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1 CMSC 630 February 10, Propositional and Predicate Logic Sources J. Gallier. Logic for Computer Science, John Wiley and Sons, Hoboken NJ, revised edition available on line at jean/gbooks/logic.html J.-Y. Girard, J.-Y., Y. Lafont and P. Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, England, 1989.

2 CMSC 630 February 10, Formal Logic... the study of truth and inference. Aristotle: syllogisms All humans are mortal. Socrates is a human. Therefore Socrates is mortal. Frege, Peirce, et al.: symbolic logic Propositional calculus Predicate calculus This course: applications of symbolic logic to software verification

3 CMSC 630 February 10, Components of Symbolic Logic Syntax. Defines the form of formulas Semantics. Defines how formulas should be interpreted Axioms. Defines what formulas are assumed to be true Inference Rules. Defines how truth of formulas may be inferred from truth of other statements

4 CMSC 630 February 10, Propositional Calculus: Syntax Syntax Let (p, q,... )P be a set of propositional variables. Then the set Φ of propositions (= formulas) is defined inductively as follows. φ ::= p if p P atomic ( φ) negation (φ φ) disjunction Note Parentheses often omitted, with assumption that binds more tightly than. So p q is equivalent to (( p) q).

5 CMSC 630 February 10, Derived Operators tt ff = p p true = tt false φ 1 φ 2 = (( φ1 ) ( φ 2 )) conjunction φ 1 φ 2 = ( φ1 ) φ 2 implication φ 1 φ 2 = (φ1 φ 2 ) (φ 2 φ 1 ) bi-implication

6 CMSC 630 February 10, Propositional Calculus: Semantics Semantics of logics involve models. Models are structures used to assign meaning to formulas. Semantics then indicates when a model satisfies a formula, i.e. makes it true. Models for propositional calculus are often called states, or truth assignments. Definition Let B = {0, 1} be the set of boolean truth values, and P a set of propositional variables. Then a state σ is any function in P B. Define Σ = P B to be the set of all states. Note Σ is isomorphic to 2 P = the set of subsets of P. Define f Σ 2 P by f(σ) = { p P σ(p) = 1 }. Define g 2 P Σ by g(s)(p) = 1 iff p S. Then g(f(σ)) = σ, and f(g(s)) = S all σ Σ, S 2 P.

7 CMSC 630 February 10, Propositional Calculus: Semantics (cont.) The semantics of the propositional calculus is given in terms of a relation = Σ Φ. Notation Write: σ = φ instead of σ, φ = σ = φ instead of σ, φ =. Intuition σ = φ: σ satisfies φ Definition = is defined inductively as follows. σ = p P if σ(p) = 1 σ = φ if σ = φ σ = φ 1 φ 2 if σ = φ 1 or σ = φ 2

8 CMSC 630 February 10, Terminology Definition Let φ Φ be a proposition. φ is satisfiable if there is a σ Σ such that σ = φ. φ is a tautology if for every σ Σ, σ = φ. φ is falsifiable if there is a σ Σ such that σ = φ. φ is inconsistent if for every σ Σ, σ = φ.

9 CMSC 630 February 10, Truth Tables What are they? p q q p p (q p) A means of enumerating (proposition-relevant parts of) states A technique for computing satisfiability / tautology / falsifiability / inconsistency information

10 CMSC 630 February 10, SAT Solvers... tools for determining whether or not a proposition is satisfiable, and if so computing a satisfying state / truth assignment. Most SAT solvers use variants of the Davis-Putnam-Logemann-Loveland (DPLL) algorithm developed in 1962, which uses backtracking to search for satisfying states of formulas in conjunctive normal form (CNF). Literal. Atomic proposition (positive literal) or negation (negative literal) of atomic proposition Clause. Disjunction of multiple literals. CNF. Conjunction of clauses. Fact Fact Every proposition may be converted into CNF (why?) The satisfiability problem (SAT) is NP-complete. Stephen Cook proved this in 1971 and won the 1982 Turing Award as a result. (He still did not get tenure at Berkeley in 1970.)

11 CMSC 630 February 10, Davis-Putnam-Logemann-Loveland Basic step of DPLL: 1. Pick a variable, assign it Remove positive literal from all clauses (why?) 3. Remove clauses containing negative literal for that variable (why?) 4. Recurse 5. If no satisfying instance found, reassign variable value 1 6. Remove clauses containing variable as positive literal 7. Remove negative literals for that variable 8. Recurse 9. If no satisfying instance found, report unsatisfiable Step 5 involves backtracking.

