Foundations of DKS Foundations of DKS Outline Foundations of Data and Knowledge Systems EPCL Basic Training Camp 2012 3. Foundations of Automated Theorem Proving Part Two 3.1 Substitutions and Unification 3.2 Transformation into Clause Form 3.3 Herbrand Interpretations Thomas Eiter and Reinhard Pichler 3.4 Semantic Trees and Herbrand’s Theorem 3.5 Proof of Several Fundamental Theorems Institut für Informationssysteme Technische Universität Wien 19 December, 2012 Thomas Eiter and Reinhard Pichler 19 December, 2012 1/36 Thomas Eiter and Reinhard Pichler 19 December, 2012 2/36 Foundations of DKS 3. Foundations of ATP Foundations of DKS 3. Foundations of ATP 3.1 Substitutions and Unification Roadmap Outline Motivation This part of the lecture is based on the following book: Alexander Leitsch: The Resolution Calculus , Texts in Theoretical Computer Science, Springer-Verlag Berlin, Heidelberg, New York, 1997. 3. Foundations of Automated Theorem Proving 3.1 Substitutions and Unification Several fundamental results on First-Order Predicate Logic have been 3.2 Transformation into Clause Form stated without proof in the first part of this lecture, like the Completeness 3.3 Herbrand Interpretations Theorem, the Compactness Theorem, and the Löwenheim-Skolem Theorem. 3.4 Semantic Trees and Herbrand’s Theorem We proceed in the spirit of Automated Theorem Proving and first prove 3.5 Proof of Several Fundamental Theorems Herbrand’s Theorem. It is then easy to prove the other results. In the article of Bry et al., the argumentation is in the opposite direction: Herbrand’s Theorem is obtained as an easy consequence of the Compactness Theorem which in turn follows easily from the Completeness Theorem (which is stated without proof). Thomas Eiter and Reinhard Pichler 19 December, 2012 3/36 Thomas Eiter and Reinhard Pichler 19 December, 2012 4/36
Foundations of DKS 3. Foundations of ATP 3.1 Substitutions and Unification Foundations of DKS 3. Foundations of ATP 3.1 Substitutions and Unification Substitutions (Ground) Instances Definition (Ground substitution, ground instance) A ground substitution is a substitution whose codomain consists of ground Definition (Substitution) terms only. A grounding substitution for a term t is a ground substitution σ A substitution is a function σ , written in postfix notation, that maps terms to whose domain includes all variables in t , such that tσ is ground. A ground terms and is instance of t is an instance of t that is ground. homomorphous, i.e., f ( t 1 , . . . , t n ) σ = f ( t 1 σ, . . . , t n σ ) for compound terms and cσ = c for constants. identical almost everywhere, i.e., { x | x is a variable and xσ � = x } is finite. The domain of a substitution σ is the finite set of variables on which it is not identical. Its codomain is the set of terms to which it maps its domain. A substitution σ is represented by the finite set { x 1 �→ x 1 σ, . . . , x k �→ x k σ } where { x 1 , . . . , x k } is its domain and { x 1 σ, . . . , x k σ } is its codomain. Thomas Eiter and Reinhard Pichler 19 December, 2012 5/36 Thomas Eiter and Reinhard Pichler 19 December, 2012 6/36 Foundations of DKS 3. Foundations of ATP 3.1 Substitutions and Unification Foundations of DKS 3. Foundations of ATP 3.1 Substitutions and Unification (Ground) Instances (Ground) Instances Definition (Ground substitution, ground instance) Definition (Ground substitution, ground instance) A ground substitution is a substitution whose codomain consists of ground A ground substitution is a substitution whose codomain consists of ground terms only. A grounding substitution for a term t is a ground substitution σ terms only. A grounding substitution for a term t is a ground substitution