HCFOL and Backward Chaining Concepts
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Questions and Answers

What is implied if an entity is classified as 'hostile' in the context of HCFOL?

  • The entity is neutral towards America.
  • The entity is an enemy of America. (correct)
  • The entity is a friend of America.
  • The entity does not possess any weapons.

    • What is the consequence of a criminal prediction in HCFOL when certain conditions are met?

  • It suggests the entity sells weapons. (correct)
  • It leads to an assumption of hostility towards America.
  • It automatically implies ownership of weapons.
  • It indicates that the entity is American.

    • In backward chaining, what results in an infinite loop where G unifies with C1’s head?

  • C1 conditions are never fulfilled.
  • G contains no relevant links to C1.
  • G is a direct consequence of C2.
  • C1 and C2 both refer to the same variables. (correct)

    • What does the expression 'missile(W) -> weapon(W)' in HCFOL denote?

    <p>Missiles are a type of weapon.</p> Signup and view all the answers

    What is the key advantage of using breadth-first search in backward chaining?

    <p>It eliminates infinite recursion issues.</p> Signup and view all the answers

    What does abduction primarily involve?

    <p>Inferring that a specific cause may explain a given observation.</p> Signup and view all the answers

    Which of the following accurately describes induction?

    <p>It is derived from a dataset of co-occurrence observations to induce a general causal rule.</p> Signup and view all the answers

    What is the key characteristic of analogy in inference?

    <p>It infers new knowledge based on the similarity between instances.</p> Signup and view all the answers

    In the context of machine learning, how is all ML reasoning characterized?

    <p>As inherently uncertain.</p> Signup and view all the answers

    What does the acronym NaF refer to in relation to Prolog?

    <p>Negation as Failure, which establishes abductive reasoning.</p> Signup and view all the answers

    What is a key design decision in resolution-based inference?

    <p>Choosing the resolution strategy</p> Signup and view all the answers

    Which resolution strategy gives priority to unit clauses?

    <p>Unit resolution</p> Signup and view all the answers

    What limitation exists in First-Order Classical First-Order Logic (FCFOL)?

    <p>No quantification over predicate names</p> Signup and view all the answers

    What does the selective linear definite (SLD) resolution strategy involve?

    <p>Pushing query clauses onto a goal stack</p> Signup and view all the answers

    In the context of resolution, what is meant by 'linear resolution'?

    <p>Picking a clause from the KB and one from the last resolvent ancestors</p> Signup and view all the answers

    What does the set of support strategy utilize during resolution?

    <p>A restricted set of support that often includes the negated query</p> Signup and view all the answers

    Which of these strategies is incomplete in general but complete with Horn KB?

    <p>Unit resolution</p> Signup and view all the answers

    What approach can simulate high-order relations in FCFOL?

    <p>Using first-order terms inside predicates</p> Signup and view all the answers

    What is the key result of unifying two constant symbols?

    <p>It succeeds and returns both substitutions.</p> Signup and view all the answers

    In FCFOLF normal forms, what happens to existentially quantified variables?

    <p>They must be skolemized by a function of universally quantified variables.</p> Signup and view all the answers

    What is true about the relationship between Many Sorted Full FOL (MSFFOL) and FCFOL?

    <p>MSFFOL is an extension of FCFOL that includes types.</p> Signup and view all the answers

    What is the purpose of the unification lift in HCFOL?

    <p>To enable forward and backward chaining of the Horn clauses.</p> Signup and view all the answers

    In the FOL normal forms, how are quantifiers treated?

    <p>Universal quantifiers are factored out to the left.</p> Signup and view all the answers

    What principle does the modus ponens inference rule rely on in HCFOL?

    <p>It is a sound and complete method for proving HCFOL theorems.</p> Signup and view all the answers

    What conclusion can be drawn about unification in the example provided of unify(p(X,f(t(a,X),Y), p(b,f(Z,c)))?

    <p>The correct substitutions lead to $ heta = {X/b, Y/c, Z/t(a,X)}$.</p> Signup and view all the answers

    What does it mean when a Horn clause is said to be in a forward chaining process?