12 CMSC 630 February 10, Propositional Calculus: Axioms and Inference Rules Recall Formal logics deal with truth and inference. Symbolic logics consist of syntax, semantics, axioms and inference rules. For the propositional calculus, we have defined: Syntax: Φ Semantics: = What about axioms, inference rules (= proof system) for the propositional calculus? First, need to identify judgments (i.e. what statements proofs manipulate) Then axioms, inference rules specify true judgments, how judgments can be inferred

13 CMSC 630 February 10, The (Propositional) Sequent Calculus... a proof system for propositional (and predicate) calculus Invented by Gerhard Gentzen, German mathematician and logician, in 1934 as part of his program to formalize natural deduction [Gentzen died at age 36 in 1945 from malnutrition in the wake of World War II.] Intended to prove tautologies Judgments. In the sequent calculus, judgments have form Γ where Γ, Φ are sequences of propositions; is called turnstile. Intuition In Γ, Γ is a list of assumptions, and is a list of (hoped for) conclusions.

14 CMSC 630 February 10, Sequent Notation Notation Γ 1, Γ 2 empty sequence sequence concatenation φ, Γ prefixing Γ, φ postfixing Γ length (number of elements in Γ) Γ, Γ conjunction, disjunction of Γ Aside Formally: Γ = tt φ if Γ = if Γ = φ Φ φ ( Γ ) if Γ = φ, Γ and φ Φ, Γ 1

15 CMSC 630 February 10, Semantics of Sequents Semantics of sequents given in terms of validity. Definition A sequent Γ is valid if and only if ( Γ) ( ) is a tautology. Note Since = tt, φ for φ Φ is valid iff φ is a tautology. So sequent calculus s intended use includes proving that individual formulas are tautologies.

16 CMSC 630 February 10, Axioms and Inference Rules for Propositional Sequent Calculus The sequent calculus uses a uniform format for both axioms and inference rules: premises conclusion (name) premises is a finite set of sequent templates conclusion is a single sequence template name is name of rule (no semantic content) Idea If (instance) of each premise has been proved, then rule may be used to prove conclusion. Axiom(s): empty premise list We will focus first on sequents whose propositions only involve atomic propositions,, (no derived operators).

17 CMSC 630 February 10, Rules Classification, and Identity / Cut There are also two miscellaneous rules / axioms: Identity axiom. φ φ (I) Cut rule. Γ 1 1, φ φ, Γ 2 2 Γ 1, Γ 2 1, 2 (Cut) The rest of the rules are classified along two dimensions. Left / Right. Which side of does the rule focus on? Logical / Structural. Does the rule introduce logical operators or re-arrange lists of propositions?

18 CMSC 630 February 10, Left and Right Logical Rules Γ 1, φ 1 1 Γ 2, φ 2 2 Γ 1, Γ 2, φ 1 φ 2 1, 2 ( L) Γ φ 1, Γ φ 1 φ 2, ( R 1) Γ φ 2, Γ φ 1 φ 2, ( R 2) Γ φ, Γ, φ ( L) Γ, φ Γ φ, ( R)

19 CMSC 630 February 10, Left and Right Structural Rules Γ Γ, φ (WL) Γ Γ φ, (WR) Γ, φ, φ Γ, φ (CL) Γ φ, φ, Γ φ, (CR) Γ 1, φ 1, φ 2, Γ 2 Γ 1, φ 2, φ 1, Γ 2 (PL) Γ 1, φ 1, φ 2, 2 Γ 1, φ 2, φ 1, 2 (PR) Notes W = Weaken C = Contract P = Permute

20 CMSC 630 February 10, Proofs in Sequent Calculus... are trees. Nodes are sequents Leaves are instances of axiom (I) Parent is result of applying proof rule to children Trees sometimes grow up (math style), with parents below children, or down (CS style), with parents above children. Confusingly, the former proof-construction method is often called bottom up ; the latter is called top down. Proofs often annotated with proof rules used to establish parenthood. Definition 1. A sequent is provable is there is a proof whose root is labeled with the sequent. 2. A proposition φ is provable if the sequent φ is provable.

21 CMSC 630 February 10, Sample Proof (Bottom-up, Leaves at Top) Here is a proof of p p. I p p R 1 p p p R p, p p R 2 p p, p p CR p p

22 CMSC 630 February 10, Sample Proof (Top-Down, Root at Top) Here is a proof of (p q) p. Because proof rules only work on,, (p q) p must be translated to ( p q) p. p p CR p p, p p R 1 p, p p R p p p R 1 p p I ( p q) p p ( p q) p p p I Cut p p ( p q) p L p ( p q) p R 2 Cut p p q p q ( p q) R 1 R p p p q, ( p q) I L p q p q I

23 CMSC 630 February 10, Observations about Proofs 1. Proof in symbolic logic is very mechanistic: small steps of reasoning using rigid rules. This is hard for humans, easier for machines. 2. The proof system just given is for a small language (atomic propositions,, ), so interesting statements get big. 3. Big statements require big (tedious) proofs! 4. (Partial) solution: add derived operators, derived rules.

