# Questions tagged [logic]

Questions related to mathematical logic and its use in computer science

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### What is the theoretical result of flattening a list containing only itself?

Consider the following python code X = [None] X = X This has many fun properties such as X == X and ...
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### Understanding $\lambda \mu$-calculus in more programming way

I am learning $\lambda \mu$-calculus (self-study). I learned it because it seems very useful for understanding Curry-Howard correspondence (e.g understanding the connection between classical logic ...
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### Z-Specification = Routes

Im trying to make an invariant for this Z schema about routes. 1) The invariant should express that each route should contain at least 20 different places. First of all i thought of doing a universal ...
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### how to construct a formula with an opposite SAT value of another one

given a formula S (that is built of binary variables and "or", "not", "and" gates). IS there a polynomial algorithim that builds S' that satisfies: S' satisfied if and only if S doesn't satisfied. ...
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### proving program equivalence

I understand that the general problem of program equivalence is undecidable, but I'm wondering what approaches exist to tackle the problem? I am familiar with Hoare-style verification, but are there ...
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### Natural deduction: understanding bottom elimination (¬e)

I am new to natural deduction and upon reading about various methods online, I came across the rule of bottom-elimination in the following example. I do not understand the step in line 10. Upon ...
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### Why do logic gates behave the way they do?

I am a Software Developer but I came from a non-CS background so maybe it is a wrong question to ask, but I do not get why logic gates/boolean logic behave the way they do. Why for example: ...
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### When can the coinduction hypothesis be used?

We can use the induction hypothesis when we are proving a property for a structure that is well-ordered. I am aware that there is a proof for this. When it comes to coinduction, I'm confused. One of ...
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### For given reduction f, can show "if f(x) in 4NAE then x in 3SAT", but not "if x is not in 3SAT then f(x) not in 4NAE"

Claim: $3SAT \le_p 4NAE$, where reduction $f$ is defined as such: given a 3CNF formula $\varphi$, add to each clause a new literal $z$ (where $z$ is same literal for each clause), and return new ...
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### Compiling an impure language into a pure stack-based language

For a personal learning and fun project, I build an abstract virtual machine based on a stack. The instructions are simple and act on the top of the stack only. There are also stack operators such as <...
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### Correctness of Simple Programs

For a homework assignment I am asked to provide claim sequences to verify that the given conditional does indeed satisfy its speciﬁcation (to compute in r the absolute value of x). ...
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### Hoare Logic for Factorial

I came across this hoare logic for factorials but I don't quite understand it. We multiply F and X but we're not adding up all values of F so how do we get the sum/factorial at the end? Precondition: ...
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### Large Conjunctive Normal Form Examples

I'm currently learning about conjunctive normal form in a course on logic for Computer Science. I was reading the Wikipedia entry on the subject and encountered this: Typical problems in this case ...
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### How is it possible that an equational theory be terminating?

I'm a bit tripped up by this fundamental notion of an equational theory with respect to how we can possibly get termination if we have that we can always orient a set of equations either right to left ...
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### Injectivity not required for unification algorithms?

When learning about a general unification algorithm, we learned the rule decompose, which states unifying $$G \cup \{f(a_0,...a_k)=f(b_0,...,b_k)\} \Rightarrow G \cup \{a_0=b_0,...a_k=b_k\}.$$ The ...
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### Sample applications based on First Order Logic

I often hear about benefits of FOL, but I wonder what are some of its real world applications? Could someone please provide samples/case studies of applications of FOL that address real world ...
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### Is Event Calculus applied for program verification?

Recently I've read wiki page about Robert Kowalski (Prolog author) and stumbled upon interesting concept of Event Calculus. The wiki article mentions only few applications of this logical language. I ...
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### What is Herbrand interpreration and model?

I was reading the book "Foundations of Logic Programming" written by J.W. Lloyd. In the book, there were definitions of interpretation and model, and when it comes to herbrand interpretation and model,...
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### Prove that exists PARITYn fuction with size O(n^2)

Have to find Boolean circuit formula that solves PARITYn problem with complexity O(n^2)? (Function PARITYn(x1, . . . , xn) is equal to 1 if and only if the number of variables x1, . . . , xn equal ...
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### How come correctness proofs aren't tautological?

Consider the following function on binary trees, which is supposed to tell whether a given int is a member of a binary tree t: <...
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### Representing a sentence using propositional logic

I am confused regarding a propositional logic representation of a sentence. Please note that this sentence is not realistic: "A person who is male (M) is smart (S) if he is tall (T), but otherwise ...
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### How to show that a partial function is recursive?