    <p>It derives new facts from existing premises.</p> Signup and view all the answers

    What does Herbrand’s Theorem state about a set of clauses that is unSAT?

    <p>There exists a finite subset of these clauses that is also unSAT.</p> Signup and view all the answers

    What is the implication of Gödel’s incompleteness theorem for FCFOL entailments?

    <p>Some FCFOL entailment problems cannot be proven systematically.</p> Signup and view all the answers

    Which of the following techniques enhances the efficiency of FOL inference rules over those based on propositionalization?

    <p>Making relevant instantiations for proofs.</p> Signup and view all the answers

    What happens during the occur-check in the unification process?

    <p>It ensures the variables do not create an infinite tree.</p> Signup and view all the answers

    What condition causes unify(f, g) to fail during its execution?

    <p>The function symbols at any level are different.</p> Signup and view all the answers

    In the context of Herbrand's and Gödel's theorems, what is true regarding FCFOL entailment?

    <p>Certain entailments may go undecided based on the constraints.</p> Signup and view all the answers

    What describes an FOLAF in relation to its structure?

    <p>It is a tree rooted by a predicate symbol.</p> Signup and view all the answers

    What can be inferred about the propositionalization method in FCFOL?

    <p>It creates many groundings, some of which are unused.</p> Signup and view all the answers

    What is the primary extension that Full Classical First-Order Logic (FCFOL) provides over Classical Propositional Logic (CPL)?

    <p>It incorporates tree-structured sentences.</p> Signup and view all the answers

    In the context of Knowledge Representation Languages (KRL), what does it mean for a KRL to be compositional?

    <p>It uses syntactic operators to build complex sentences.</p> Signup and view all the answers

    Which property distinguishes a declarative KRL from a procedural one?

    <p>It allows an inference engine to derive new sentences.</p> Signup and view all the answers

    Which of the following is a key limitation of Classical Propositional Logic (CPL)?

    <p>It has no structure for complex data representations.</p> Signup and view all the answers

    What is a characteristic of Horn Classical First-Order Logic (HCFOL)?

    <p>It provides a practical trade-off between expressivity and explainability.</p> Signup and view all the answers

    In logic representation, what do 'ground terms' refer to?

    <p>Terms that can represent specific individuals in a domain.</p> Signup and view all the answers

    What does it imply for a KRL to be relational?

    <p>It can express universal relations concisely.</p> Signup and view all the answers

    What is the implication of the fact that first-order logic extends CPL?

    <p>It allows for more expressive power in logic statements.</p> Signup and view all the answers

    Which inference method is NOT typically utilized in inference engines for declarative KRL?

    <p>Procedural programming</p> Signup and view all the answers

    Why is Full Classical First-Order Logic (FCFOL) considered more concise than FCPL?

    <p>It represents logical relationships more compactly.</p> Signup and view all the answers

    Which of the following methods can apply composition in Knowledge Representation Languages?

    <p>Using syntactic operators to combine sentences.</p> Signup and view all the answers

    What does the logical representation 'bel(safe(X,Y,T), T)' express in FCFOL?

    <p>A belief about safety in a specific location at a given time.</p> Signup and view all the answers

    Which aspect of Horn Classical First-Order Logic (HCFOL) provides a benefit in real-world applications?

    <p>Its balance between complexity and modularity.</p> Signup and view all the answers

    Which of the following best describes the role of an inference engine (IE) in a KRL?

    <p>It derives new sentences based on existing relations.</p> Signup and view all the answers

    Study Notes

    Introduction to "Symbolic" Artificial Intelligence

    • Course: INF4067 – S8 - MAJ - IADS 2024-2025
    • Section: 3 - Predicate Logic
    • Authors: Jacques Robin and Camilo Correa
    • Copyright: 2022-2025

    Section Outline

    • Full Classical First-Order Logic (FCFOL): A relational extension of FCPL
    • Horn Classical First-Order Horn Logic (HCFOL): A practical trade-off between expressivity, explainability, and inference scalability

    Section Readings

    • First-Order logic: AIMA4 chapters 8 and 9

    KRL Properties: Procedural vs. Declarative

    • Procedural:
      • Knowledge is split into application-specific data structures or objects representing the agent's environment state space and its preferences.
      • Also includes application-specific functions in a functional language, procedures in a procedural language, or methods in an object-oriented language.
      • For each application, manually program multiple code units to derive environment state properties from percepts and agent action history.
    • Declarative:
      • Application-specific sentences in a KRL that an Inference Engine (IE) can use to infer new sentences.
      • IE is independent, fully reusable to perform inference.
      • Specify in the knowledge base (KB) only the relations between properties.
      • Let the IE (e.g., a SAT solver or resolution, forward, or backward chaining) figure out how to infer some properties based on those relations.