24 CMSC 630 February 10, (Derived) Logical Rules for, Γ, φ 1 Γ, φ 1 φ 2 ( L 1) Γ 1 φ 1, 1 Γ 2 φ 2, 2 Γ 1, Γ 2 φ 1 φ 2, 1, 2 ( R) Γ, φ 2 Γ, φ 1 φ 2 ( L 2) Γ 1 φ 1, 1 Γ 2, φ 2 2 Γ 1, Γ 2, φ 1 φ 2 1, 2 ( L) Γ, φ 1 φ 2, Γ φ 1 φ 2, ( R) Note There are similarities between: L 1 and R 1 L 2 and L 2 L and R

25 CMSC 630 February 10, Relating Proofs and Semantics Recall Sequent Γ is valid iff ( Γ) ( ) is a tautology. Validity is a semantic notion (why?). Provability is a syntactic one: application of proof rules requires only syntactic manipulations. Logicians identify two notions when studying the relationship between validity and provability. Soundness. Is everything that is provable also valid? Completeness. Is everything that is valid also provable?

26 CMSC 630 February 10, Soundness, Completeness of Sequent Calculus Theorem (Soundess) Let Γ be a provable sequent. Then Γ is valid. Proof Proof relies on showing that each axiom is valid and each proof rule preserves validity: if the premises are valid, then the conclusion is guaranteed to be valid. Theorem (Completeness of Sequent Calculus) Let Γ be a valid sequent. Then Γ is provable. Proof Harder. Usually relies on induction over sequents. In studying completeness, Gentzen proved one of his famous results: the Cut elimination theorem. Theorem (Cut Elimination (Hauptstatz)) Let Γ be provable. Then there is a proof of Γ that does not use the Cut rule. Note the following corollary to the Completeness Theorem. Corollary Every tautology in the propositional calculus is provable!

27 CMSC 630 February 10, The Predicate Calculus The second symbolic logic in this review Sometimes also called first-order logic Extends the propositional calculus with data and quantification (existential / universal) Like the propositional calculus, we will define the predicate calculus via a syntax, semantics and proof system. First, the notion of data theory (a.k.a. first-order structure, structure ) needs defining: these provide the mathematical API for the data used in the predicate calculus.

28 CMSC 630 February 10, Data Theories: Syntax The syntactic specification of a data theory consists of: Constants. A set (c, c 1,... )C of data values (sometimes called the carrier set) Example For natural numbers, C = N = {0, 1, 2,...} Variables. A countably infinite set (x, x 1,...) X of data variables Terms. A set (t, t 1,... )T of data terms such that C X T Example For natural numbers: 3x + y, etc. (expressions that can evaluate to numbers) Predicates. A set (A, A 1,... )A of atomic predicates Syntactic functions. Two functions for computing free variables and substitutions. 1. FV T,A (T A) 2 X computes free variables of predicates, terms 2. subst (T A) T X (T A) where subst(a, t, x) A if and only if A A. subst(a, t, x) returns the result of replacing x by t in A, and similarly for subst(t, t, x). Example For natural numbers, x y + 3, etc. Taking A to be x y + 3: FV T,A (A) = {x, y} subst(a, w + 1, y) is x w

29 CMSC 630 February 10, Data Theory Semantics The semantics of a data theory interprets terms and atomic predicates with respect to data states. Definition The set of data states is defined as (σ,... )Σ = X C. The semantics of a data theory consists of: Term Evaluation. A function [[ ]] T Σ C. [[t]]σ C is the result of evaluating t in σ. Example Suppose σ(x) = 4, σ(y) = 6. Then [[x + 3]]σ = 7. Predicate Satisfaction. A relation = A Σ A. σ = A A holds if state σ satisfies A. Example Take σ as before. Then σ = A x y, σ = A x > y 1. Notation We will write a data theory D as F, I, where F ( formulas ) is the syntax part: and I ( interpretation ) is the semantic part: F = C, X, T, A, FV T,A, subst I = [[ ]], = A

30 CMSC 630 February 10, Predicate Calculus: Syntax Suppose we have a data theory D = F, I as described previously. Then the formulas Φ F of the predicate calculus are as follows. φ ::= A atomic predicate φ negation φ φ disjunction x. φ (x X) existential Derived operators include, and x. φ = x. φ. In x. φ, x is said to be bound (as opposed to free). x. φ may be thought of as indexed disjunction, i.e. x C index variable. φ; then x may be seen as the x. φ may similarly be interpreted as x C φ.

