Many Sorted First-order Logic

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Many Sorted First-order Logic

• Student Liuxing Kan
• Instructor William Farmer
• Dept. of Computing and Software
• McMaster University, Hamilton, CA

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Contents

• Introduction What is many sorted FOL
• Syntax Languages of many sorted FOL
• Syntax Terms and Formulas of many sorted FOL
• Example of Many-sorted FOL Semantics
• Semantics of many sorted FOL
• Proof Theory of Many-sorted FOL
• The completeness of many-sorted logic
• Reduction many-sorted logic to one-sorted logic
• Reference

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Introduction what is many-sorted FOL

• Sometimes people want to express properties of
  structures of different sorts (types). There are
  plenty of examples of subjects where the
  semantics of formulas are many-sorted structures.
  For instance, in geometry, to take a simple and
  ancient example, we use universes of pointes,
  lines, angles, triangles, rectangles, polygons,
  etc.
• By adding to the formalism of first-order logic
  the notion of sort, we can obtain a flexible and
  convenient logic called many-sorted first order
  logic, which has the same properties as first
  order logic.
• In contrast to FOL, in many-sorted FOL, the
  arguments of function and predicate symbols may
  have different sorts, and constant and function
  symbols also have some sorts.
• Uses of many-sorted logic abstract data types,
  semantics and program verification, definition of
  programming languages, algebras for logic,
  databases, dynamic logic, semantics of natural
  languages, computer-aided problem solving,
  knowledge representation of design, logic
  programming and automated deduction.

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Syntax Languages of many sorted FOL

• In contrast to standard first-order languages, in
  many-sorted first order languages, the argument
  of function and predicate symbols may have
  different sorts, and constant and function
  symbols also have some sort. Technically, this
  means that a many-sorted alphabet is a
  many-sorted ranked alphabet. (An S-ranked
  alphabet is pair(S,r) consisting of a set S
  together with a function r S?SS assigning a
  rank (u, s) to each symbol f in S )
• Alphabet of many-sorted first order language
• - S U bool of sorts containing the
  special sort bool
• - Connectives ?,?,?,?, all of rank
  (bool.bool, bool), (not) of rank (bool, bool),
  
• -(of rank (e, bool))
• - Quantifiers ? , ? , each of rank (bool,
  bool)
• - Variables a countable set
  VsX0,X1,X2, each variable Xi being of rank
  (e,s).
• - Auxiliary symbols ( and )
• - Equality symbol ?, of rank (ss, bool)

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Syntax Languages of many sorted FOL

• A (S U bool)-ranked alphabet L of nonlogical
  symbols consisting of
• - FS function symbols, FS?SS, assigning a
  pair r(f)(u,s) called rank to every function
  symbol f. The string u is called the arity of f,
  and the symbol s is the sort of f.
• - CSs constants, For every sort s?S, a set
  CSs of symbols c0,c1,, each of rank (e,s). The
  family of sets CSs is denoted by CS
• -PS predicate symbols, A set PS of symbols
  P0,P1,,and a rank function rPS?Sbool,
  assigning a pair r(P)(u,bool) called rank to
  each predicate symbol P. The string u is called
  the arity of P. If ue, P is a propositional
  letter.
• It is assumed that the sets Vs, FS, CSs, and PS
  are disjoint for all s?S. We will refer to a
  many-sorted first order language with set of
  nonlogical symbols L as the language L.
  Many-sorted first order languages obtained by
  omitting the equality symbol are referred to as
  many-sorted first order languages without
  equality.

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Syntax Terms and Formulas of M-S FOL

• Let L(CS,FS,PS) be a language of many sorted FOL
• Terms atomic formulas of L are defined as
  follows
• - Each constant and each variable of sort
  sis a term of L.
• - If t1,,tn are terms, each ti of sort ui,
  and f is function symbol of rank (u1un,s), then
  ft1tn is a term of sort s
• - Each propositional letter is an atomic
  formula.
• - If t1,,tn are terms, each ti of sort ui,
  and P is a predicate symbol of arity u1un, then
  Pt1tn is an atomic formula
• Formulas are defined as follows
• - Every atomic formula is a formula
• - For any two formula A and B, (A?B), (A?B),
  (A ?B), (A?B ), A are also formula
• - For any variable xi of sort s and any
  formula A, ?sxiA and ?sxiA are also formulas

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Example of Many-sorted FOL

• Let L be following many-sorted first order
  language for stacks, where Sstack, integer,
  CSinterger0, CSstack?, FSSucc, , ,
  push, pop, top, and PSlt
• The rank functions are given by
• - r(succ)(integer, integer)
• - r()r()(integer.integer, integer)
• - r(push)(stack.integer, stack)
• - r(pop)(stack, stack)
• - r(top)(stack, integer)
• - r(lt)(integer.integer, bool)
• Then, the following are terms
• Succ 0
• top push ? Succ 0
• The following are formulas
• lt 0 Succ 0
• ?integerx ?stacky?stack pop push y x y

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Semantics of many sorted FOL

• Many-sorted first order structures
• - First, let define many-sorted S-algebra A
  is a pair ltA,Igt, where A(As)s?S is an S-indexed
  family of nonempty sets, each As being called a
  carrier of sort s, and I is an interpretation
  function assigning functions to the function
  symbols.
• - Give a many-sorted first order language
  L, a many-sorted L-structure M is a many-sorted
  L-algebra, such that the carrier of sort bool is
  the set BOOLT,F
• Semantics of formulas
• Given a many-sorted L-structure M and an
  assignment v V?M, the function tM V?M?Ms
  defined by a term t of sort s is the function
  such that for every assignments v in V?M, the
  value tMv is defined recursively as follows
• 1. For a variable x of sort s, xMvvs(x)
• 2. For a constant c, cMv cM
• 3. Let tf(t1tn), then (ft1tn)MvfM((t1)
  Mv,(tn)Mv)
• 4. Let AP(t1tn), then (Pt1tn)MvPM((t1)
  Mv,(tn)Mv)
• 5. If (t1)Mv(t2)Mv then (?st1t2)MvT
  otherwise F
• 6. Let denotes ?, ??, ?, then
  (AB)MM(AM,BM)
• 7. (?xisA)MvT iff AMvximT, for
  all m?Ms and (? xisA)MvT iff AMvximT,
  for some m?Ms

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Proof Theory of Many-sorted FOL

• Gentzen System G for Many-sorted Languages
  Without Equality
• Gentzen System G for languages With Equality

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The completeness of many-sorted logic

• There are two degrees of completeness
• Weak completeness Each validity is a
  theorem. That is, Fimplies -F for every
  formula F.
• Strong completeness Every consequence of a
  set of formulas is also derivable from it. That
  is, for every set of formulas G and formulas F ,
  when every GF then also F can be deduced from
  G.
• Henkins theorem G consistent ? G has a model.

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Reduction to one-sorted logic

• There is translation of many-sorted logic into
  one-sorted logic. Such a translation is described
  in Enderton, 1972, and you can read it for
  details. The essential idea to convert a
  many-sorted language L in to a one-sorted
  language L is to add domain predicate symbols Ds,
  one for each sort, and to modify quantified
  formulas recursively as follows
• Every formula A of the form ?x sB (or ? sB)
  is converted to the formula A ? x(Ds(x)?B ),
  where B is the result of converting B

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Reference

• Many-Sorted Logic And its Applications, K.Meinke
  and J.V.Tucker
• Logic for Computer Science Foundations of
  Automatic Theorem Proving, Jean Gallier