    KRL Properties: Compositionality, Structure, and Relation

    • Compositional:
      • Has syntactic operators to build complex sentences from atomic sentences (e.g., logical connectives in CPL).
      • Semantics of complex sentences are derived from the semantics of their atomic components (e.g., truth tables in CPL).
    • Structured:
      • Atomic sentences specify properties over complex data structures, rather than just symbols.
      • One key limitation of CPL is its lack of structure.
    • Relational:
      • Can express, intentionally and concisely, in a single sentence, universal relations between all individuals from a given class.
      • Another key limitation of CPL is its lack of relationality.
      • SQL database schema (not the data itself) is declarative and relational.

    FCFOL

    • Full Classical First-Order Logic (FCFOL):
      • Defines the syntax and semantics of composite formulas, connected formulas, and binary formulas.
      • Includes quantifier expression, binary FCFOL connectives, unary FCFOL connectives, negative literals, and logical variables.
    • Examples:
      • X, Y, T safe(X, Y, T) pit(X, Y)
      • (wumpus(X,Y) ) wumpusAlive(T))
      • X, Y, T bel(safe (X, Y, T), T)
      • bel(noPit(X, Y), T) (bel(noWumpus(X, Y), T)

    FCFOL vs FCPL as WW Agent KRL

    • FCFOL representation is exponentially more concise for specifying and explaining agent knowledge.

    KR with FCFOL

    • Ground Terms: Represent application domain individuals, concept instances, or objects with properties. Examples: pit(1, 2, 3, 7), gold(4, 4).
    • Constant Terms: Represent values of unstructured properties. Example: 1, 2, 3, 7 in pit(1, 2, 3, 7).
    • Logical Variables: Represent intended properties of concepts and individuals. Examples: Id, X, Y and T in pit(Id, X, Y, T).
    • Non-Ground Terms: Represent application domain concepts or classes with logical variable instantiations (binding). Example: pit(Id, X, Y, T) with the substitution {Id = 1, X = 2, Y = 3, T = 4}.

    FCFOL Semantic Models

    • Atomic FCPL formulas logically connect unstructured symbols. Semantics or meaning of a FCPL formula is solely derived from the cognitive association between symbols and the IA's environment properties. The logical connectives' truth table.
    • FCFOL makes the same epistemological commitment about degrees of belief as FCPL. Degrees of belief for a FCFOLF can be derived from FOLAF using truth tables.

    FCFOL Semantic Models

    • FCFOL makes a richer, relational, ontological commitment than FCPL, classifying symbols into four categories (Constants, Variables, Function Symbols, Predicate Symbols).
    • Constants in FOLAF denote unstructured objects in the IA's environment or property values.
    • Variables in FOLAF denote sets of unstructured objects or properties.
    • quantifiers specify whether the property holds for all possible values of variables
    • Function symbols denote objects as trees.
    • Predicate symbols represent relations among objects.

    FCFOL Semantic Example

    • Shows semantic relations (e.g., person, king, brother, onHead, crown), function, predicate, and variable symbols, illustrated within the context of an example.

    FCFOL Denotational Semantic Models

    • To compute the (Herbrand) model of a FCFOL formula f, construct the Herbrand Universe H(f) (all ground terms), the Herbrand Base H(f) (conjunction of all clauses), Skolemize clauses with existentially quantified variables into ground clauses, and then apply propositionalization to derive the model's truth values.

    FCFOL Inference by Propositionalization

    • To answer a query of the form "f |= g," propositionalize f and g using the Herbrand Universe and Herbrand Base, then determine if propositionalized f entails propositionalized g using a propositional inference system like resolution. Function symbols will likely mean an infinite Herbrand base.

