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A.Rakesh Babu (^) M.C.A., M.Tech. 69
Chapter 7
SYMBOLIC REASONING UNDER UNCERTAINITY
Introduction to Nonmonotonic Reasoning:
Before we can explain non-monotonic reasoning, we can consider the following example. Ex: Suppose we take, there are 3 persons to be suspects in a murder case. Let Abbott, Babbitt, Cabot be suspects in a murder case. Abbott was an alibi in the register of respected hotel in Albany. Babbitt also has alibi for his brother-in-law testified that Babbitt was visiting him in Brooklyn at the time. Cabot pleads alibi too, claiming to have been watching a ski meet in Catskills. Based on the above statements, we believe
- That Abbott did not commit the crime,
- That Babbitt did not,
- That Abbott or Babbitt or Cabot did,
- That Cabot did not. The explanation of the statements can be considered “Who might commit the crime?” we cannot identify that who has committed the crime. Because of that knowledge base is not enough. There are two systems to explain the statements and avoid the uncertainty is Nonmonotonic Reasoning Statistical Reasoning
Nonmonotonic Reasoning, in which the axioms and/or the rules of inferences are extended to
make it possible to reason with incomplete information.
Statistical Reasoning, in which each of the rules is inferences, should be assigned with some
numerical measure. It consider either true or false but not both.
Nonmonotonic system principles:
There are three principles in Nonmonotonic reasoning. They are as follows
- Knowledge base is complete.
- Consistent- we are able to add the new statement which does not lead to contradiction.
- Each statement will be true or false but not both.
A.Rakesh Babu (^) M.C.A., M.Tech. 70 Logics for Nonmonotonic reasoning Monotonicity is fundamental to the definition of first-order predicate logic, we need some logics to support nonmonotonicity. The formal approach should posses all the features given, Defines the set of possible words according to the facts we have. Provides a way to say, it is preferable to believe in some model rather others. Provides the basis for implementation.
Nonmonotonic Reasoning:
It is basically extension of first order predicate logic to include model operator say M. the purpose of this is allow for consistency. Ex: (x): play-instrument(x) ^ Mimprovise (x) musician (x) It states that for all x if play an instrument and the fact that x can improve consistent with all other knowledge then we can infer that x is musician. How we can define consistency? One common solution is, to show that fact p is true attempt to prove, &p. If we fail we may say that p is consistent. Ex: x: Republican(x) ^ M pacifist(x) pacifist(x)
DEFAULT REASONING:
Default reasoning is one type of non-monotonic reasoning, which treats conclusions as, believed until a better reason is found to believe something else. Two different approaches to deal with non-monotonic system are: Nonmonotonic logic Default logic
Nonmonotonic logic: Nonmonotonic logic is one in which the language of first order predicate
logic is augmented with a modal operator M, which can be read as “is consistent”. Non-monotonic Logic defines the set of theorems that can be derived from a set of wff’s A to be the intersection of the sets of theorems that results from the various ways in which the wff’s of A might be combined. A Λ MB B ¬A Λ MB B We conclude: MB B.
A.Rakesh Babu (^) M.C.A., M.Tech. 72
Inheritance:
Inheritance is a basis for inheriting attribute values from a prototype description of a class to individual entities that belong to the class. Inheritance says that “An object inherits attribute values from all the classes which it is member unless doing so leads to contradiction, in which case a value from a more restricted class has precedence over a value from a broader class”. Ex : Base Ball Game In this we can combine more than one statement in the following way.
1. Adult Male (x) : height (x, 5’10”) Height (x, 5’10”) 2. Baseball player (x) : height(x, 6 ’ 1 ” ) Height(x, 6’ 1”) If it is considered to assume that, x is not a baseball player then x height is 5’ 10”. x: Adult Male (x) : ¬Baseball player (x) ^ Height (x, 5’10”) Height (x, 5’10”)
MINIMALIST REASONING:
Minimalist reasoning follows the idea that “there are many fewer true statements than false ones. If something is true and relevant it makes sense to assume that is has been entered into our knowledgebase. Therefore, assume that the only true statements are those that necessarily must be true in order to maintain the consistency of knowledge base”. Two kinds of minimalist reasoning are: (1) Closed world assumption (CWA). (2) Circumscription.
Closed world assumption: Another form of assumption made with regard to incomplete
knowledge, is more global in nature than single defaults. This type of assumption is useful in application where most of the facts are and it is, therefore, reasonable to assume that if a proposition cannot be proven, it is false. This is known as the closed world assumption with failure as negation. This means that in a kb if the ground literal p(a) is not provable, then ~p(a) is assumed to hold true. CWA is another form of non-monotonic reasoning. CWA is essentially the formalism under which prolog operates and prolog has been shown to be effective in numerous application. Disadvantages:
- A knowledge base augmented with CWA is not consistent.