I try to prove that this function is recursive: $$f(x_1,x_2)= \begin{cases} 2x_1-x_2 & \text{if x_1 \geqslant \sqrt{x_2}} \newline \bot & \text{otherwise} \end{cases}$$ I think that I need ...
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### Is true * true = true in Separation Logic?

I am trying to show that the following interference is unsound in terms of Separation Logic: $$(p_0 \implies p_1) \implies ((p_0 * q) \implies (p_1 * q))$$ I came up with the following values for ...
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### What algorithms for unification over (multidimensional) array terms?

I am looking for references on implementing unification over multidimensional array terms. Are there specialized unification algorithms for those cases? I wasn't able to find satisfactory literature ...
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### Model for formula $\Box (\psi \vee \phi) \Rightarrow (\Box \psi) \vee (\Box \phi)$

I am new to LTL and I am trying to understand how it works. My question is: is there such $\sigma$ that: $\sigma \models [\Box (\psi \vee \phi) \Rightarrow (\Box \psi) \vee (\Box \phi)]$ I know ...
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### Proving a first order logic theorem in equational logic with a term rewriting system

I am trying to translate and prove a theorem, originally written in first order logic (FOL), into a combination of equational logic (EL) and Boolean logic (BL) (more precisely a model of Boolean ...
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### In what sense the computer program (Turing machine) can be considered as the complex system and its IIT Phi can be measured and improved?

I am reading https://global.oup.com/academic/product/a-world-beyond-physics-9780190871338?cc=us&lang=en& about one approach of complex systems' theory for the emergence of the life. It is ...
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### Deciding Intuitionistic Logic via QBF

I want an algorithm to decide whether a theorem holds in propositional Intuitionistic logic ($IL$). We know $IL$ is $PSPACE$-complete, so we should be able to reduce $IL$ to $QBF$. In Literature i ...
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### Does the callback concept of programming have any basis in computer science?

Although I seriously code with computer languages in general since 2010 and as an amateur programmer with programming languages in particular since 2015 (primarily Bash and JavaScript imperative ...
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### Proof strategy to show that an algorithm cannot be implemented using just hereditarily terminating procedures

I am taking my question here from there. Consider the following scenario: You are given a fixed programming language with no nonlocal control flow constructs. In particular, the language does not ...
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### Is it possible to prove Tarski's Undefinability theorem from Turing's Halting Problem?

In the Wikipedia page for Turing's Halting problem, they mention that you can prove Gödel's First Incompleteness theorem from it. This is the relevant quote: Assume that we have a sound (and hence ...
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### Simplifying a logic function using Boolean algebra

Consider this Logic function : D = A B C + A B’ C + A’ B’ C + A B C’ + A’ B C’ + A B’ C’ I am trying to simplify it using Boolean algebra , I am stuck in this step : D= AB +B'C+ A’ B C’ + A B’ C’ So ...
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### Are these two sensible and related or unrelated ways of regarding a logic system as a programming language?

When I am trying to understand logic programming languages e.g. Prolog, I am immediately confused by the following two ways of relating logic systems and programming languages or type systems. In ...
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### Is there a model of ZF¬C where some program always terminates but has no loop variant?

Wikipedia has a proof that every loop that terminates has a loop variant—a well-founded relation on the state space such that each iteration of the loop results in a state that is less than the ...
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### How to prove following two statements are equivalent in Hilbert System?

statement 1: $Γ$ is satisfiable implies $Γ$ is consistent. statement 2: If $Γ$ derives $α$ then $Γ$ entails $α$. I can easily prove statement 1 from 2 , but not 2 from 1 (without using strong ...
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### Meaning of $a\lor b \to b' \lor c'$

So I have done part a) but I have no clue what I am supposed to do for part b), I have been trying for days to wrap my head around and even asked my fellow course mates, none of which seem to know ...
Let $x_{ij} \in \{0,1\}$, $1 \leq i \leq M$ (typically, $M = 2000$), $1 \leq j \leq N$ (typically, $N = 10$), be Boolean variables. If possible at all, I would like to simplify the following ...
Assuming you have a formula in first order logic like $$(\forall_x p(x) \land \forall_x q(x)) \rightarrow \forall_x(p(x) \land q(x))$$ (which seems valid?) Converting the formula to ...