Chapter 1 Logics Course “Model checking” Volker Stolz, Martin Steffen Autumn 2019
Section Algebraic and first-order signatures Chapter 1 “Logics” Course “Model checking” Volker Stolz, Martin Steffen Autumn 2019
Intro IN5110 – Verification and specification of parallel systems Algebraic and first-order signatures First-order logic Syntax Semantics Proof theory Modal logics Introduction Semantics Proof theory and axiomatic systems Exercises References 1-3
Signature IN5110 – Verification and specification of parallel systems • fixes the “syntactic playground” Algebraic and first-order • selection of se signatures • functional and First-order logic • relational Syntax Semantics Proof theory symbols, together with “arity” or sort-information Modal logics Introduction Semantics Proof theory and axiomatic systems Exercises References 1-4
Sorts IN5110 – Verification and specification of parallel systems • Sort • name of a domain (like Nat ) Algebraic and • restricted form of type first-order signatures • single-sorted vs. multi-sorted case First-order logic Syntax • single-sorted Semantics Proof theory • one sort only Modal logics • “degenerated” Introduction • arity = number of arguments (also for relations) Semantics Proof theory and axiomatic systems Exercises References 1-5
Terms IN5110 – Verification and specification of • given: signature Σ parallel systems • set of variables X (with typical elements x, y ′ , . . . ) Algebraic and first-order signatures t ::= x variable (1) First-order logic | f ( t 1 , . . . , t n ) f of arity n Syntax Semantics Proof theory • T Σ ( X ) Modal logics Introduction • terms without variables (from T Σ ( ∅ ) or short T Σ ): Semantics Proof theory and axiomatic ground terms systems Exercises References 1-6
Substutition IN5110 – Verification and specification of parallel systems Algebraic and • Substitution = replacement , namely of variables by first-order signatures terms First-order logic • notation t [ s/x ] Syntax Semantics Proof theory Modal logics Introduction Semantics Proof theory and axiomatic systems Exercises References 1-7
First-order signature (with relations) IN5110 – Verification and specification of parallel systems • add relational symbols to Σ Algebraic and • typical elements P , Q first-order signatures • relation symbols with fixed arity n -ary predicates or First-order logic Syntax relations) Semantics • standard binary symbol: . Proof theory = (equality) Modal logics Introduction Semantics Proof theory and axiomatic systems Exercises References 1-8
Section First-order logic Syntax Semantics Proof theory Chapter 1 “Logics” Course “Model checking” Volker Stolz, Martin Steffen Autumn 2019
Syntax IN5110 – Verification and specification of parallel systems • given: first order signature Σ Algebraic and first-order signatures First-order logic P ( t, . . . , t ) | ⊤ | ⊥ ϕ ::= atomic formula Syntax | ϕ ∧ ϕ | ¬ ϕ | ϕ → ϕ | . . . formulas Semantics Proof theory | ∀ x.ϕ | ∃ x.ϕ Modal logics Introduction Semantics Proof theory and axiomatic systems Exercises References 1-10
First-order structures and models • given Σ IN5110 – • assume single-sorted case Verification and specification of parallel systems first-order model model M Algebraic and first-order signatures First-order logic M = ( A, I ) Syntax Semantics Proof theory Modal logics • A some domain/set Introduction • interpretation I , respecting arity Semantics Proof theory and axiomatic ] I : A n → A systems • [ [ f ] Exercises ] I : A n • [ [ P ] References • cf. first-order structure 1-11
Giving meaning to variables IN5110 – Verification and specification of parallel systems Variable assignment Algebraic and first-order • given Σ and model signatures First-order logic σ : X → A Syntax Semantics Proof theory • other names: valuation , state Modal logics Introduction Semantics Proof theory and axiomatic systems Exercises References 1-12
(E)valuation of terms IN5110 – Verification and specification of parallel systems Algebraic and • σ “straightforwardly extended/lifted to terms” first-order signatures • how would one define that (or write it down, or First-order logic Syntax implement)? Semantics Proof theory Modal logics Introduction Semantics Proof theory and axiomatic systems Exercises References 1-13