- CWA assumptions are not always true in the world.
A.Rakesh Babu (^) M.C.A., M.Tech. 73
- CWA forces completion of a knowledge base by adding the negation assertion ¬P whenever it is consistent to do so. But the assignment of a property to some predicate P and its complement to the negation of P may be arbitrary. Example: Define a predicate SINGLE and create knowledge base. SINGLE (john) SINGLE (Mary) Now if we ask about john then answer may be ¬ SINGLE (john). Suppose we choose the predicate married. Then the knowledge base may be ¬Married (john) ¬Married (Mary) If we ask about john the CWA may give the answer as ¬ Married (john)
Circumscription:
Circumscription is a procedure in which we add new axioms to the existing knowledge base and the effect of these is to force a minimal interpretation on a selected portion of the knowledge base. Each specific axiom describes a way that the set of values for which a particular axiom of the original theory is true is to be circumscribed. CWA does not capture all of the idea. It has two limitations: CWA operates on individual predicates with out considering the interactions among predicates that are there in the knowledge base. CWA assumes that all predicates have all of their instances listed. But this is not true in many knowledge-based systems. Circumscription overcomes that above limitations. Example: Suppose we have a university world situation in which there are two known students. We may wish to state that the known students are the only students to circumscribe the students. i.e. Student (a) Student (b) We have the circumscriptive inferences as even Student (a) & Student (b) infer as, X: Student (x) (x= a v x = b)
A.Rakesh Babu (^) M.C.A., M.Tech. 75 Example: The dependency directed backtracking can use in finding the route such as search for a path between two locations first on a map that includes only major cities and inter state highways, then refine paths to get to the inter state highways
2. Justification Based Truth Maintenance System
The idea of truth maintenance system or TMS arose as a way of providing the ability to do dependency directed backtracking and to support Nonmonotonic reasoning. The main job of the TMS is to maintain consistency of the knowledge being used by the problem solves and not to perform any inference functions. The TMS also gives the inference component the latitude to perform Nonmonotonic inferences. When new discovers are made, this more recent information can displace previous conclusions that are no longer valid .in this way the set of beliefs available to the problem solver will continue to be current and consistent. Diagram: Tell Ask Fig: Architecture of the problem solver with a TMS The role played by the TMS a part of the problem solver. the inference engine solves domain problems based on its current belief set, while the TMS maintains the currently active belief set .the updating process is incremented .after each inference ,information is exchanged between the 2 components .the IE tells the TMS what the deductions has made. The TMS in turn, asks questions about current beliefs and reasons for failures. It maintains a consistent set of beliefs for the I.e. to work with even if now knowledge is added & removed. The TMS maintains complete records of reasons of justifications for beliefs. Each proposition having at least 1 valid justification is made a part of current belief set. Statements lacking acceptable justifications are excluded from this set. When a contradiction is discovered the statements responsible for the contradiction are identified & an appropriate one is retracted. This in turn may result in other retractions & additions. The procedure used to perform this process is called dependency back tracking. The TMS maintains records to reflect retractions and additions & that will always know its current belief set. The records are maintained in the form of dependency network. The node in the network represents KB entries such as premises, conclusions, inference rules and the like. Attached
Knowledge
Inference engine TMS
A.Rakesh Babu (^) M.C.A., M.Tech. 76 to the nodes are justifications, which represent steps from which the node was derived. Nodes in the belief set must have valid justifications. a premise is a fundamental belief which is assumed to be always true. Premises need no justifications. They form a base from which all other currently active nodes can be explained in terms of valid justifications. Example: ABC murder story. Initially Abbottt is believed to be the primary suspect the reason is non-monotonic. The three assertions believed initially are. Suspect Abbottt (Abbottt is the primary suspect) Beneficiary Abbott (Abbottt is a beneficiary of the victim) Alibi Abbottt (Abbottt was at on Albany hotel at that time.) Representation in TMS: A TMS dependency network offers a purely syntactic, domain-independent way to represent belief and change it consistently. Suspect Abbott [IN] ] supported belief Justification Beneficiary Abbott Alibi Abbott Justification:
- The assertion “Suspect Abbott” has an associated TMS justification. An arrow to the assertion it supports connects the justification.
- Assertions in a TMS dependency network are believed when they have a valid justification.