    Herbrand's and Gödel's Theorems

    • Herbrand's Theorem (1930): If a set of clauses is unsatisfiable, a finite subset of those clauses is also unsatisfiable. Propositionalization of FCFOL can use this.
    • FCFOL entailment is semi-decidable. Algorithms can guarantee a proof for F |= G if F does actually entail G.

    FOLAF Unification

    • unifies FOL formulas by comparing their structures and finding substitutions.
    • uses an algorithm by traversing both trees synchronously in a depth-first manner.

    FCFOLF Normal Forms

    • Explains how FCFOLs can be put in specific normal forms.
    • All quantifiers are factored out to the left for normalization.
    • Existentially quantified variables are skolemized into constants to remove existential quantification.

    FOL Normal Forms

    • Expands the scope of FOL normal form rules, by introducing types to each predicate argument.

    Horn CFOL (HCFOL)

    • HCFOL is a subset of FCFOL focusing on single-conclusion clauses.
    • Unification lifts HCLP clause chaining (forward and backward) into HCFOL.
    • Forward chaining unifying modus ponens inference is sound and complete.

    Horn FOL as a Specialization of FOL

    • Shows a specific version of sorted full FOL (MSFFOL) tailored to Horn logic.

    HCFOL Forward Chaining Example

    • Demonstrates a natural language problem about a crime, expressed in HCFOL, and solved using forward chaining.

    HCFOL Backward Chaining

    • Explains HCFOL backward chaining.
    • Presents a method potentially incomplete if searching depth first because of an infinite loop scenario.
    • Demonstrates a method that is complete if searching breadth first or using iterative deepening.

    Non-Horn FCFOLF Resolution Example

    • Illustrates non-Horn FCFOL resolution example with predicates (e.g., loves, kills).

    Resolution Strategies

    • Explains various strategies for resolution, including unit preference, set of support, and linear resolution.
    • Specific strategies include strategies for a resolution-based Inference Engine to address the problem of selecting pairs of clauses to attempt resolution.
    • Explains resolution selection strategies like unit preference and set of support that favor efficient resolution and completeness within the context of Horn clauses.

    Limitations of FCFOL

    • Explains limitations of FCFOL such as lack of quantification over predicate names, which limits high-order relationships and the inability to directly handle issues like transitive closure.
    • Reasoning regarding truth value of a formula over time is limited to monotonic reasoning.
    • Simulating changes in truth values can be addressed with axiomatization.

    Example of Axiomatization and Reification

    • Uses the Event Calculus to illustrate how a monotonic IE can be used to implement non-monotonic reasoning that handles state change in an inference Engine.
    • Introduces application-independent axioms for predicates like "happens" and "terminates" to manage non-monotonic reasoning in a structured way, and relates these concepts to AIMA4 10.3.
    • Demonstrates reification in the event calculus to represent relationships about events, states, and times when they exist.

    Four Top-Level Classes of IA Inferences

    • Explains the four main categories of inference methods in AI, Deduction, Abduction, Induction, and Analogy.

    Diversity of Logic KRLs and IEs

    • Outlines various types of knowledge representation languages (KRLs) and Inference Engines (IEs) used in AI.

    IE Diversity

    • Presents a more detailed overview of different categories of Inference Engines (IEs) found in AI, highlighting the diversity of approaches across these methods.
    • Shows the diverse types of reasoning algorithms that exists with different approaches that are often used in AI Inference engines (IEs).

    KRL Diversity

    • Illustrates the range of Knowledge Representation Languages (KRLs).
    • Categorizes these languages based on aspects like symbolic representation, epistemological commitment, and ontological commitments.

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    Predicate Logic PDF - INF4067 - S8 - MAJ - IADS 2024-2025

    Description

    Test your understanding of key concepts in Higher-Order Class Logic (HCFOL) and backward chaining. This quiz covers implications of hostility, predictions in HCFOL, and important strategies in logic and reasoning, including abduction and induction. Challenge your knowledge on the inference techniques used in machine learning and Prolog.

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