Logic and Language Satoshi Tojo JAIST.

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1 Logic and Language Satoshi Tojo JAIST

2 Human intelligence = computation？ Language = grammar + logic？
The left-hand side: Turing machine. Can human intelligence be simulated by computation? Yes in principle, but practically no because of computational complexity.

3 - intuitionistic logic
Grammar Theory - Chomskian theory - Categorial grammar - HPSG Formal semantics - tense and aspect implicature - negation Logic - modal logic - temporal logic - intuitionistic logic Logic of Knowledge and Belief - negation causality and implication legal application Analysis of music score The top three boxes: my original background. Bottom four boxes: my current research fields. Multi-agent language evolution language acquisition Agent communication belief revision communication channel

4 Specialité Agent Communication Language Acquisition
Legal Document Analysis Grammatical Analysis of Music Now, we are tackling the following four themes.

5 Agent Communication (1/3)
φ ￢ψ，φ→ψ Inform(φ) Agent communication (1): belief revision. If the woman informs the man of Φ, he needs to revise his belief. He also needs to consider the reliability of information source. How can we maintain consistency? ψ，φ

6 Agent Communication (2/3)
𝐾 𝑎 𝑡 𝜑 𝐾 𝑎 𝑡+1 𝐾 𝑏 𝑡+1 𝜑 𝐾 𝑏 𝑡+1 𝜑 𝐾 𝑏 𝑡+1 𝐾 𝑎 𝑡 𝜑 Inform(φ) b a Agent communication (2): If the woman informs the man of Φ, he comes to know (K) Φ at time t+1, also he knows the woman knows Φ at t. Also, she knows he knows Φ at t+1, ..., and so on. Which information is valuable? Who knows what at which time? 𝐾 𝑐 𝑡+2 𝜑 c

7 Agent Communication (3/3)
Upper left is an analysis of recipe and lower right is an orchestra score. Both is common: vertically agents are aligned, and horizontally time-axis. Multiple agents’ knowledge (who knows what) should be aligned in such a time map.

8 Language acquisition (1/5)
Mother speaks, baby listens to the utterances and arbitrarily generalizes and finds rules. Then the baby becomes a mother in the following generation.

9 Language acquisition (2/5)
Mary eats an apple. John eats an apple. John swims. John walks. X eats an apple. John Y. X={Mary, John} Y={eats an apple, swims, walks} S→X Y Since the baby has generalized mother’s utterances, he can speak such a new sentence as `Mary swims’ which was never heard from his mother.

10 Language acquisition (3/5)
S/read(john, book) → ivnre S/read(mary, book) → ivnho ↓chunk S/read(x, book) → ivn N/x N/john → re N/mary → ho N/john → re B/john → re ↓merge The methods of generalization: from `ivnre’ and `ivnho’, a common part `ivn’is chunked, and a compositional rule is found.

11 Language acquisition (4/5)
Through generations, the number of rules decreases since rules become more compositional, while the number of utterable sentences increases.

12 Language acquisition (5/5)
Why babies can learn language so rapidly? It is said that they employ `symmetry bias.’ Seeing a vision of an apple, they learn the word `apple.’ Then, when a baby hears `apple,’ he can imagine a concrete apple. Because this is the reverse implication, we adult know that it is generally prohibited. Chimps are said not to have such a bias.

13 Legal Document Analysis (1/7)
Legal documents and/or anaphora Verification multiple If--Then From natural language text to logical formula, we have two problems: (i) long, complicated `or’ structure and anaphora (it, that, and so on) becomes vague. (ii) There are multiple kinds of `if-then’ structures. From logical formula we can verify the consistency of original law code. Logical formulae

14 Legal Document Analysis (2/7)
Vehicles are not admitted to the park. ∀x[vehicle(x) → ￢admitted(x)] Baby cars are admitted to the park. ∀x[babycar(x) → admitted(x)] A baby car is a vehicle. ∀x[babycar(x) → vehicle(x)] An example verification: if a fact `babycar(a)’ is added to the set of law code, we can find the code is inconsistent.