- Each justification has two parts: a. An IN-list [connected to justification by ‘+’] b. An OUT-list [connected to justification by ‘-‘]
- If the assertion corresponding to the node should be believed, then in the TMS it is labeled IN.
- If there is no reason to believe the assertion, then it is labeled OUT. Premise Justification: Premise justifications are always considered to valid. Premises need no justifications. Labeling task of a TMS: The labeling task of a TMS is to label each node so that three major criteria of dependency network are met.
- Consistency
- Well-founded-ness
- Resolving contradictions.
A.Rakesh Babu (^) M.C.A., M.Tech. 78 Well-Founded ness criterion: (1) It is defined as the proper grounding of a chain of justifications on a set of nodes that do not themselves depend on the nodes they support. (2) For example: Cabot justification for his alibi that he was at a ski show is hardly valid. The only support for the alibi of attending the ski show is that Cabot is telling the truth. ------ (1) The only support for his telling the truth would be if we knew he was at the ski show. --------(2) Above (1) and (2) statements show a chain of IN- List links to support the “Alibi Cabot “ Node. So, in such cases the node should be labeled OUT for well-founded ness. Suspect Cabot [IN]
Beneficiary Cabot [IN] Alibi Cabot [OUT] +
- TellsTruthCabot [OUT] Cabot Justification --- Well Founded ness. Resolving Contradictions Criterion: (1) A contradiction in TMS is a state of the database explicitly declared to be undesirable. (2) Example: In our example, a contradiction occurs if we do not have at least one murder suspect or if there are suspects other than Abbott, Babbitt, and Cabot. Contradiction [out] Suspect Abbott Suspect Babbit Suspect Cabot Other suspect
A.Rakesh Babu (^) M.C.A., M.Tech. 79 (3) Initially there is no valid justification for other suspects so, contradiction is labeled OUT. (4) Suppose Cabot was seen on T.V that he was at the ski slopes, and then is causes “Alibi Cabot” node to be labeled IN. So, it makes ‘Suspect Cabot’ node to be labeled OUT. (5) The above point gives a valid justification for contradiction and hence is labeled IN. Contradiction [IN] Alibi Abbott Alibi Babbit Alibi Cabot [IN] Other suspects (6) The job of TMS is to determine how the contradiction can be made OUT. I.e. The justification should be made invalid. (7) Non monotonic justifications can be invalidated, by asserting some fact whose absence is required by the justification (8) That is we should install a justification that should be valid only as long as it needs to be. (9) A TMS have algorithms to create such justifications, which is called Abductive justification.
3. Logic Based Truth Maintenance System
LTMS stands for logic based truth maintenance system. It is combination of both DDB and JTMS. In ABC murder story example in an LTMS system, we would not have created an explicit contradiction corresponding to the assertion that there was no suspect.
BFS:
The breadth first search proceeds by employing all the nodes at a given depth before proceeding to the next level. Here all immediate children of nodes are explored before any of the children. The Assumption-based truth maintenance system (ATMS) is an alternative way of implementing Nonmonotonic reasoning. In both JTMS and LTMS systems a single line of reasoning is pursued at a time, and dependency directed backtracking occurs whenever it is necessary to change the system’s assumptions. In an ATMS, alternative paths are maintained in parallel. Backtracking is avoided at the expense of maintaining multiple contexts
A.Rakesh Babu (^) M.C.A., M.Tech. 81 A semantic Net can be further classified into different techniques as follows.
- Intersection search.
- Representing Nonbinary Predicates.
- Making some important distinctions.
- Partitioned Semantic Nets.
- The evolution into frames.
Intersection search
Semantic nets were used to find relation ship among objects by spreading activation out from each of two nodes seeing where the activation meets.
