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Home , Mathematical proof

Deciding

Problem Is there an algorithm to determine whether a formula φ is the logical consequence of a set of formulas Γ?

Mathematical Na¨ıvesolution Apply directly the definition of logical consequence i.e., for all Introduction to Reasoning and Automated Reasoning. possible interpretations I determine if I |= Γ, if this is the Hilbert-style Propositional Reasoning. case then check if I |= A too. This solution can be used when Γ is finite, and there is a finite Chiara Ghidini number of relevant interpretations.

FBK-IRST, Trento, Italy

Chiara Ghidini Chiara Ghidini Mathematical Logic Complexity of deciding logical consequence in Deciding logical consequence is not always possible Propositional Logic

The table method is Exponential Propositional The problem of determining if a formula A containing n primitive , is a logical consequence of the empty set, i.e., the problem The truth table method enumerates all the possible interpretations of a of determining if A is valid, (|= A), takes an n-exponential number of formula and, for each formula, it computes the relation |=. steps. To check if A is a tautology, we have to consider 2n interpretations in the truth table, corresponding to 2n lines. Other logics For first order logic and modal logics there is no general algorithm to More efficient algorithms? compute the logical consequence. There are some algorithms computing Are there more efficient algorithms? I.e. Is it possible to define an the logical consequence for first order logic sub-languages and for algorithm which takes a polinomial number of steps in n, to determine sub-classes of structures (as we will see further on). the of A? This is an unsolved problem

P =? NP Alternative approach: decide logical consequence via reasoning. The existence of a polinomial algorithm for checking validity is still an open problem, even it there are a lot of in favor of non-existence

Chiara Ghidini Mathematical Logic Chiara Ghidini Mathematical Logic Reasoning What is it to ?

Reasoning is a process of deriving new statements (conclusions) from other statements () by . What the dictionaries say: For reasoning to be correct, this process should generally reasoning: the process by which one judgement is deduced preserve truth. That is, the should be valid. from another or others which are given (Oxford English How can we be sure our arguments are valid? Dictionary) Reasoning takes place in many different ways in everyday life: reasoning: the drawing of or conclusions through Word of Authority: we derive conclusions from a source that the use of reason we trust; e.g. religion. reason: the power of comprehending, inferring, or thinking, Experimental science: we formulate hypotheses and try to esp. in orderly rational ways (cf. intelligence) confirm them with experimental . (Merriam-Webster) Sampling: we analyse many pieces of evidence statistically and identify patterns. : we derive conclusions based on mathematical . Are any of the above methods valid?

Chiara Ghidini Mathematical Logic Chiara Ghidini Mathematical Logic What is a Proof? (I) What is a ?

We can be sure there are no hidden premises by reasoning according to alone. For centuries, mathematical proof has been the hallmark of Example logical validity. Suppose all men are mortal. Suppose Socrates is a man. But there is still a social aspect as peers have to be Therefore, Socrates is mortal. convinced by argument. The validity of this proof is independent of the meaning of This process is open to flaws: e.g. Kempes proof of the Four “men”, “mortal” and “Socrates”. Colour . Indeed, even a nonsense gives a valid sentence: To avoid this, we require that all proofs be broken down to Suppose all borogroves are mimsy. Suppose a mome rath is a their simplest steps and all hidden premises uncovered. borogrove. Therefore, a mome rath is mimsy. General pattern: Suppose all Ps are Q. Suppose x is a P. Therefore, x is a Q.

Chiara Ghidini Mathematical Logic Chiara Ghidini Mathematical Logic Symbolic Proof Propositional reasoning: Proofs and deductions (or derivations)

proof

The modern notion of symbolic formal proof was developed in A proof of a formula φ is a sequence of formulas φ1, . . . , φn, with φn = φ, such that the 20th century by logicians and mathematicians such as each φk is either Russell, Frege and Hilbert. an or it is derived from previous formulas by reasoning rules The benefit of formal logic is that it is based on a pure φ is provable, in symbols ` φ, if there is a proof for φ. : a precisely defined symbolic language with procedures

for transforming symbolic statements into other statements, Deduction of φ from Γ based solely on their form. A deduction of a formula φ from a set of formulas Γ is a sequence of formulas No intuition or interpretation is needed, merely φ1, . . . , φn, with φn = φ, such that φk applications of agreed upon rules to a set of agreed upon is an axiom or formulae. it is in Γ (an assumption) it is derived form previous formulas bhy reasoning rules φ is derivable from Γ, in symbols Γ ` φ, if there is a proof for φ from formulas in Γ.