Free and bound occurrences of variables IN5110 – Verification and specification of parallel systems • quantifiers bind variables Algebraic and • scope first-order signatures • other binding, scoping mechanisms First-order logic • variables can occur free or not (= bound ) in a formula Syntax Semantics • careful with substitution Proof theory Modal logics • how could one define it? Introduction Semantics Proof theory and axiomatic systems Exercises References 1-14
Substitution IN5110 – Verification and specification of parallel systems • basically: • generalize substitution from terms to formulas • careful about binders especially don’t let substitution Algebraic and first-order lead to variables being “captured” by binders signatures First-order logic Syntax Example Semantics Proof theory Modal logics ϕ = ∃ x.x + 1 . = y θ = [ y/x ] Introduction Semantics Proof theory and axiomatic systems Exercises References 1-15
Satisfaction IN5110 – Verification and specification of parallel systems Definition ( | = ) M, σ | = ϕ Algebraic and first-order signatures • Σ fixed First-order logic Syntax • in model M and with variable assignment σ formula ϕ Semantics Proof theory is true (holds Modal logics • M and σ satisfy ϕ Introduction Semantics • minority terminology: M, σ model $ ϕ Proof theory and axiomatic systems Exercises References 1-16
Exercises IN5110 – Verification and specification of parallel systems • substitutions and variable assignments: Algebraic and similar/different? first-order signatures • there are infinitely many primes First-order logic • there is a person with at least 2 neighbors (or exactly) Syntax Semantics • every even number can be written as the sum of 2 Proof theory Modal logics primes Introduction Semantics Proof theory and axiomatic systems Exercises References 1-17
Proof theory IN5110 – • how to infer, derive, deduce formulas (from others) Verification and specification of parallel systems • mechanical process • soundness and completeness Algebraic and • proof = deduction (sequence or tree of steps) first-order signatures • theorem First-order logic • syntactic: derivable formula Syntax Semantics • semantical a formula which holds (in a given model) Proof theory Modal logics • (fo)-theory: set of formulas which are Introduction • derivable Semantics Proof theory and axiomatic • true (in a given model) systems Exercises • soundness and completeness References 1-18
Deductions and proof systems IN5110 – Verification and A proof system for a given logic consists of specification of parallel systems • axioms (or axiom schemata ), which are formulae assumed to be true, and Algebraic and first-order • inference rules, of approx. the form signatures First-order logic Syntax ϕ 1 . . . ϕ n Semantics Proof theory Modal logics ψ Introduction Semantics Proof theory and axiomatic systems • ϕ 1 , . . . , ϕ n are premises and ψ conclusion. Exercises References 1-19
A simple form of derivation Derivation of ϕ IN5110 – Verification and specification of Sequence of formulae, where each formula is parallel systems • an axiom or • can be obtained by applying an inference rule to Algebraic and first-order formulae earlier in the sequence. signatures First-order logic Syntax • ⊢ ϕ Semantics Proof theory • more general: set of formulas Γ Modal logics Introduction Γ ⊢ ϕ Semantics Proof theory and axiomatic systems Exercises References • proof = derivation • theorem: derivable formula (= last formula in a proof) 1-20
Proof systems and proofs: remarks • “definitions” from the previous slides: not very formal IN5110 – Verification and in general: a proof system: a “mechanical” (= formal and specification of parallel systems constructive) way of conclusions from axioms (= “given” formulas), and other already proven formulas Algebraic and • Many different “representations” of how to draw first-order signatures conclusions exists, the one sketched on the previous First-order logic slide Syntax • works with “sequences” Semantics Proof theory • corresponds to the historically oldest “style” of proof Modal logics systems (“Hilbert-style”), some would say outdated . . . Introduction • otherwise, in that naive form: impractical (but sound & Semantics Proof theory and axiomatic systems complete). Exercises • nowadays, better ways and more suitable for computer References support of representation exists (especially using trees). For instance natural deduction style system 1-21
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