Representing Nonbinary Predicates:
Semantic nets are a natural way to represent nonbinary predicates. It would be similar to isa and instance. Some of the arcs in the below example are as follows. isa Visiting team score home-team fig: A semantic Net for an n-place Predicate isa (G23, Game) home-team (G23, Dodgers) visiting team (G23, Cubs) Score (G23, 5- 3 ) In this example, we can represent an arc more than 2 objects is Score (Cubs, Dodgers, 5-3) The above statement is binary predicate. Suppose we take a unary predicate Man (Marcus), it can be converted into a binary predicate as Instance (Marcus, Man) Game Cubs^ G23^5 -^3 Dodgers
A.Rakesh Babu (^) M.C.A., M.Tech. 82
Making some important distinctions
In this we can show difference between two semantic nets. Ex:
- John height is 72. height
- John is taller than Bill. Height height Greater than The above two statements can we represent in only one form is Height height Greater than
Partitioned Semantic net
Suppose we want to represent simple quantified expressions in semantic nets. One way to do this is to partition the semantic net into a hierarchical set of spaces, each of which corresponds to the scope of one or more variables. This net corresponds to the statement The dog bit the mail carrier. John 72 JOHN Bill H1 H JOHN Bill H1 H 72
A.Rakesh Babu (^) M.C.A., M.Tech. 84 In this net, the node c representing the victim lies out side the form of the general statement. Thus it is not viewed as an existentially quantified variable whose value may depend on the value of d, instead it is interpreted as standing for a specific entity. (In this case, a particular constant), just as do other nodes in a standard, non partitioned. Every dog has bitten every mail carrier In this case, g has two links, one pointing to d, which represents any dog, and one pointing to m, representing any mail carrier. SA isa isa isa SI assailant victim Dogs Bite Mail-carrier d b m GS g Gs Dogs Bite Constables Town-Dogs g d b c isa isa isa assailant victim SI SA isa form isa
A.Rakesh Babu (^) M.C.A., M.Tech. 85 Frames: Frames were first introduced by Marvin Minsky (1975) and a data structure to represent a mental model of a stereotypical situation such as driving a car, attending a meeting or eating in a restaurant. Frames are general record like structures, which consist of a collection of slots and slot values. The slots may be of any size and any type. Slots typically have names and any number of values. A frame can be defined as a data structure that has slots for various objects and collection of frames consists of expectations for a given situation. A frame structure provides facilities for describing objects, facts about situations, procedures on what to when a situation is encountered because of these facilities a frame provides, frames are used to represent the two types of knowledge. A general structure of a frame system as follows. Person Isa: Mammal Cardinality: 6,000,000, *handed: Right Adult-Male Isa: Person Cardinality: 2,000,000, *height: 5 - 10 ML-Base-ball Player Isa: Adult-Male Cardinality: 624 *height: 6 - 1 *Bats: equal to handed *batting-avg:. *team: *uniform Color: Fielder Isa: ML-Base-ball Player Cardinality: 376 *batting-avg:. Pee-Wee-Reese Instance: Fielder Height: 5 - 10
A.Rakesh Babu (^) M.C.A., M.Tech. 87 Pee-Wee-Reese Instance: Brooklyn-Dodgers Instance: Fielder Batting-avg:. Uniform-Color: Blue Suppose we have to represent the relationship among the classes as follows. Isa isa isa isa isa Instance instance ML-Baseball-Player Is-covered-by: {pitcher. Catcher, Fielder}, {American-Leaguer, National-Leaguer} Pitcher Isa: ML-baseball-Player Mutually-disjoint-with: {Catcher, Fielder} Catcher: Isa: ML-Baseball-Player Mutually-disjoint-with: {pitcher, Fielder} Fielder: Isa: ML-Baseball-Player Mutually-disjoint-with: {pitcher, Fielder} American-Leaguer Isa: ML-Baseball-Player Mutually-disjoint-with: {National-Leaguer} ML-Baseball-Player American- Leaguer National Leaguer Three-Finger-Brown Pitcher Catcher Fielder
A.Rakesh Babu (^) M.C.A., M.Tech. 88 National-Leaguer Isa: ML-Baseball-Player Mutually-disjoint-with: {American-Leaguer} Three-finger-brown Instance: pitcher Instance: National-leaguer Reasoning using frames: The task of action frames is to provide facility for procedural attachment and help transforming from initial to goal state. It also helps in breaking the entire problem in to sub-tasks, which can be described as top-down methodology. It is possible for one to represent any tasks using these action frames. Reasoning using frames is done by instantiation. Instantiation process begins when the given situation is batches with frames that already exist. The reasoning process tries to match the frame with the situation and latter fills up slots for which values must be assigned. The values assigned to the slot depict a particular situation and but this reasoning process tries to move from one frame to another to match the current situation. This process builds up a wide network of frames, there by facilitating one to build a knowledge base for representing knowledge about common sense. Frame-based representation language: Frame representations have become popular enough that special high level frame-based representation languages have been developed. Most of languages use LISP as the host language. They typically have functions to create access, modify updates and display frames. Implementation of frame structures: One way to implement frames is with property lists. An atom is used as the frame name and slots are given as properties. Facts and values with in slots become lists of lists for the slot property. Another way to implement frames is with an association list ( an-a-list), that is, a list of sub lists where each sub list contains a key and one or more corresponding values. It is also possible to represent frame like structures using Object oriented programming extensions to LISP languages such as Flavors.