Note: the sequence φ1, . . . , φn, with φn = φ is finite!

Chiara Ghidini Mathematical Logic Chiara Ghidini Mathematical Logic Hilbert for classical propositional logic Hilbert proofs and deductions

Axioms Hilbert proof

Inference rule(s) A proof of a formula φ is a sequence of formulas φ1, . . . , φn, with φn = φ, such that A1 φ ⊃ (ψ ⊃ φ) each φk is either an axiom or A2 (φ ⊃ (ψ ⊃ θ)) ⊃ ((φ ⊃ ψ) ⊃ (φ ⊃ θ)) φ φ ⊃ ψ MP it is derived from previous formulas by MP A3 (¬ψ ⊃ ¬φ) ⊃ ((¬ψ ⊃ φ) ⊃ ψ) ψ φ is provable, in symbols ` φ, if there is a proof for φ.

Hilbert Deduction of φ from Γ Why there are no axioms for ∧ and ∨ and ≡? A deduction of a formula φ from a set of formulas Γ is a sequence of formulas φ1, . . . , φn, with φn = φ, such that φk The connectives ∧ and ∨ are rewritten into equivalent formulas is an axiom or containing only ⊃ and ¬. it is in Γ (an assumption) it is derived form previous formulas by MP A ∧ B ≡ ¬(A ⊃ ¬B) φ is derivable from Γ in symbols Γ ` φ if there is a deduction for φ from Γ. A ∨ B ≡ ¬A ⊃ B

A ≡ B ≡ ¬((A ⊃ B) ⊃ ¬(B ⊃ A)) Note: the sequence φ1, . . . , φn, with φn = φ is finite!

Chiara Ghidini Mathematical Logic Chiara Ghidini Mathematical Logic Deduction and proof - example Deduction and proof - other examples

Example (proof of ¬A ⊃ (A ⊃ B)) We prove that A, ¬A ` B and by deduction theorem we have that ¬A ` A ⊃ B and that ` ¬A ⊃ (A ⊃ B) Example (Proof of A ⊃ A) We label with Hypothesis the formula on the left of the ` sign. 1. A1 A ⊃ ((A ⊃ A) ⊃ A) 2. A2 (A ⊃ ((A ⊃ A) ⊃ A)) ⊃ ((A ⊃ (A ⊃ A)) ⊃ (A ⊃ A)) 1. hypothesis A 3. MP(1, 2) (A ⊃ (A ⊃ A)) ⊃ (A ⊃ A) 2. A1 A ⊃ (¬B ⊃ A) 4. A1 (A ⊃ (A ⊃ A)) 3. MP(1, 2) ¬B ⊃ A 5. MP(4, 3) A ⊃ A 4. hypothesis ¬A 5. A1 ¬A ⊃ (¬B ⊃ ¬A) 6. MP(4, 5) ¬B ⊃ ¬A 7. A3 (¬B ⊃ ¬A) ⊃ ((¬B ⊃ A) ⊃ B) 8. MP(6, 7) (¬B ⊃ A) ⊃ B 9. MP(3, 8) B

Chiara Ghidini Mathematical Logic Chiara Ghidini Mathematical Logic Logical consequence and derivability Soundness & Completeness of a proof system

A formula A is a logical consequence of a set of formulas Γ, in symbols

Γ |= A Soundness if and only if for any interpretation I that satisfies all the formulas in Γ, I satisfies A, The proof system proves only logical consequences (or, analogously, does not enable us to prove non logical consequeces).

A formula A is derivable from a set of formulas Γ, in symbols Formally, if Γ ` A then Γ |= A.

Γ ` A Completeness if and only if there is a deduction of A with assumptions in Γ The proof system enables us to prove all logical consequences. Formally, if Γ |= A then Γ ` A.

How can we be sure that derivability exactly mimics logical consequence?