31 CMSC 630 February 10, Predicate Calculus: Semantics Let D = F, I be a data theory, and Σ F = X C the associated data states. The semantics of the predicate calculus is specified using a relation = D Σ F Φ F A notion of state updating is handy. Definition If σ Σ F, x X and c C, then σ[x c] Σ F is the state defined as: c if y = x σ[x c](y) = σ(y) otherwise Definition = D is defined inductively as follows. σ = D A if σ = A A σ = D φ if σ = D φ σ = D φ 1 φ 2 if σ = D φ 1 or σ = D φ 2. σ = D x. φ if σ[x c] = D φ for some c C. A formula φ is a D-tautology if for every σ Σ F, σ = D φ. The definitions of D-satisfiable, etc., carry over from the propositional calculus in the obvious way.

32 CMSC 630 February 10, Free and Bound Variables, and Substitution To define a proof system for predicate calculus, we will need to be able to replace variables by terms inside formulas. There are some subtleties: Some variables are bound in formulas, and should not be subject to these replacements (e.g. the x occurrences in x. φ). Example Consider φ = x. x y. Replacing y by 3 should yield φ[3/y] = x. x 3. Replacing x by 3 should have no effect: φ[3/x] = φ. Some terms have free variables that should not be captured by quantifiers. Example Consider φ as above, t = x + 3. Replacing y by t naively yields φ[t/y] = x. x x + 3, but this is wrong (x outside x is different than x inside x). Solutions Only substitute for free occurrences of a variable in a formula. When defining substitution, rename bound variables in order to avoid capture.

33 CMSC 630 February 10, Free Variables A variable is free in a formula if its value can be affected by a state. Definition Let φ Φ F be a formula. Then the set FV (φ) X of free variables in φ is defined inductively as follows. FV (φ) = FV T,A (φ) if φ A FV (φ ) if φ = φ FV (φ 1 ) FV (φ 2 ) if φ = φ 1 φ 2 FV (φ ) {x} if φ = x. φ FV can be extended to a sequence of formulas in the obvious manner: if Γ = φ 1,...,φ n then n FV (Γ) = FV (φ i ). i=1 The function new returns a fresh variable (i.e. one not in the set given as an argument). Definition Let new (2 X {X}) X be such that for any X X, new(x) X.

34 CMSC 630 February 10, Defining Substitution Definition Let φ Φ F, t T and x X. Then the result, φ[t/x], of substituting t for x in φ is as follows. subst(φ, t, x) if φ A φ[t/x] = (φ [t/x]) if φ = φ (φ 1 [t/x]) (φ 2 [t/x]) if φ = φ 1 φ 2 z. ((φ [z/y])[t/x]) if φ = y. φ and z = new (FV (φ ) FV T,A (t))

35 CMSC 630 February 10, Predicate Calculus: Axioms and Inference Rules The Sequent Calculus may be adapted! There need to be left and right logical rules for. Γ, φ Γ, x. φ (x FV (Γ) FV ( )) ( L) Γ φ[t/x], Γ x. φ, ( R) Note The L rule has a side condition restricting its application. There needs to be a proof system (non-logical axioms / inference rules) for the data theory. (The regular rules are, confusingly, called the logical axiom and inference rules, even though we were using logical for a subset of these rules also.) Relative Soundness The Sequent Calculus for predicate calculus is sound if the non-logical axioms, inference rules are sound. What about completeness?

36 CMSC 630 February 10, Logical Completeness Some kind of relative completeness result would be nice, but is hard to formulate. Kurt Goedel (he also has a famous Incompleteness Theorem) defined a notion of logical completeness and in 1929 proved completeness of a proof system equivalent to the Sequent Calculus. Definition Let D = F, I be a data theory. Then a formula φ Φ F is logically valid if for all states σ Σ F and all interpretations I, σ = F,I φ. Theorem (Goedel Completeness) only the logical axiom and inference rules. Suppose φ is logically valid. Then φ is provable using

37 CMSC 630 February 10, Derived Rules for Γ, φ[t/x] Γ, x. φ ( L) Γ φ, Γ x. φ, (x FV (Γ) FV ( )) ( R) Note symmetries between ( L) and ( R), and between ( R) and ( L). Decidability What about decision procedures (SAT solvers, etc.)? Alas, thanks to Church and Turing in 1936, none can exist. Theorem (Undecidability) Logical validity of formulas in the predicate calculus is undecidable.

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