σ whose domain includes all variables in t , such that tσ is ground. A ground whose domain includes all variables in t , such that tσ is ground. A ground instance of t is an instance of t that is ground. instance of t is an instance of t that is ground. Definition (Instance of a formula) Definition (Instance of a formula) Let ϕ be a formula and σ a ground substitution. Then ϕσ is the formula Let ϕ be a formula and σ a ground substitution. Then ϕσ is the formula obtained from ϕ by replacing each free variable occurrence x in ϕ by xσ . obtained from ϕ by replacing each free variable occurrence x in ϕ by xσ . Definition (Ground instance of a formula) Let ϕ be a formula. Let ϕ ′ be a rectified form of ϕ . Let ϕ ′′ be obtained from ϕ ′ by removing each occurrence of a quantifier for a variable. A ground instance of ϕ is a ground instance of ϕ ′′ . Thomas Eiter and Reinhard Pichler 19 December, 2012 6/36 Thomas Eiter and Reinhard Pichler 19 December, 2012 6/36
Foundations of DKS 3. Foundations of ATP 3.1 Substitutions and Unification Foundations of DKS 3. Foundations of ATP 3.1 Substitutions and Unification Unification Unification Definition (Unification) Definition (Unification) Two terms s and t are unifiable, if there exists a substitution σ with sσ = tσ . In Two terms s and t are unifiable, if there exists a substitution σ with sσ = tσ . In this case σ is called a unifier of s and t . this case σ is called a unifier of s and t . A most general unifier or mgu of s and t is a unifier σ , s.t. for any other unifier A most general unifier or mgu of s and t is a unifier σ , s.t. for any other unifier σ ′ of s and t , there exists a substitution ϑ with σ ′ = σϑ . σ ′ of s and t , there exists a substitution ϑ with σ ′ = σϑ . If σ is a most general unifier of s and t , then the term sσ is called a most If σ is a most general unifier of s and t , then the term sσ is called a most general common instance of s and t . general common instance of s and t . Remarks For any two terms s and t , if they are unifiable, then there exists an mgu of s and t . In this case, the most general common instance is unique up to variable renaming. Testing if s and t are unifiable and, if so, computing an mgu can be done efficiently. However, care is required concerning the used data structures (e.g., dag representation rather than string representation of terms). Thomas Eiter and Reinhard Pichler 19 December, 2012 7/36 Thomas Eiter and Reinhard Pichler 19 December, 2012 7/36 Foundations of DKS 3. Foundations of ATP 3.1 Substitutions and Unification Foundations of DKS 3. Foundations of ATP 3.1 Substitutions and Unification Example Example s = h ( x 1 , x 2 , . . . , x n ) s = h ( x 1 , x 2 , . . . , x n ) t = h ( f ( x 0 , x 0 ) , f ( x 1 , x 1 ) , . . . , f ( x n − 1 , x n − 1 )) t = h ( f ( x 0 , x 0 ) , f ( x 1 , x 1 ) , . . . , f ( x n − 1 , x n − 1 )) Unification of s and t yields the mgu Unification of s and t yields the mgu { x 1 �→ f ( x 0 , x 0 ) , x 2 �→ f ( f ( x 0 , x 0 ) , f ( x 0 , x 0 )) , . . . } { x 1 �→ f ( x 0 , x 0 ) , x 2 �→ f ( f ( x 0 , x 0 ) , f ( x 0 , x 0 )) , . . . } Remarks The mgu maps each x i to a complete binary tree of height i . The size of the mgu (represented as a string or a tree) is exponential in the size of the input due to copying (or duplication) of (sub)terms. Better alternative. Represent terms as directed acyclic graphs. Intuition. Build up a substitution by collecting a list of bindings without duplicating terms, i.e. { x 1 �→ f ( x 0 , x 0 ) , x 2 �→ f ( x 1 , x 1 ) , . . . } Thomas Eiter and Reinhard Pichler 19 December, 2012 8/36 Thomas Eiter and Reinhard Pichler 19 December, 2012 8/36
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