Chiara Ghidini Mathematical Logic Chiara Ghidini Mathematical Logic The Hilbert axiomatization is sound & complete Hilbert axiomatization

Minimality The main objective of Hilbert was to find the smallest set of Theorem (Soundness of Hilbert axiomatization) axioms and rules from which it was possible to derive all the tautologies. If Γ ` A then Γ |= A. Unnatural Theorem (Completeness of Hilbert axiomatization) Proofs and deductions in Hilbert axiomatization are awkward and If Γ |= A then Γ ` A. unnatural. Other proof styles, such as Natural Deductions, are more intuitive. As a matter of facts, nobody is practically using Hilbert for deduction.

We will not prove the . Why it is so important Providing an Hilbert style axiomatization of a logic describes with simple axioms the entire properties of the logic. Hilbert axiomatization is the “identity card” of the logic.

Chiara Ghidini Mathematical Logic Chiara Ghidini Mathematical Logic The deduction theorem The deduction theorem

Theorem Proof. Step case If An = B is either an axiom or an element of Γ, then we can reason Γ, A ` B if and only if Γ ` A ⊃ B as the previous case. If B is derived by MP form Ai and Aj = Ai ⊃ B. Then, Ai and Proof. Aj = Ai ⊃ B, are provable in less then n steps and, by induction hypothesis, Γ ` A ⊃ Ai and Γ ` A ⊃ (Ai ⊃ B). Starting from the =⇒ direction (⇐= is easy) deductions of these two formulas from Γ, we can build a deduction If A and B are equal, then we know that ` A ⊃ B (see previous example), and by of A ⊃ B form Γ as follows: monotonicity Γ ` A ⊃ B. Suppose that A and B are distinct formulas. Let π = (A1,..., An = B) be a . deduction of Γ, A ` B, we proceed by induction on the length of π. By induction . deduction of A ⊃ (Ai ⊃ B) form Γ Base case n = 1 If π = (B), then either B ∈ Γ or B is an axiom. Then A ⊃ (Ai ⊃ B) . Axiom A1 B ⊃ (A ⊃ B) By induction . deduction of A ⊃ Ai form Γ B ∈ Γ or B is an axiom B A ⊃ Ai by MP A ⊃ B A2 (A ⊃ (Ai ⊃ B)) ⊃ ((A ⊃ Ai ) ⊃ (A ⊃ B)) MP (A ⊃ Ai ) ⊃ (A ⊃ B) is a deduction of A ⊃ B from Γ or from the empty set, and therefore MP A ⊃ B Γ ` A ⊃ B.

Chiara Ghidini Mathematical Logic Chiara Ghidini Mathematical Logic ....Let’s go back to symbolic proofs Automated Reasoning

But... Formal proofs are bloated and over expanded! Automated Reasoning (AR) refers to reasoning in a computer I find nothing in [formal logic] but shackles. It does not using logic. help us at all in the direction of conciseness, far from it; AR has been an active area of research since the 1950s. and if it requires 27 equations to establish that 1 is a It uses to tackle problems such as: number, how many will it require to demonstrate a real constructing formal mathematical proofs; theorem? (Poincar´e) verifying programs meet their specifications; Can automation help? modelling human reasoning.

Chiara Ghidini Mathematical Logic Chiara Ghidini Mathematical Logic Different Forms of Reasoning More efficient reasoning systems?

Deduction: Given a set of premises Γ and a conclusion φ show that indeed Γ |= φ (this includes Validity: Γ = ∅) Automate Hilbert style reasoning Checking if Γ |= φ by searching for a Hilbert-style deduction of φ from Γ is not an easy Abduction/Induction: given a theory T and an observation task for computers. Indeed, in trying to generate a deduction of φ from Γ, there are to φ, find an explanation Γ such that T ∪ Γ |= φ many possible actions a computer could take: Satisfiability Checking: given a set of formulae Γ, check adding an instance of one of the three axioms (infinite number of possibilities) applying MP to already deduced formulas, whether there exists a model I such that I |= φ for all φ ∈ Γ? adding a formula in Γ Model Checking: given a model I and a formula φ, check whether I|=φ More efficient methods to check if a formula is not satisfiable SAT DP, DPLL to search for an interpretation that satisfies a formula Automated reasoning attempts to mechanise all of these forms of Tableaux search for a model of a formula guided by its structure reasoning for different logics: propositional or first-order, classical, intuionistic, modal, temporal, non-monotonic, . . .

Chiara Ghidini Mathematical Logic Chiara Ghidini Mathematical Logic