15 Legal Document Analysis (3/7)
厚生労働大臣は、連続する三保険年度中の各保険年度において次の各号のいずれかに該当する事業であって当該連続する三保険年度中の最後の保険年度に属する三月三十一日（以下この項において「基準日」という。）において労災保険に係る保険関係が成立した後三年以上経過したものについての当該連続する三保険年度の間における労災保険法の規定による業務災害に関する保険給付（労災保険法第十六条の六第一項第二号の場合に支給される遺族補償一時金、特定の業務に長期間従事することにより発生する疾病であって厚生労働省令で定めるものにかかった者（厚生労働省令で定める事業の種類ごとに、当該事業における就労期間等を考慮して厚生労働省令で定めるものに限る。）に係る保険給付（以下この項及び第二十条第一項において「特定疾病にかかったものに係る保険給付」という。）及び労災保険法第三十六条第一項の規定により保険給付を受けることができることとされた者（以下「第三種特別加入者」という。）に係る保険給付を除く。）の額（年金たる保険給付その他厚生労働省令で定める保険給付については、その額は、厚生労働省令で定めるところにより算定するものとする。第二十条第一項において同じ。）に労災保険法第二十九条第一項第二号に掲げる事業として支給が行われた給付金のうち業務災害に係るもので厚生労働省令で定めるものの額（一時金として支給された給付金以外のものについては、その額は、厚生労働省令で定めるところにより算定するものとする。）を加えた額と一般保険料の額（第一項第一号の事業については、前項の規定による労災保険率（その率がこの項の規定により引き上げまたは引き下げられたときは、その引き上げまたは引き下げられた率）に応ずる部分の額）から非業務災害率（労災保険法の適用を受けるすべての事業の過去三年間の通勤災害に係る災害率及び二次健康診断等給付に要した費用の額その他の事情を考慮して厚生労働大臣の定める率をいう。（労働保険の保険料の徴収等に関する法律12条3項） A hideous long sentence of law. This is one sentence! We’d like to divide into two sentences, analyzing complicated `or’ structures.

16 Legal Document Analysis (4/7)
((A or B) or their C) of (D or E) 販売の──┐　 用に──┐ 供し、若しくは<P>─┐ 営業上──┐　 　│　 使用する<P>─PARA──┐ 器具若しくは<P>─┤　 容器──┐　　　│ 包装若しくは<P>─┤　 これらの──┐　　　│　 原材料に<P>─PARA──┐　 つき──┐ 規格を──┤ 定め、又は<P>─┤ A Japanese example. 「販売用(D) もしくは 営業上(E)」使用する「「器具(A)もしくは容器(B)」もしくはこれらの:原材料(C)」

17 Legal Document Analysis (5/7)
If P is negatively proved, ￢P. Unless P is not proved, P ？ Assuming P, we obtain contradiction. Therefore ￢P. Assuming ￢P, we obtain contradiction. Therefore P？ (Reductio ad Absurdum) Either P or ￢P？ (Law of Excluded Middle) Legal reasoning is intuitionistic. We should employ intuitionistic logic.

18 Legal Document Analysis (6/7)
Transitivity (Syllogisms) A → B ＆ B → C ⇒ A → C If it rained hard, it rained. If it rained, it didn’t rain hard. Therefore it rained hard, it didn’t rain hard？ Contraposition A → B ⇒ ￢B → ￢A If she wrote a letter to Santa Claus, she didn’t get an answer. Therefore, if she got an answer from Santa Claus, she didn’t write a letter to him？ Strengthening (Weakening) A → C ⇒ A＆B → C If Betty had been at the party, Bill would have a good time. Therefore, if Betty had been at the party and Bill had broken his leg, he would have had a good？ In legal texts, counterfactuality often appears, which invalidates ordinary syllogisms, contraposition, and weakening.

19 Legal Document Analysis (7/7)
If P were the case, Q would be the case. Counterfactuality in branching time. Now ~P and ~Q. If P, then Q is more preferable to ~Q. ~P and Q is least probable. ￢P⋀￢Q ＞ P⋀Q ＞ P⋀￢Q ＞￢P⋀Q

20 Analysis of Music (1/4) Music can have context free grammar since it is heard by human language devices such as ear and hippocampus in our brain, which works as a push-down stack. Actually there is a long distance dependency between chords: Tonic (ドミソ) – Dominant (ソシレファ) motion implies to result in Tonic again. W. A. Mozart: Piano Sonata No.11 in A major, K.331

21 Analysis of Music (2/4) Generative Theory of Tonal Music (GTTM) reduces comparatively non-important notes, and constructs such tree structures, called time-span tree.

22 Analysis of Music (3/4) T’C 4 ⊔TD6 T’C4 T’D TC 5 ⊔TD6 TB TA TD6 TC5
Since time-span trees are aligned by reduction, we can calculate join (∪) and meet (∩); thus trees organize a lattice. We can compose an interpolation of two music, that is a melody morphing. TD6 TC5 TA ⊓TB

23 Analysis of Music (4/4) Grammatical analysis of chords progression written in Head-driven Phrase Structure Grammar (HPSG).