Z (Y, X) Rule 4
Page 682 and 683: 23 Case Study I: Building a System
Page 684 and 685: features in human faces and their r
Page 686 and 687: Definition 23.4: A linear edge segm
Page 688 and 689: lock containing edge. The productio
Page 690 and 691: Definition 23.13: The measure of di
Page 692 and 693: Partitioning Fe into Blocks Feature
Page 694 and 695: The fuzzy membership functions used
Page 696 and 697: ∆ • Fig. 23.4: The delta and th
Page 698 and 699: Thus we have altogether 6 different
Page 700 and 701: cross-section of the vocal tract, w
Page 702 and 703: 23.4.2 Training a Multi-layered Neu
Page 704 and 705: information with their normalized v
Page 706 and 707: SR (Y, X) OR (M (X) ∧ M (Y) ∧ L
Page 708 and 709: procedure, however, no attempts hav
Page 712 and 713: that the inferences will be minimal
Page 714 and 715: the formation of the FPN. The FPN o
Page 716 and 717: is then initiated for detecting the
Page 718 and 719: initial fuzzy beliefs at all places
Page 720 and 721: at all feasible x, y. Then estimate
Page 722 and 723: [13] Malmberg, B., Manual of Phonet
Page 724 and 725: 24.1 Mobile Robots Mobile robots ar
Page 726 and 727: about its world. There exists ample
Page 728 and 729: the process, else it attempts to fi
Page 730 and 731: Depending on the type of planning p
Page 732 and 733: WEST NORTH SOUTH Fig. 24.5: Represe
Page 734 and 735: NW child B NE child 'C' SW child 'D
Page 736 and 737: As observed from the node status, t
Page 738 and 739: In their evolutionary planner algor
Page 740 and 741: tuning of the path, such as avoidin
Page 742 and 743: S S= starting point, G= Goal point,
Page 744 and 745: Once the training is over, the syst
Page 746 and 747: N o Table 24.2: Training pattern sa
Page 748 and 749: 24.9.1 Finite State Machine Finite
Page 750 and 751: occurrence of a state, but also in
Page 752 and 753: 24.10 An Application in a Soccer Pl
Page 754 and 755: Exercises 1. Devise an algorithm fo
Page 756 and 757: [15] Meng, H. and Picton, P. D.,
Page 758 and 759: 24 + The Expectations from the Read
Page 760 and 761:
Appendix A How to Run the Sample Pr
Page 762 and 763:
Predicate: LSR Argument1: S Argumen
Page 764 and 765:
ANTECEDENT 1: Predicate: END conclu
Page 766 and 767:
Rules properly arranged MAIN MENU:
Page 768 and 769:
Level POSITION_FROM_LEFT NODE 1 1 H
Page 770 and 771:
calculating membership values enter
Page 772 and 773:
C > File currently in progress = ci
Page 774 and 775:
When the goal is within concave obs
Page 776 and 777:
path selected 160,160 and 240,240 T
Page 778 and 779:
(a) Robot entering the room (b) Aft
Page 780 and 781:
where tr and Outr denote the target
Page 782 and 783:
∂E/∂Wsp = ∑r (∂E/∂Outr) (
Page 784 and 785:
Proof of theorem10.2: For an oscill
Page 786 and 787:
Thus, we get analogously , and 1 0
Page 788:
= Q 'f m T o Rf m o ( Q ' f m o I )
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