Many Sorted First-order Logic

Student: Liuxing Kan

Instructor: William Farmer

Dept. of Computing and Software

McMaster University, Hamilton, CA

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

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.

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(Σ,r) consisting of a set Σ together with a function r: Σ→S*×S assigning a rank (u, s) to each symbol f in Σ )

• Alphabet of many-sorted first order language:

－ S U {bool} of sorts containing the special sort bool

－ Connectives: Λ,ν,→,↔, 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 Vs={X0,X1,X2…}, each variable Xi being of rank (e,s).

－ Auxiliary symbols: “(” and “)”

－ Equality symbol , of rank (ss, bool)≒

Syntax: Languages of many sorted FOL

• A (S U {bool})-ranked alphabet L of nonlogical symbols consisting of: － FS: function symbols, FS→S+×S, assigning a pair r(f)=(u,s) called rank to every fun

ction 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 r:PS→S+

×{bool}, assigning a pair r(P)=(u,bool) called rank to each predicate symbol P. The string u is called the arity of P. If u=e, 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 t∈o 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.

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 (u1…un,s), the

n ft1…tn 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 u1…un, th

en Pt1…tn is an atomic formula:

• Formulas are defined as follows: － Every atomic formula is a formula － For any two formula A and B, (AΛB), (AνB), (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

Example of Many-sorted FOL

• Let L be following many-sorted first order language for stacks, where S={stack, integer}, CSinterger={0}, CSstack={Λ}, FS={Succ, +, *, push, pop, top}, and PS={<}

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(<)=(integer.integer, bool); Then, the following are terms: Succ 0 top push Λ Succ 0 The following are formulas: < 0 Succ 0 ∀integerx ∀stacky≒stack pop push y x y

Semantics of many sorted FOL• Many-sorted first order structures

－ First, let define many-sorted Σ-algebra A is a pair <A,I>, 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 BOOL={T,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 tM[v] is defined recursively as follows:

1. For a variable x of sort s, xM[v]=vs(x)

2. For a constant c, cM[v]= cM

3. Let t=f(t1…tn), then (ft1…tn)M[v]=fM((t1)M[v],…(tn)M[v])

4. Let A=P(t1…tn), then (Pt1…tn)M[v]=PM((t1)M[v],…(tn)M[v])

5. If (t1)M[v]=(t2)M[v] then (≒st1t2)M[v]=T otherwise F

6. Let * denotes {Λ, ν→, ↔}, then (A*B)M=*M(AM,BM)

7. ( x∀ i:sA)M[v]=T iff AM[v[xi:=m]]=T, for all m∈Ms and ( x∃ i:sA)M[v]=T iff AM[v[xi:=m]]=T, for some m∈Ms

Proof Theory of Many-sorted FOL

• Gentzen System G for Many-sorted Languages Without Equality

• Gentzen System G= for languages With Equality

The completeness of many-sorted logic

• There are two degrees of completeness:

Weak completeness: Each validity is a theorem. That is, |=Φimplies |-Φ for every formula Φ.

Strong completeness: Every consequence of a set of formulas is also derivable from it. That is, for every set of formulas Γ and formulas Φ , when every Γ|=Φ then also Φ can be deduced from Γ.

• Henkin’s theorem: Γ consistent → Γ has a model.

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

Reference

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

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